a dissertation submitted to the department of …nq917rf4604/24... · 2011-09-22 · a dissertation...

TRADING MECHANISMS FOR FINANCIAL EXCHANGES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ECONOMICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Patricia Lassus

June 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/nq917rf4604

© 2010 by Patricia Macri-Lassus. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/nq917rf4604

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Jonathan Levin, Primary Adviser


Muriel Niederle


Michele Tertilt

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

ABSTRACT

The �rst part of this thesis discusses the structure of the US equities market and the

regulatory challenges involved, with respect both to particular venues and order types,

and to the market as a whole. This part addresses the role of public equities markets,

their objectives, and the market-design concepts relevant to evaluating them, it describes

the market structure before and after the signi�cant regulatory reforms of 2005, goes on

to detail some speci�c market-design challenges and current proposals from the SEC, and

concludes with an analysis and evaluation of several trading mechanisms and three order

types, from a mechanism-design perspective.

The second part of the thesis presents a theoretical model of an exchange and analyzes

optimal order submission in a one-shot game in which a buyer and seller, each of whom can

have two (privately known) types, can trade up to two units of an asset through an order

book, which is empty at the beginning of the game. In both a setting with private values

and interdependent values, equilibria in four games are characterized: a basic game with

limit and market orders, and games involving one of the three order types from the �rst part

of the thesis- iceberg, discretionary, and volume orders. The e¤ect of the introduction of

the di¤erent order types is analyzed with respect to volume and transparency. Each setting

also includes an analysis of the buyer-optimal mechanism and its relation to equilibria of

the games involving orders.

iv

ACKNOWLEDGEMENTS

I would like to thank all the people who made this dissertation possible. Above all,

I would like to thank my two co-advisers Jon Levin and Michele Tertilt for the valuable

comments they provided with respect to the thesis, and for their continued support. I feel

very fortunate and am forever grateful to both of them for all their help, especially over

the last year. I would also like to thank Michele for her enthusiasm and good advice on all

sorts of general aspects of academic life.

I would like to thank my committee members Jeremy Bulow, John Hat�eld and Muriel

Niederle for their comments and input on my thesis. Paul Milgrom and Ilya Segal provided

valuable insights, and I am also grateful for all their support during my time at Stanford.

I would also like to thank my department chair Larry Goulder.

Over the years, I was fortunate to be able to learn from other faculty at Stanford,

and would like to especially thank Susan Athey, Darrell Du¢ e, George Papanicolaou, Ken

Singleton, and Steve Tadelis. It was a pleasure to take their classes and I learned more than

I could have hoped. I would also like to thank the organizers of the �nancial mathematics

seminar that I have enjoyed attending.

I would like to thank Mark Tendall, both for helping me improve my teaching over the

years that I worked as a teaching assistant for him, and for his continued support. And I

would like to thank Clyde Wilson for what I learned in his nutrition class, which was the

best I took outside of my �eld.

I would like to thank Joshua Thurston-Milgrom and Amy Scott for helping copy-edit

this thesis.

I am grateful for the friends and fellow students I got to know at Stanford. All of my

friends changed my life and my time at Stanford for the better. I would especially like

to thank Andres Angel, Aaron Bodoh-Creed, Jose Bento, Adam Cagliarini, James Chen,

Silvia Console Battilana, Stephanie Fullen, Nick Haber, Samanatha and Zak Holdsworth,

Malte Jung, Pauline Larmaraud, Yuanchuan Lien, Nan Li, Michael Lipnowski, Isidora Milin,

Christine McBride, Moritz Meyer-ter-Vehn, Dan Quint, Tomas Rodriguez, Yuliy Sannikov,

and Alessandra Voena. I am also thankful for the comments some of my friends gave on the

thesis, and feel fortunate to have been able to learn with and from some of them in classes

and seminars.

Finally, I am grateful to my family, my stepfather, and especially my wonderful mother

and brother, for always having been so very loving and supporting, even if it had to be from

afar during my time here at Stanford.

v

Contents

Contents vi

List of Tables ix

List of Figures x

1 Introduction 1

2 Theory and Practice of the US Equities Market 42.1 Roles of the Public (Equities) Market and Relevant Theoretical Concepts . 4

2.1.1 Roles of the Public (Equities) Market . . . . . . . . . . . . . . . . . 4

2.1.2 Three Theoretical Concepts:

Liquidity, Transparency and Price Discovery . . . . . . . . . . . . . 5

2.1.3 A First Look at the Open Questions in Practice . . . . . . . . . . . 13

2.2 Historical Situation (until around 2005) and Problems . . . . . . . . . . . . 14

2.3 Transition Period (around 2005 onwards) till Today . . . . . . . . . . . . . 17

2.3.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 General Mechanism Design Questions . . . . . . . . . . . . . . . . . 20

2.4 Speci�c venues and mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Transaction Fees, Liquidity Rebates and Access Fees . . . . . . . . . 30

2.4.2 Sample ATS and Internalization Pools . . . . . . . . . . . . . . . . . 32

2.4.3 Iceberg Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Discretionary Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.5 Volume Orders, Discretionary Reserve Orders, Hidden Limit Orders,

Dark Reserve Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.6 Theoretical Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Overview of the Theory Sections 453.1 Background for the Di¤erent Order Types . . . . . . . . . . . . . . . . . . . 45

3.2 Overview of the Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Aside: Perfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Overview of the Results for Private Values . . . . . . . . . . . . . . . . . . . 50

3.4.1 Limit Orders and Market Orders Only (Basic Game) . . . . . . . . . 50

3.4.2 Game with Iceberg Orders . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.3 Game with Discretionary Orders . . . . . . . . . . . . . . . . . . . . 50

3.4.4 Game with Volume Orders . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.5 Optimal Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Overview of the Results for Interdependent Values . . . . . . . . . . . . . . 56

3.5.1 Limit and Market Orders Only (basic game) . . . . . . . . . . . . . 56

vi

3.5.2 Game with Iceberg Orders . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.3 Game with Discretionary Orders . . . . . . . . . . . . . . . . . . . . 57

3.5.4 Game with Volume Orders . . . . . . . . . . . . . . . . . . . . . . . 58

3.5.5 Optimal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Overview Table for the Results . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 Private Values 634.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.3 E¢ cient Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Limit Orders Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.2 Payo¤s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.3 Solving for Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.4 Characteristics of the Equilibria . . . . . . . . . . . . . . . . . . . . 68

4.3 Iceberg Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 Discretionary Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.2 Volume and Transparency . . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Volume Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


4.6 Buyer-Optimal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6.1 Principal-Agent-Game Setup . . . . . . . . . . . . . . . . . . . . . . 89

4.6.2 Characterization of Optimal Mechanisms . . . . . . . . . . . . . . . 93

4.6.3 Optimal Mechanisms Allocations as Equilibrium Outcomes . . . . . 94

5 Interdependent Values 965.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Limit Orders Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Iceberg orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.3.2 Volume and transparency . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Discretionary Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


5.5 Volume Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5.1 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106


vii

5.6 Buyer-Optimal Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6.1 Principal-Agent-Game Setup . . . . . . . . . . . . . . . . . . . . . . 117

5.6.2 Characterization of Optimal Mechanisms . . . . . . . . . . . . . . . 117

5.6.3 Optimal Mechanisms Allocations as Equilibrium Outcomes . . . . . 118

6 Appendix for Private Values 1226.1 For Section 4.2 (Private Values with Only Limit Orders) . . . . . . . . 122

6.2 For Section 4.4 (Private Values with Discretionary Orders) . . . . . . . 122

6.3 For Section 4.5 (Private Values with Volume Orders) . . . . . . . . . . . 126

6.4 For Section 4.6 (Private Values and Optimal Mechanisms) . . . . . . . 152

7 Appendix for Interdependent Values 1587.1 For Section 5.2 (Interdependent Values with Only Limit Orders) . . . . 158

7.2 For Section 5.3 (Interdependent Values with Iceberg Orders) . . . . . . 159

7.3 For Section 5.4 (Interdependent Values with Discretionary orders) . . . 161

7.4 For Section 5.5 (Interdependent Values with Volume Orders) . . . . . . 165

7.5 For Section 5.6 (Interdependent Values and Optimal Mechanisms) . . . 179

References 185

viii

List of Tables

1 Post-trade transparency with LO and discretionary equilibrium . . . . . . . 53

2 Overview table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3 Potentially optimal limit orders in private values setting. . . . . . . . . . . . 68

4 Execution probabilities for orders in equilibrium classes A through D. . . . 74

5 Player�s expected trade volume in equilibrium classes B and C. . . . . . . . 77

6 Post-trade transparency in equilibrium classes B and C. . . . . . . . . . . . 78

7 Post-trade transparency in limit order and class C equilibrium. . . . . . . . 80

8 Post-trade transparency in limit order and class C equilibrium. . . . . . . . 81

9 Post-trade transparency in limit order and iceberg order equilibrium of class

E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

10 Player�s Expected Trade Volume in Discretionary Order Equilibrium Classes

A, B, C and D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11 Equilibrium orders in equilibirum classes B and C. . . . . . . . . . . . . . . 123

12 Post-trade transparency in limit order equilbrium and equilibirum of class C. 125

13 Post-trade transparency with limit orders. . . . . . . . . . . . . . . . . . . . 126

14 Case B.2.1 with Volume Condition, Seller Responses (2 and 6) and (2 and 7). 142

15 Case B.2.1 with Volume Condition, Seller Responses (6 and 7). . . . . . . . 143


17 Case B.2.1, Seller Responses (2 and 7). . . . . . . . . . . . . . . . . . . . . . 144




21 Case B.2.2 with Volume Condition, Seller Responses (1 and 2). . . . . . . . 146





26 Post-trade Transpareny with Limit Ordes and in Equilibrium Class C. . . . 152

27 Possibe combination of trade volume per seller type, for a given buyer. . . . 157

28 Equilibirum Orders in Iceberg Equilibrium Classes E and F. . . . . . . . . . 159

29 Post-Trade Transparency in Limir Order Equilibrium and Equilibrium E. . 161

30 Equilibrium Orders in Discretionary Order Equilibria A Through D. . . . . 162

31 Post-Trade Transparency in Equilibria A and D. . . . . . . . . . . . . . . . 163




ix

List of Figures

1 Example Limit and Iceberg Order. . . . . . . . . . . . . . . . . . . . . . . . 45

2 Example Discretionary and Volume Order. . . . . . . . . . . . . . . . . . . . 46

3 Example Discretionary Order Equilibrium. . . . . . . . . . . . . . . . . . . . 52

4 Example Volume Order Equilibrium. . . . . . . . . . . . . . . . . . . . . . . 54

5 Example Iceberg Order Equilibrium. . . . . . . . . . . . . . . . . . . . . . . 57

6 Example Volume Order Equilibrium. . . . . . . . . . . . . . . . . . . . . . . 59

7 Example Buyer and Seller Valuations. . . . . . . . . . . . . . . . . . . . . . 64

8 Buyer orders in candidate discretionary equilibria A through D. . . . . . . . 74

9 Volume Order Equilibrium Class G. . . . . . . . . . . . . . . . . . . . . . . 85

10 Volume Order Equilibrium Class J. . . . . . . . . . . . . . . . . . . . . . . . 86

11 Volume Order Equilibrium Class Kb. . . . . . . . . . . . . . . . . . . . . . . 86

12 Volume Order Equilibrium Class B. . . . . . . . . . . . . . . . . . . . . . . 87

13 Volume Order Equilibrium Class C. . . . . . . . . . . . . . . . . . . . . . . 87

14 Iceberg Equilibrium Class E. . . . . . . . . . . . . . . . . . . . . . . . . . . 99

15 Iceberg Equilibrium Class F. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

16 Buyer Orders in Discretionary Equilibrium Classes A, B, C, and D. . . . . . 103

17 Discretionary Order Equilibrium Class D. . . . . . . . . . . . . . . . . . . . 104

18 Volume Order Equilibrium Class G. . . . . . . . . . . . . . . . . . . . . . . 108

19 Volume Order Equilibrium Class H. . . . . . . . . . . . . . . . . . . . . . . 108

20 Volume Order Equilibrium Class I. . . . . . . . . . . . . . . . . . . . . . . . 109

21 Volume Order Equilibrium Class J. . . . . . . . . . . . . . . . . . . . . . . . 109

22 Volume Order Equilibrium Class Kb. . . . . . . . . . . . . . . . . . . . . . . 110

23 Volume Order Equilibrium Class A. . . . . . . . . . . . . . . . . . . . . . . 110

24 Volume Order Equilibrium Class B. . . . . . . . . . . . . . . . . . . . . . . 111

25 Volume Order Equilibrium Class C. . . . . . . . . . . . . . . . . . . . . . . 111

26 Volume Order Equilibrium Class D. . . . . . . . . . . . . . . . . . . . . . . 112

27 Volume Order Equilibrium Class L. . . . . . . . . . . . . . . . . . . . . . . . 112

28 Volume Order Equilibirum Class M. . . . . . . . . . . . . . . . . . . . . . . 113

29 Volume Order Equilibrium Class N. . . . . . . . . . . . . . . . . . . . . . . 114

30 Volume Order Equilibrium Class O. . . . . . . . . . . . . . . . . . . . . . . 114

31 Discretionary Equilibrium B, Volume Equilibrium N and Fictitious Equilib-

rium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

32 Discretionary Order Equilibrium Classes B and C. . . . . . . . . . . . . . . 122

33 Volume Order Equilibrium Candidate Classes, Cases A.1.1 and A.1.2. . . . 128




37 Volume Order Equilibrium Candidate Classes, Cases B.1.1, B.1.2 and B.1.3. 137

x

38 Volume Order Equilibrium Candidate Classes, Cases B.1.2 and A.1.2. . . . 139

39 Volume Order Equilibrium Candidate Classes, Cases B.1.3 and A.2.2. . . . 141

40 Non-Monotonic Volume Order Equilibria. . . . . . . . . . . . . . . . . . . . 141

41 Iceberg Equilibria Classed E and F. . . . . . . . . . . . . . . . . . . . . . . 159

42 Buyer Orders in Discretionary Equilibrium Classes A, B, C, and D. . . . . . 161





47 Volume Order Equilibrium Candidate Classes, Cases B.1.1. . . . . . . . . . 174

48 Non-Monotonic Volume Order Equilibria. . . . . . . . . . . . . . . . . . . . 176

xi

1 Introduction

This thesis presents an analysis of trading mechanisms in US equities markets, focusing on

order types that have recently been introduced in exchanges.

Up until 2005, the manual trading �oor of the NYSE handled about 80% of the volume

in NYSE listed stocks; as of late 2009, its share has dropped to 15% (with another 14%

traded on Arca, the electronic version of the NYSE). Currently, a number of exchanges

exist, along with a multitude of other trading platforms, many of which employ trading

mechanisms that are geared towards large orders and o¤er little to no transparency about

available quotes (orders). In addition, exchanges have introduced new order types that are

also designed for large quantities and that provide less transparency, as part of the order

information is hidden (rather than being displayed in the order book upon submission of

the order). Regulators are concerned that the overall reduction in transparency may hinder

price discovery. On the other hand, requiring more transparency may increase trading costs

for large investors and/or long-term investors.

The thesis consists of two main parts. The �rst part of this thesis, in Section 2, dis-

cusses the structure of the US equities market and the regulatory challenges involved, with

respect both to particular venues and order types, and to the market as a whole. This

part addresses the role of public equities markets, their objectives, and the market-design

concepts relevant to evaluating them, it describes the market structure before and after the

signi�cant regulatory reforms of 2005, goes on to detail some speci�c market-design chal-

lenges and current proposals from the SEC, and concludes with an analysis and evaluation

of several trading mechanisms and three order types, from a mechanism-design perspective.

The three order types �iceberg, discretionary, and volume orders�are also at the focus of

the theoretical analysis of the second part of the thesis, in Sections 4 and 5.

The theoretical analysis starts with a simple game theory model of an exchange and

analyzes a one-shot game involving a buyer and a seller, each of who can have two (privately

known) types and trade up to two units of an asset through an order book, which is empty

at the beginning of the game. For ease of exposition, the player submitting the initial

order is assumed to be a buyer, and the player responding to the order is a seller (this is

without loss of generality, as the opposite case would be analogous). By starting o¤ with

an empty book, the analysis focusses on a buyer�s problem of submitting an optimal order

to be posted in the book. In practice, a buyer would �rst have to decide whether to trade

against any sell orders already displayed in the book. A player�s type de�nes his valuation

on each of the units; the analysis is carried out for both a setting with private values and

with interdependent values.

Each setting analyzes four games, which di¤er in the set of admissible orders. In the

basic game, the buyer can submit only limit orders to the book (limit orders consist of

a price and a quantity, and are displayed in their entirety), and the seller responds with

either limit orders or market orders (market orders consist of a quantity only). In the other

1

three games, both players�s set of admissible orders also includes iceberg orders (part of the

quantity is hidden), discretionary orders (a discretionary price is hidden, which is above the

visible price for buy orders and below it for sell orders), or volume orders (consisting of a

visible quantity at a price, and a hidden quantity at another price, as well as an optional

minimum execution size, or bundle size, for the hidden quantity). Note that while volume

orders are not currently used on exchanges, similar orders, namely entirely hidden orders,

do exist.

Equilibria in the four games characterized for each of the settings, and the e¤ect of the

new order types with respect to volume and transparency are investigated (by comparing

equilibria involving one of the three order type with the corresponding equilibrium in the

basic game).

Finally, each setting includes an analysis of the buyer-optimal mechanism, which is

the mechanism-design analogue of the optimal buy order for a buyer (namely the optimal

contract proposed by a buyer in a principal-agent game in which he is the principal and

the seller is the agent). It is investigated whether any optimal mechanism�s allocation

(that is, quantities and transfers exchanged by each buyer-seller pair ) can be replicated as

equilibrium trade quantities and prices in an equilibrium of a game with orders.

The main �ndings are as follows. Volume and transparency are not monotonically

linked: for some parameter combinations, say, discretionary order equilibria may exist that

have higher expected trading volume than the corresponding limit order equilibrium, while

having lower pre-trade and post-trade transparency (de�ned as what is known about the

buyer�s type before and after the trade, respectively). It is also possible that, say, volume

orders lead to equilibria in which pre-trade transparency and trading volume is the same

as with limit orders, but the trade reveals the seller as opposed to the buyer type. This

result may be an advantage in practice, as a large buyer wishing to trade more units over

time would prefer to not have his type revealed (inferred) so that the price he receives on

any following trades is not a¤ected.

Iceberg orders are used only in the interdependent values setting, where both buyer types

pool onto a visible order in equilibrium, while one buyer type hides a unit and bene�ts from

a lower price on the visible part of his order than if he separated and revealed his type.

Discretionary orders allow the buyer type who hides a price on a given unit (while again

pooling onto a visible order with the other buyer type) to screen the seller types with respect

to their valuation on that unit. If values are interdependent, there is an additional bene�t

resulting from pooling on the visible part, as with iceberg orders.

Volume orders can lead to equilibria that are very similar to those with discretionary

orders (and involve screening seller types with respect to their valuations on a given unit).

In addition, volume orders allow the buyer to submit supply schedules, and to bundle units.

In practice supply and bundling features of volume orders would render it easier for buyers

to engage in price discrimination, by providing a static alternative to the common practice

2

of splitting of large orders, which involves submitting sequences of small orders at increasing

prices, thus buying up liquidity at each price point along the way. (Note that another main

motivation for order splitting is that it reduces information leakage, as discussed in detail

in the �rst part of the thesis). Bundling as such also has the bene�t of reducing information

leakage, by increasing the cost of ��shing�for hidden liquidity (a trader would have to sell

the entire bundle to �nd out if quantity was hidden).

When values are interdependent, volume orders have the added bene�t of pooling. More-

over, equilibria then exist, in which volume orders allow the buyer to separate seller types

by o¤ering lotteries across units. When choosing between such a menu of lotteries, the seller

types�relative magnitude of the valuations on the �rst and second unit, or the �slope�of

the supply line de�ned by the seller types�valuation, is relevant. Since the slope of the

supply line represents the depth on that side of the market, volume orders would thus allow

the buyer to screen the opposing side of the market with respect to depth.

In the private values setting, the buyer-optimal mechanism is shown to simplify to the

full information program (as in Maskin and Tirole (1990) who consider a similar model

with two-sided private information and private values). Thus, each buyer type�s optimal

mechanism can be calculated independently. Moreover, the transfers associated with opti-

mal mechanisms are equivalent to either a demand schedule or a supply schedule (which

may involve bundling), implying that all mechanisms can be implemented with volume or-

ders. In the interdependent values setting, the optimal mechanism may involve pooling of

buyers onto one contract (as suggested by Maskin and Tirole (1992)). Some, but not all

optimal mechanism can be replicated as equilibria of games with orders. Thus, buyer types

may do better under the optimal mechanism than in equilibria with any of the games. The

analysis involves grouping mechanisms into equivalence classes (each class comprising mech-

anisms that have the same traded quantities and expected transfers for each type of buyer

and seller, but di¤erent transfers for a given buyer-seller-type pair). Intuitively, not all

equivalence classes of mechanisms can be implemented because sellers have more deviations

available in the sequential games than in the principal-agent game.

The outline of the remainder of the thesis is as follows. Section 2 gives a mechanism-

design-focused overview of the current structure of the US equities market, including a

discussion of the order types that are part of the theoretical analysis. Section 3 provides a

brief overview of the theory models and result. Section 4 contains the theoretical analysis

for the private values case, while Section 5 contains the analysis for the interdependent

values case. Most proofs are found in the appendix.

3

2 Theory and Practice of the US Equities Market

This section discusses the structure of the US equities market and the regulatory challenges

involved, with respect both to particular venues and order types, and to the market as a

whole. The �rst subsection addresses the role of public equities markets, their objectives,

and the market-design concepts relevant to evaluating them. The second subsection de-

scribes the structure of the US equities market up until the signi�cant regulatory reforms

of 2005. The third describes the market since 2005, and goes on to detail some speci�c

market-design challenges and to consider current proposals from the SEC, made in antici-

pation of the upcoming regulatory review. The fourth subsection presents an analysis and

evaluation of several trading mechanisms and order types, including the three at the focus

of the theoretical analysis in Sections 4 and 5, from a mechanism-design perspective .

2.1 Roles of the Public (Equities) Market and Relevant Theoretical Con-cepts

The section �rst introduces essential roles and desirable features of public markets; then

presents the major theoretical concepts relevant for market design� liquidity, transparency,

and price discovery; and concludes by adumbrating some of the open questions upon which

the SEC has recently solicited comments, in preparation for a broad regulatory review of

this market.1

2.1.1 Roles of the Public (Equities) Market

Despite a number of open questions, the SEC has laid out objectives (or the �essential role�,

as in Brigagliano (2009)) for the national market system (itself mandated as �multiple com-

peting markets that are linked through technology.�2 These objectives comprise �e¢ cient

price discovery, fair competition, and investor protection and con�dence.�3 E¢ cient price

discovery implies that asset prices re�ect (discover) all relevant available information about

the value of the asset. Fair competition includes access to markets, to insure that no traders

are excluded nor any given special rights. Investor protection and con�dence includes legal

issues like fraud protection, and is a concern primarily for regulators, rather than for market

design theory.4

Similarly, the Securities Exchange Act of 1934 speci�es �ve objectives well-functioning

markets are to assure. In slightly simpli�ed form, these are the following: e¢ cient execu-

tion of transactions, fair competition among all trading venues, availability of information

1See SEC (2010).2See page 10 of SEC (2010).3See for example Brigagliano (2009).4While this is not the focus of this thesis, it is a very current topic, with the House Committee passing

an Investor Protection Act in November of 2009, in response to the latest �nancial crisis.

4

with respect to quotations and transactions, best execution of customer orders by their

intermediaries, and direct access to trading for investors.5

Market features that have been suggested to help meet these three objectives include

�lower commissions, tighter spreads, faster execution speeds, greater system capacity,�as

well as less systemic risk. Systemic risk will be discussed in more detail below in the context

of liquidity. In this section, systemic risk mostly refers to the risk of the temporary collapse

of the market or market system as a whole (that is, the inability to execute trades). Note,

though, that certain market structures or trading mechanisms may lead to more short-

term price volatility, which can also pose a threat to the �nancial system, even though the

physical market and ability to execute trades remains intact. These features are suggested

to be desirable for their intrinsic bene�t to retail and institutional investors, as well as for

providing an implicit metric for evaluating how well the objectives are met.

The question what are the best indicators of market health, though, remains open. It

is possible, for example, that the reduction in spreads over the past �ve years� generally

considered a success� was linked to an increase in the overall cost of the execution of a

given order. The SEC in fact points out that the diverse objectives of a well-functioning

market are �di¢ cult to reconcile.�6 Decisions about optimal market design and regulation

thus have a signi�cant policy aspect, as will become evident in the analysis provided in

Section 2.3.2.

2.1.2 Three Theoretical Concepts:Liquidity, Transparency and Price Discovery

These theoretical concepts are crucial to the study of equities markets: liquidity, trans-

parency, and price discovery. They are intimately linked to the basic objectives of public

markets, to the features thought to best support those objectives, and to the possibility of

measuring them. As this section illustrates, the theoretical relationship among the three

concepts is complex� another factor that makes market design challenging in practice. In

addition to this relationship, this section discusses the relationship between liquidity and

fragmentation, and that between transparency and information leakage.

Liquidity

Liquidity can be de�ned in several ways. One states that, in liquid markets, traders have

the ability to sell large quantities quickly with little price impact. More formal de�nitions

invoke three measures of liquidity: depth, tightness, and resilience. Depth is a market�s

ability to absorb large trade volumes without signi�cant impact on prices. Tightness indi-

cates the general transaction cost in a market, irrespective of prices, and is measured by the

average spread between bids and asks. Resilience is a measure of how quickly the market

5See Section 11.A (a) (1) on pages 69-70 of the Securities Exchange Act, SEC (1934).6See page 11 of SEC (2010).

5

�corrects�transitory price �uctuations arising from trades.7

As is apparent from this de�nition, several of the features proposed to support market

objectives� spreads, execution speed, and system capacity� are precisely those of liquid

markets. Liquid markets in which all traders have fair access are also bound to result in

lower commissions

Liquidity in the equities market in the US has increased over time: along with the NYSE,

a number of other, newer, venues have seen volume growth and tighter spreads. Still, the

equities market as a whole remains insu¢ ciently liquid to accommodate the everyday needs

of many traders, especially for institutional investors and others wishing to trade quantities

that are large relative to the average execution size or the typical volume traded by retail

investors. One of the major long-standing challenges in market design thus remains: to �nd

a market structure that accommodates both institutional and retail order �ow, meeting the

needs of both large and small traders.

Fragmentation/Dispersion of Liquidity

Until recently, the NYSE was the only US equities exchange, but the loosening of reg-

ulation over the past �ve years has lead to the creation of a large number of alternative

trading venues. Some of these are exchanges much like the NYSE, while others, realiz-

ing the di¤ering needs of institutional and retail investors, cater speci�cally to large order

from institutions and brokers. The current situation, then, is one of signi�cant market

fragmentation, and thus Fragmentation or Dispersion of liquidity.

This fragmentation (or dispersion) of liquidity, due to the coexistence of a multitude of

trading platforms, may have both advantages and disadvantages; the SEC has used both

terms, noting that fragmentation has a negative connotation, while no overall conclusion

has been made as to whether its disadvantages outweigh its bene�ts.8

There are currently over 200 alternative trading venues,9 with varying volume traded

and varying trading mechanisms. This fragmentation o¤ers several bene�ts. Allowing

traders to choose the trading mechanism that best �ts their need is bene�cial to them and,

even if all venues employed the same mechanism, competition among venues for order �ow

would likely lead reduced commissions (i.e., reduced transaction costs for traders). In fact,

in recent years, competition in commissions has heated up, with a number of venues paying

a �liquidity rebate�to certain orders routed to them10.

Fragmentation of liquidity among trading venues may also have the advantage of lower-

ing systemic risk within the equities market as a whole: in case an individual venue were to

temporarily fail, order �ow could be quickly diverted to other venues in the system without

signi�cant impact. The failure of a single centralized system, by contrast, would be much

7See page 95 of IMF (2006).8See page 21 of SEC (2010).9See page 15 of SEC (2010).10See for example Ku (2009).

6

worse for the market. Though the SEC has acknowledged most of these points11, regulators

continue to view fragmentation with concern.

The main disadvantage of fragmentation is that it may impede e¢ cient price discovery in

any of the individual markets, compared to all orders interacting in one centralized trading

venue. Aggregating order �ow may thus lead to less volatility, and less price impact from a

given order in a given exchange.

In addition, investors in a fragmented market may �nd it di¢ cult to get a clear pic-

ture of market activity (e.g. volume, prices, and trends) when they have to keep track of

a number of di¤erent venues that execute and report trades according to di¤erent rules.

One may thus think of such a fragmented market informally as less �transparent� to in-

vestors. The formally de�ned theoretical concept of transparency, however, does not quite

apply here, since it usually refers to the characteristics of a speci�c trading mechanism

(see below). The di¢ culties investors may encounter in a fragmented market are related to

transparency, though, in that the structure of the market as a whole a¤ects the accessibility

(and, potentially, the availability) of real-time information about prices and trades. This

link is also recognized by the SEC, which has referred to transparency also as �the extent

to which prices are visible and understandable to market participants.�12

Transparency

When formally de�ning market transparency, it is necessary to distinguish between

pre-trade transparency and post-trade transparency. Pre-trade transparency refers to the

amount of information available before a trade. The order book of a highly transparent

market, for example, would show all bids and asks, as well as the volumes available at each

of these prices. A less transparent market might show only the best bid and ask, with

their respective volumes. A completely opaque market would have no information available

before the trade. Post-trade transparency refers to the information made available after

trades are executed. Transparent markets would make both the execution price and the

quantity traded available immediately after the trade is executed, while an opaque market

may publish no information after the trade.

Transparency has long been considered a key element of fair, e¢ cient markets, since it

allows investors to make well-informed choices and tends to level the playing �eld between

retail and institutional investors. The di¢ culty is in �guring out how much transparency is

optimal in a given market. Complicating this question even more in the case of equities in

the US is the lack of a single market, and the existence instead of a collection of markets with

varying levels of pre- and post-trade transparency. The question thus becomes how much

transparency to require (both pre- and post-trade) for individual trading venues, when the

collection of venues jointly competes for order �ow, and this order �ow is a¤ected by the

degree (both absolute and relative) of transparency at each venue.

11See for example SEC (2010).12See Levitt (1999).

7

The SEC has recently been especially concerned with the issue of transparency. Some

current questions are discussed in detail in Section 2.3.2 below, along with an analysis of

the current market structure. Even for a centralized market, though, choosing the optimal

degree of transparency is di¢ cult, both in practice and in theory.

Over the last ten years, US regulation has gradually mandated less pre-trade trans-

parency, allowing for new trading venues with di¤erent mechanisms and less pre-trade

transparency than the NYSE or other classic (i.e., order-book-driven) exchanges. In addi-

tion, new order types introduced within exchanges have less pre-transparency than simple

limit orders. All these market design innovations are discussed in Section 2.4 below, with

emphasis on the mechanism design perspective. Note, though, that, current regulation

notwithstanding, regulators�

views on the optimal level of transparency remain ill-de�ned, prompting the SEC�s recent

solicitation of comments in connection with a broad review of the equities market, and its

emphasis of the question whether the last milestone in regulation (RegNMS from 2005) has

led to too little transparency in the market, or whether the current level of transparency

should be reduced. These points are addressed below in the �Open questions�subsection,

as well as in Section 2.3.2.

Transparency and Information Leakage

A market�s pre- trade and post-trade transparency signi�cantly a¤ects order �ow, as

market participants can (and likely will) condition their trading strategies on any market

data that is made publicly available. The academic literature has introduced the term infor-

mation leakage, referring to information revealed (or �leaked�) to all market participants by

publicly displayed orders or published trades. Information can be either information about

future order �ow or information about the value of the traded asset. Since price discovery

is a central market objective, it is neither surprising nor undesirable that some information

is revealed by any displayed order or published trade. The term �information leakage� is

thus used mostly in conjunction with relatively large orders. Other market participants�

knowing about the existence of a large order will likely have detrimental consequences for

the trader trying to �ll the order, be it in the form of a lower execution probability or a

worse average price. This is the next important point.

As stated above, the information leaked to the market through a displayed order may

theoretically be about the value of an asset, about future order �ow, or both. All this

information is bound to be valuable, whether in a pure common-values scenario, a pure

private-values scenario, or an intermediate case of interdependent values (in which the value

of a given unit of the stock has both a private-values and a common-values component).

Consider the case in which a displayed order suggests a substantial buy interest from

some anonymous trader A at some price P (this could be a large order, or just large enough

to suggest that the submitter would like to buy more in the near future). Seeing this interest,

other traders might conclude the asset is undervalued at P, update their belief about the

8

value, and reduce or withhold their supply at P. Moreover, some traders might engage in

anticipating trading strategies, also called front-running.13 Other traders may then submit

a buy order priced slightly above P, hoping to buy up all the liquidity around P and force

trader A to buy back from them at a higher price shortly afterwards. Note that anticipating

trading strategies would remain a problem even in the case of pure private values.

In practice, information leakage has increased the incidence of order splitting. Order

splitting refers both to the older practice of splitting a large order into many small pieces

that are traded sequentially, and to the more recent practice in which large �parent�orders

are split not only over time, but also into many small �child�orders, which are submitted

to a number of di¤erent trading venues simultaneously. Traders whishing to trade large

quantities face a trade o¤ between immediacy and price: having to weigh the bene�ts of

faster execution speed when trading the entire order at once or in large chunks, against the

disadvantage of reduced possibility of price discrimination. (In simpli�ed terms, a buyer

engaging in price discrimination would submit a sequence of buy orders at increasing prices,

buying up liquidity at each price point- rather than submitting a higher-priced order for that

the entire quantity that would execute at once.) Information leakage worsens the trade-o¤

as trading faster by displaying larger quantities becomes more costly.14

A reduction in pre-trade transparency would thus mitigate information leakage, but

uncertainty about available interest (i.e., liquidity at a given price) may discourage the

submission of responding orders.

The next theoretical concept is intimately linked to information leakage or, more pre-

cisely, to the price information contained in orders.

Price discoveryPrice discovery, universally recognized as a main function of public markets, refers to

the dynamic process by which markets set prices by incorporating available information

about the corresponding asset. The market price emerges through trading, that is, by

crossing orders that may contain information about the value of the asset. As such, the

price or value of the asset is revealed or �discovered.� In the academic literature, market

e¢ ciency refers to the degree to which information about the asset is incorporated into

the price. The literature on e¢ ciency distinguishes between strong e¢ ciency (where all

available information, both public and private, is re�ected in prices) and weak e¢ ciency

(where only publicly available information is incorporated into prices).15

Testing a market�s e¢ ciency with respect to price discovery is complex. Some proposed

tests investigate how e¢ ciently prices respond to information that becomes public at one

13Front running is illegal and refers to the case in which information, rather than being inferred frompublicly available data, has been privately obtained and then misappropriated by a broker, e.g. when abroker trades orders in advance of pending customer orders, or when a broker knows that his or her �rm isabout to recommend a given stock.14For a more detailed description, see pages 7-8 in Harris (1997).15For a detailed description, see, for example, page 24 in Brunnermeier (2001).

9

point in time. Gauging this e¢ ciency requires measuring both the speed of price discovery

and the depth of the market, the latter being what some academics have termed �volume

discovery.�16

Though measuring price discovery is di¢ cult, listing market characteristics that will

a¤ect price discovery is not. The following characteristics are thought to increase price

discovery: lower transaction costs (to increase the likelihood of informative transactions

like arbitrages), liquidity (to increase the number of potentially informative trades), and

the fraction of informed versus uninformed traders (where more well-informed traders are

thought to increase price discovery).

Note that all three of these market characteristics are (to varying degrees) endogenous

to the trading mechanism used. Central, then, to the projects of market design and regu-

lation, are choosing optimal trading mechanisms and/or providing a regulatory framework

for existing and potentially new mechanisms. And the greatest challenges for market (or

mechanism) design arise when considering the complex interrelationships among all three

concepts.

Transparency, Liquidity and Price discoveryThis section emphasizes the relevant basic theoretical arguments that illustrate the

complex relationships among liquidity, transparency and price discovery. More precise con-

siderations for speci�c mechanisms, along with their corresponding theoretical results from

the present research, are the topics of Section 2.4.

None of the relationships among these three concepts is well understood. Consider,

for example, the relationship between liquidity and price discovery. An increase in liquidity

theoretically corresponds to an increase in volume traded, so trade volume serves as a proxy

for liquidity in considering the interactions of the concepts. Without any traded volume,

there can be no price discovery, and, since trade is voluntary and re�ects the buyer�s and

the seller�s beliefs, more volume traded should give more �credibility�to the prices at which

trades are executed. Thus, one would expect more and better price discovery in very liquid

markets. As it turns out, though, there may be cases in which an increase in trading volume

correlates with, or even causes, a reduction in price discovery. This concern is mentioned

in the SEC�s recent concept release (discussed in more detail in Section 2.3.2). Trade

volume in the US equities market has nearly tripled over the last 5 years, accompanied by

a signi�cant reduction in spreads and speed of execution.17 Neither volume nor spreads

provides the complete picture if, say, short-term volatility signi�cantly increased at the

same time. Speci�cally, if order �ow increases as a response to the trading mechanism and

causes an increase in short-term volatility, then the average execution price of transactions

may be a less accurate indicator of an asset�s fundamental value, despite �improvements�

in volume, spreads, and execution speed.

16See Gomber, Budimir, and Schweickert (2006).17See page 6 of SEC (2010).

10

Another complication in the relationship between price discovery and trade volume is

that the concept of price discovery assumes the existence of only one market price for a given

asset, regardless of whether the transaction is for a few hundred shares, or a few hundred

thousand shares. In practice, though, execution prices for small orders often di¤er from

those on large orders, depending on the depth of the market at and around the best quoted

prices. Thus, two markets may have �discovered� the same best price, but one market

may have much more depth at that price than the other market. Such di¤erences in depth

would have signi�cant practical impact for traders trying, for example, to liquidate a given

position in the stock. Similar considerations have inspired recent e¤orts in the academic

literature on �nance to distinguish the mark-to-market value of a position (the product of

the quantity held and the market price) from, say, a (distressed) liquidation value for the

position (the product of the quantity held and the expected average price from its expedient

liquidation under current market conditions).18 These considerations suggest that a more

comprehensive concept of price discovery should link better to the concepts of liquidity and

market depth.

The concept of price discovery grows even more complex when liquidity is fragmented:

one must then distinguish between price discovery in the market as a whole, and the contri-

bution to price discovery made by each individual trading venue. At some venues, like the

NYSE, prices are discovered as trades are executed at the best bids and asks. Other venues,

though, may trade signi�cant volume (larger by orders of magnitude on an individual trade

level) without forming an �own� price (this is discussed in detail below). That is, these

venues collect orders that consist of quantity only, and then execute trades at the current

NYSE price. While these venues neither form their own price nor contribute directly to

the formation of the NYSE price, they still contribute to price discovery in the market as

a whole in the sense that the volume they execute gives more weight or credibility to the

NYSE price. Moreover, these venues would contribute to price discovery under a more

comprehensive de�nition, that included a depth component.

Finally, the di¢ cult relationship between price discovery and transparency is illustrated

by the example of several venues crossing orders at the NYSE�s price. The venues above

often o¤er little or no pre-transparency as to orders that have been submitted. More-

over, there are other venues that do form their own prices, but also have low transparency

(and signi�cant volume). In setting more �accurate� prices, price discovery depends on

the market�s incorporating the price information revealed by the submission and execution

of orders made by well-informed traders. In a venue that requires more pre-trade trans-

parency, traders may be more concerned about exposure (being front-run, or otherwise

having market impact), which would reduce the volume and price-aggressiveness of orders,

thereby impairing price discovery in that venue as less-informed order volume is incorpo-

rated into its prices. Conversely, lower pre-trade transparency may improve price discovery

18See, for example, Brunnermeier and Pedersen (2005).

11

in practice. The theoretical arguments that have just been presented are similar to those

relevant to the analysis of the relationship between trade volume and transparency, which

is considered next.

With respect to the relationship between transparency and trade volume, consider the

following points. Some degree of transparency is generally considered necessary to encourage

order submission for two major reasons. First, transparency may encourage order submis-

sion by increasing traders�con�dence in the fairness prices they will receive on an individual

transaction.19 Consider, for example, the case of a very opaque inter-dealer market: even

if traders have a sense of the price a broker will o¤er, they may be hesitant to trade if they

do not know whether the prices they will receive are fair.

Second, transparency may encourage order submission by increasing traders�certainty

about the actual prices they will receive on a given order. Traders in an exchange, for

example, may be more likely to submit a buy market-order (that is, an order to buy a

speci�c quantity) if the book o¤ers more information about the prices at which that order is

likely to be executed. In a perfectly transparent book that displayed all available quantities

at all prices, a trader could perfectly predict the execution price (assuming no other order

possibly impacting the trade was submitted or cancelled at the same fraction of a second

that his order was submitted). Some recently-introduced order types render the order book

more opaque, and their impact on total volume is subject of debate among both regulators

and practitioners (as detailed in Section 2.4 below).

Contrary to these considerations, though, is signi�cant evidence that less transparency

may sometimes encourage more order submission. As mentioned in the next Section, 2.2,

some hidden liquidity has always existed in equities markets and other �nancial markets,

meaning that not all orders and latent demand were continuously displayed 20. Many new

trading venues, like Dark pools and other Alternative Trading Systems (ATS) with relatively

little pre-trade transparency have experienced a signi�cant increase in volume since their

creation (this is discussed in Section 2.3). Moreover, as detailed in Section 2.4 below, less

transparent order types have become increasingly popular. In fact, some exchanges have

started to allow completely hidden orders (revealing no order information).

The most common argument in favor of reduced transparency is that it reduces the

information leakage associated with the submission of larger orders (see above). In other

words, prices are less likely to move against the submitter of a large order, either because

traders on the contra-side of the order have less information available to update their beliefs

about the value of the asset, or because it will be more di¢ cult for other traders to engage

in anticipating-type strategies (like front-running). Either way, the reduction in the market-

impact cost of executing large orders is argued to encourage their submission. Section 2.4

below provides an alternative argument for the positive impact of reduced transparency on

19For a brief overview of related theoretical literature as well as regulator�s view see for example, page 267of Madhavan, Porter, and Weaver (2005).20This de�nition is taken from page 2 of Brigagliano (2009).

12

trade volume, namely one that relates to traders�desire to reduce adverse selection in the

execution of their order.

The theoretical challenges involved illustrate the practical di¢ culty in choosing �opti-

mal�regulation. Sections 2.2 and 2.3 below details the structure of the US equities before

and since the regulatory changes of 2005, and the last section of this section includes an

overview of the mechanisms design results of the thesis, as well as a mechanisms-design-

centered analysis of the order types that are part of the theoretical analysis of Sections

4 and 5. (Now that the relevant theoretical concepts have been introduced, this fourth

section may be read independently of the next two.) First, however, it useful to consider

an overview of the open questions and concerns the SEC is currently trying to address.

2.1.3 A First Look at the Open Questions in Practice

The varied nature of the open questions among regulators points up the practical challenges

of optimal market design. The SEC�s recent request for comments begins by clearly stating

a focus �on the interests of long-term investors,� since �these are the market participants

who provide capital investment and are willing to accept the risk of ownership in listed

companies for an extended period of time.�21 But the heterogeneity of traders in equity

markets means that the interests of long-term investors in a given market may or may not

always align with those of other traders since, for example, �short-term professional traders

may like short-term volatility to the extent that it o¤ers more trading opportunities, while

long-term investors do not.�22

The decisions on the SEC�s table center around questions about market performance,

high-frequency trading, and undisplayed liquidity.

The �rst di¢ culty in improving the equities market is that of assessing its current

performance, including the question whether its quality has increased or decreased over the

past ten years. In fact, the SEC is seeking new market-quality measures (more on this in

Section 2.3.2).

High-frequency trading is a strategy in which a trader daily submits a very large number

of orders (in sub-second intervals, often canceling the orders within sub-seconds if they are

not executed). It evolved in response to the ever-increasing execution speed of electronic

trading venues, and seems to have been one of the main drivers of the increase in trading

volume in recent years.23

The questions surrounding undisplayed (or dark) liquidity are the focus of the present

thesis. Undisplayed liquidity refers to orders that are submitted to di¤erent trading venues

but not publicly displayed. Market-design and regulation requires a deeper understanding

of undisplayed liquidity, especially considering its increased prevalence in recent years.

21See page 33 of SEC (2010).22See page 33 in SEC (2010).23The share of high frequency trading volume is estimated at "50% or higher" as SEC (2010) notes on

page 45, also referring to Spicer and Lash (2009) and Patterson and Rogow (2009).

13

Section 2.3.2 discusses each of these questions individually. At a more general level,

though, they all concern the relationship between transparency and trade volume (or order

execution quality), as well as their interrelations with price discovery. These relationships,

even for the case of an individual market (rather than a collection of trading venues), remain

the topic of much debate among academics (as illustrated in the previous section), as well

as practitioners and regulators.

Mandating more transparency means demanding quotes to be displayed, the hope being

that displayed orders should directly re�ect buy- or sell-interest in a stock. However, as one

institutional agency brokerage put it, �quotes are not (nor have they ever been) displays of

available liquidity, but instead tactics employed to safely attract liquidity.�24

As yet, there exists virtually no empirical analysis of the interactions between trans-

parency, price discovery, and volume in the collection of trading venues that has come to

constitute the US equities market system. The SEC�s solicitation of such work acknowledges

the complexity of their impending review of the equities market structure.

The last signi�cant legislation for equities markets, called RegNMS, went into e¤ect in

2005 (NMS refers to National Market System). Earlier, in 1998, groundwork was laid with

RegATS (Alternative Trading System). Alternative trading systems, trading venues other

than traditional exchanges like the NYSE, will be described in detail in Sections 2.3 and

2.4. The next section gives a broad overview of the market structure before Reg NMS was

introduced, and illustrates some of the problems that motivated the regulation.

2.2 Historical Situation (until around 2005) and Problems

Until the signi�cant regulatory changes of the past decade, US equities were mostly traded

on one central exchange, the NYSE, which had only a manual trading �oor. The NYSE

traded about 80% of the volume of NYSE listed stocks, which in turn make up 80% of the

market capitalization of US equities25. The NASDAQ started trading electronically before

the NYSE, about ten years ago. Equity markets in other countries (in Europe and around

the world) were similar in that trading would take place mostly on one centralized exchange,

though automated trading started earlier outside the US, where exchanges were historically

order- rather than quote-driven. Order-driven markets are characterized by the existence

of one central order book in which all orders are entered (anonymously), and then crossed

with each other. In quote driven markets (like the NYSE used to be), market-makers post

quotes to which other traders can then respond.

In addition to the central exchange, a signi�cant fraction of trades both in the US and

elsewhere were executed in upstairs markets (the trading rooms of the brokers, which would

be upstairs rather than down on the �oor of the exchange), essentially through over-the-

counter trading of listed stocks. These trades were bilateral or multilateral transactions, in

24See Rosenblatt and Gawronski (2007).25See pages 4-5 in SEC (2010).

14

which the broker/dealer would try to match the client by either itself acting as a contra-

side, or as an agent by �nding a contra-side (with other client orders, or, in the case of

the US, on the �oor of the exchange). Note, though, that upstairs markets were designed

to facilitate block trading, meaning the trading of orders whose size is large relative to the

average execution size of orders on the �oor of the exchange.

As mentioned before, trading large orders can be di¢ cult on exchanges, in part due to

the negative consequences of information leakage that would result (given the transparency

requirements). That is, making the information about a large order public on the �oor of

the NYSE, or posting a large order into the order book of an exchange, would cause prices

to move against the order, as discussed in Section 2.1.2. Upstairs markets thus o¤er an

alternative to the common practice of order splitting. In many cases, brokers executing

block trades could be assured that the client�s motive for trading was not privately-held

information about the asset�s value (but rather, say, an institutional investor�s needed to

increase a position due to a larger investment in the fund).26

Conceptually, upstairs markets constitute a pool of undisclosed, or dark, liquidity. The

SEC de�nes dark liquidity as �orders and latent demand that are not publicly displayed.�27

Dark liquidity exists when traders do not publicize the total volume they would be willing

to trade at a given price. As mentioned before, whenever the good to be traded has a

common-values or interdependent-values component, traders will have incentive to hide

their demand/supply. Dark liquidity is thus a feature of many �nancial markets. Two

recent studies help to illuminate the e¤ects of dark liquidity on traders of large orders, one

with respect to trading on the NYSE, and the other with respect to secondary markets for

treasury securities.

The �rst study, Bacidore, Battalio, and Jennings (2002), analyzes the extend to which

NYSE dealers�grant price improvement on their transactions. These dealers make markets

for assets by posting a bid and an ask at each point in time, together with a volume at the

bid and ask (these may represent customer orders or their own o¤ers). Price improvement

denotes the situation in which a trader approaches the dealer with the intention to, say,

buy shares, demands a total quantity that is larger than the quantity posted by the dealer,

and is able to buy that total quantity at the posted ask price. The term price improvement

refers to the trader�s having received a better price on those units that exceeded the posted

quantity, by buying them at the posted ask price (rather than walking up the book and

selling them at the next best ask, which would be higher). Analogously, for traders that

submit a sell order with a quantity in excess of the best posted bid, the broker may agree to

buy the entire order quantity at the best bid (rather than buying some at the next best bid).

26This assumption is also common in the theoretical literature; see for example Seppi (1990).For an empirical analysis, see for example, B.Smith, Turnbull, and White (2001), who use data from the

upstairs market of the Toronto stock exchange.27See page 2 of Brigagliano (2009).

15

The study �nds that quantities demanded by traders are larger than the posted price in

about 13% of cases (one in six orders), and that conditional on this being the case, traders

receive price improvement 70% of the time. The conclusion is that a signi�cant amount of

undisclosed depth, or dark liquidity, generally exists at the best prices.

The second relevant study, Boni and Leachs (2000), investigates the US Treasury security

market, an inter-dealer market in which a number of dealers make markets, posting bids and

asks that can re�ect either their own demand, or customer order �ow. Customers approach

the dealers to trade against publicized bid and ask quantities. In many cases, orders are

worked, which refers to the following practice. Suppose a buyer approaches a dealer about

buying treasuries. The dealer will sell the posted quantity, and the trade be reported. The

dealer then can start mediating a bilateral negotiation over the phone between the buyer

and seller whose orders re�ected the posted ask. In this conversation, the dealer does not

reveal the buyer�s and seller�s identity to each other, but simply solicits proposals from each

for how much more they would like to buy/sell at the posted price. If they agree, another

trade is executed, and the process continues until either the buyer or seller declines to trade

any more quantity. When, say, the seller chooses to end this negotiation, that seller walks

away with the knowledge that more demand exists at that price. The market, though,

sees only the sequence of published trades. The study found this practice of expanding (or

�working�) orders to be very common, to the point of creating ine¢ ciencies: other traders

would have either to wait in line for a broker to �nish working up the order, or else approach

another broker, despite the latter�s inferior prices, in order to avoid the wait.

Quantitatively, this study found that about 45% of the total traded volume happened

during the negotiations part, and thus almost half the available liquidity was never posted

to the market, but only reported after the trade had been agreed on bilaterally. This may

come as a surprise, since on-the-run treasury securities are highly liquid. Moreover, the

study found that the degree of order expansion (and thus the amount of hidden liquidity)

may be linked to the level of possible information-asymmetry about the value of the asset at

the time of the trade. In particular, dealers use expandable-limit orders more often during

hours outside of the New York business day, as well as for o¤-the-run issues (which are less

liquid).

The existence and extent of hidden liquidity in a number di¤erent markets gives a sense

of the di¢ culties encountered by those who wish to trade relatively large quantities, while

trying to minimize the market-impact of their orders. Market impact increases the cost of

trading a given quantity: prices may move in a direction unfavorable to the trader, even as

orders are split (leading, in addition, to less immediacy). An article, Merrin (2002), by the

founder of liquidnet (which is trading platform for equities) suggested that the contemporary

market structure introduced an implicit hundred-billion-dollar tax on equities trading. Back

then, when 80% of the volume was traded on NYSE, with an average 700-share execution

size, and a typical institutional order around 200,000 shares, traders were forced to split

16

orders, causing temporary price movements over the course of one or a few days as the order

was executed. The argument was that alternative trading venues could give institutions a

cheaper alternative, and would also reduce price volatility in markets, assuming most of the

volatility now observed originates in these temporary price movements. Moreover, having a

platform in which all buyers and sellers interact would allow them to obtain better prices,

since evaluating broker quotes is costly, especially because of potential information leakage.

The improved understanding of dark liquidity has a¤ected the current motivations of

regulators. The SEC has stated the goal of reducing barriers to entry, promoting competi-

tion among trading venues, and improving the quality of execution. After the introduction

of Reg ATS in 199828, which allowed for the creation of alternative trading systems, trading

volume as a whole started to increase, as it did again with the introduction of RegNMS

in 2005, especially trading on the new ATSs. The main reason was the modi�cation, of

the �trade-through rule.�The trade-through rule protects the best displayed bids and asks

available for a given stock at a given point in time (across all markets that trade that stock).

Speci�cally, the rule states that no other exchange or broker can execute a trade at prices

inferior to the best bids and asks o¤ered on any other exchange. Naturally, this implies that

a large number of orders will be routed to the exchange currently displaying the best prices.

Before RegNMS, all best quotations, including manual quotations like those on the �oor of

the NYSE, were protected by the trade-through rule. RegNMS eliminated the protection

for the slower, manual quotes, amid widespread belief that protection was outdated. This

change in regulation led not only to a signi�cant increase in overall trading volume, but also

to a large number of exchanges gaining volume at the expense of the NYSE �oor, including

alternative venues whose trading mechanisms are very di¤erent from traditional exchanges

like the NYSE or NASDAQ.

The introduction of RegNMS crucially a¤ected not only volume, but also market struc-

ture, with this change in structure leading to signi�cantly less pre- and post-trade trans-

parency associated with the executed volume. The exact nature of the changes in order

�ow (volume), along with an overview of the current market structure (a taxonomy of the

trading venues), constitute the focus of the next section.

2.3 Transition Period (around 2005 onwards) till Today

2.3.1 Statistics

After a short overview of the recent changes in the US equities market, the following sub-

sections discuss mechanism-design questions at the level of the market as a whole, before

speci�c mechanisms used at given trading venues are treated in Section 2.4.

In the last �ve years, the US equities market has undergone a signi�cant transformation,

as illustrated by a few statistics presented in the SEC�s concept release of January 14th,

28See page 2 of Brigagliano (2009).

17

2009 (SEC (2010)), and restated below. In fact, note that all statistics presented in this

section are from this source (pages 6-7 and pages 14-15) unless otherwise noted.)

As of January 2005, the NYSE accounted for 80% of the trading volume in NYSE listed

stocks. By October of 2009, that share had dropped to 25%. This 25% was split about

evenly between the NYSE �oor (where trading is executed manually by brokers) and the

electronic platform NYSE Arca. This latter was integrated into the NYSE around 2006,

creating the hybrid market that the NYSE is today.

Since 2005, execution speed for small orders has been cut by a factor of more than

ten, reaching sub-second magnitudes (from 10.1 seconds on average in 2005 to 0.7 seconds

today). Average daily trading volume has more than tripled (from 2.1 billion to 5.9 billion

shares), while the size of the average trade size has fallen by almost a third (from 724 to

268 shares).

Trading is now mostly done on electronic platforms. Order �ow is fragmented in that

there are �ve large exchanges, as well as a number of other venues, that operate as Alter-

native Trading Systems. Among ATSs, one can distinguish three types: ECNs, (agency)

dark pools, and broker-dealer internalization pools.

ECN�s (Electronic Crossing Networks) are very similar to exchanges. In particular,

ECN�s publish their best prices to the market (there are two large ECN�s that together have

about the same total volume as one of the �ve large exchanges). The distinction between an

ECN and an exchange is hardly noticeable in practice, and a number of current exchanges

were started as ECN�s, �led for exchange status, and are now regulated as exchanges. (Dark

pools and internalization pools, by contrast, do not publish their best prices, and may have

mechanisms that are very di¤erent from those of a typical exchange.)

In total, exchanges and ECN�s account for about 74.6% of total volume: 63.8% on

registered exchanges and 10.8% on ECN�s. Thus, 75% of the total volume is labeled as

�displayed�in the SEC concept release, since both exchanges and ECN�s publish their best

prices. Note that exchanges and ECN�s may still allow for orders that hide part of the order

information (as discussed in detail in Section 2.4), so that there may still be hidden volume

on a given exchange or ATS at a given point in time.

The remaining 25.4% of the total volume consists of so-called undisplayed liquidity,

which is traded on dark pools and broker dealer internalization pools. Most (broker/dealer)

internalization pools were created by banks with the purpose of internalizing executions.

These pools have seed liquidity given by the banks�proprietary �ow and customer orders

(both retail and institutional), and are also open to other traders. Internalization pools may

route orders to exchanges, in which case the routed orders become part of the published

consolidated market data, and thus become visible. Within broker/dealer internalization

pools, one has to distinguish two groups: OTC market makers and block positioners. Ac-

cording to RegNMS, OTC market makers are de�ned as �any dealer that holds itself out as

being willing to buy and sell to its customers or others [...] a stock for its own account on a

18

regular or continuous basis otherwise than on a national securities exchange in amounts of

less than block size.�29 Block size is at least 10,000 shares (which is large compared to an

average execution size of below 300 on the NYSE). Block positioners execute larger trades,

whether for customers or on their own account. There are over 200 of these internalization

pools, with a combined trading volume of 17.5%.

Non-broker/dealer internalization pools, referred to below as dark pools, are mostly

agency-based. As such they do not have natural seed liquidity, and need to attract liquidity

in the same way that an exchange does. Some hybrids of internalization and agency dark

pools exist, namely dark pools that have been created by a consortium of banks and other

institutions. Dark pools do not publicize their best bids and o¤ers, which is why they are

labeled as trading undisplayed volume. There are about 32 dark pools and they make up

7.9% of the trading volume.

All dark pools operate as ATSs. In contrast to exchanges, which have to satisfy speci�c

pre-trade transparency requirements (most notably publishing the best available quotes),

dark pools may have little or no pre-trade transparency. In Section 2.4 below, I discuss the

trading mechanisms employed by some of the largest dark pools, as well as what (pre-trade)

transparency they o¤er.

Dark pools are interesting from a mechanism design perspective because the regulatory

framework (in particular with respect to transparency) leaves a large degree of freedom as

to the trading mechanism the pools employ. It is important to note, though, that there have

also been signi�cant changes in the mechanisms employed by the more heavily regulated

exchanges. Speci�cally, a number of exchanges, including the NYSE and NASDAQ, have

recently introduced new order types (some of which are discussed in detail in Section 2.4

below). Interestingly, most of these new order-types reduce transparency by hiding part or

all of the order information.

The recent statistics provided by the SEC about trading volume are interesting both in

terms of the numbers themselves, and because of the trend they demonstrate. Undisplayed

volume has reached a signi�cant 25.4% (as of September 2009), after strongly and steadily

increasing over the last few years.30 These changes have recently attracted the interest of

regulators, who are starting to become concerned about the fraction of trading volume exe-

cuted at venues with little or no (pre- and post-trade) transparency. The SEC commission

is currently trying to investigate the e¤ects of dark liquidity on order execution quality,

public price discovery, and fair access (as some of the ATSs are accessible only to a select

group of investors, whether formally or in practice). The next section gives a mechanism-

design analysis of some of the regulatory concerns the current US equities market structure

and addresses, and the following, Section 2.4, discusses some (speci�c) mechanisms used in

29See page 20 of SEC (2010).30Recently, more than 20% of all trades in the NYSE-listed stocks have been funneled through dark pools,

up from 3-5% in 2005. See Peek (2007).

19

speci�c trading venues.

2.3.2 General Mechanism Design Questions

This section analyzes mechanism design at the market level as a whole, leaving speci�c trad-

ing rules for the di¤erent venues aside. The �rst part of the section focuses on mechanism-

design issues at a meta-level: goals like fair access, and ability to evaluate the mechanisms.

The second part address the question of whether and why one may want to have a

market structure in which venues with di¤erent trading mechanism compete for liquidity,

or whether di¤erent types of order �ow (that is, types of investors) may bene�t from,

or need, di¤erent trading mechanisms. This question naturally raises issues like market

fragmentation, and regulatory requirements on transparency.

The third part focuses on the link between transparency and price discovery (distinguish-

ing between pre- and post-trade transparency, and addressing transparency�s interdepen-

dence with volume). The section begins with the theoretical arguments and the empirical

data on the link between transparency and price discovery, then explores how this link re-

lates to pricing, which naturally leads to an evaluation of the trade-through rule. Finally,

this third part re�ects on whether the current system is conducive to achieving the broader

mechanism-design goals, or whether and how the current order �ow re�ects systemic and

regulatory problems.

Mechanism design meta-level considerations Fair access is a desirable characteristic

of any trading mechanism. For exchanges, fair access is mandated; that is, any individual

should have relatively simple access to the exchange and be able to buy and sell any stock.

Dark pools and internalization pools do not always grant access to all investors. In

many cases, this has historical reasons: some dark pools were created in order to facilitate

the execution of large trades by institutional investors, and were closed to sell-side traders

like hedge funds. The argument was that buy-side investors would thus be protected from

the relatively more-informed order �ow of the sell side. One example is Liquidnet, which

was initially closed to hedge funds.31 Liquidnet has since opened to the sell side, arguing

that the additional liquidity they can provide outweighs the concerns about the quality

of information they process.32 Some pools still have formal access restrictions, and much

debate among practitioners centers around whether that is acceptable for any trading venue,

even if it is not an exchange.

More than formal or legal de�nitions, though, what matters is who has access to a given

dark pool in practice. Two points stand out. First, for some pools, like Pipeline, the mech-

anism employed implies, in itself, that not all traders can access this venue. Pipeline�s min-

imum order size of 100,000 shares for liquid stocks (and 25,000 shares for illiquid stocks)33

31See for example Schmerken (2005).32See, for example Schack (2006) and later Mehta (2009b).33See for example Pipeline (2010).

20

makes it inaccessible to individual investors with limited capital (and thus order size).

Second, the new subsecond execution speeds may also be a practical form of access

restriction, especially when coupled with decreased transparency, as is the case for dark

pools. An individual investor will likely end up with suboptimal or poor execution for a

given order, if the investor lacks the sophisticated tools necessary, �rst, to analyze trade

data from recently executed orders on all of the di¤erent pools and exchanges and, then, to

optimally split and route a parent order to the di¤erent pools in subsecond time. The more

complex the market microstructure, the harder it is for an unsophisticated individual to

make good use of the trading venues available to him, and achieve good execution-quality

for an order. This, in turn, links the goal of fair access to the second desirable characteristic

of trading mechanisms: good executions.

Ideally, a good trading mechanism should yield good execution-quality for all traders

that use it. As noted above, though, more-sophisticated investors may fare better than

less-sophisticated investors with a given mechanism. But before execution quality can be

evaluated, it must be measured. This turns out to be a nontrivial task (as the SEC�s

solicitation of possible measures illustrates).

Historically, the execution quality provided by a given trading platform has been mea-

sured mostly by spreads. One has to distinguish three types of spreads: quoted spreads,

e¤ective spreads, and realized spreads. A quoted spread is the di¤erence between the best

bid and best ask displayed for an asset at the venue at any point in time. E¤ective spreads

are calculated by doubling the distance between the midpoint of the market at the time

an order is submitted and the execution price received on that order. Realized spreads are

calculated by doubling the di¤erence between the execution price and the midpoint �ve

minutes after the an order executed. Realized spreads thus capture whether the market

moves against the order after the it is executed. Over the last �ve years, all three types of

spreads have decreased throughout the market.

Another long-standing measure of execution quality has been the speed of execution,

which is thought to be valuable to traders since they generally have demand for immediacy.

Moreover, fast execution speeds seem to indicate improved price discovery, meaning that

markets are e¢ ciently and quickly re�ecting any available information. As mentioned be-

fore, execution speeds have fallen signi�cantly over the last �ve years to subsecond levels,

putting traders with less-sophisticated tools at a signi�cant disadvantage.

A relatively new measure of market quality is short-term volatility. Volatility at larger

time scales is natural, and captures the uncertainty about the value of the stock, or any

changes in the estimated value of the stock as new information arrives and is priced in.

On a smaller time frame (of, say, minutes) though, price volatility may be largely driven

by the trading mechanism itself. For example, the temporary price movements caused by

a large order may depend on the trading mechanism at the venue where it is placed. In

their concept release, the SEC proposes variance ratios as a measure of short-term volatility.

21

These refer to the ratio of two variances, for example, the ratio of the variance of a stock

over a 5 minute interval to its variance over a one-hour interval (or the one-day return

variance to the 4-week return variance34. The general idea is that by dividing the variance

over a short horizon by the variance over a longer horizon, one can net out the long-term,

fundamental value related uncertainty, and be left with a measure that says something

about the micro-structure of trading instead.

Finally, another common measure of the broad quality of a mechanism is whether it

leads to market depth. measures of depth proposed by the SEC include �ll rates for limit

orders (that is, the probability that the limit order is �lled), and how much volume is

quoted at the best bid and ask, as well as at any other visible levels of the book. Despite

trade volume�s having almost tripled in the last �ve years, average execution sizes have

been reduced, much more volume is traded at venues with little transparency (and thus no

visible depth), and many orders are cancelled within a few seconds of submission.

It is important to note that all of the above quality measures are meaningful mostly

for small orders (normally under 10,000 shares) that are executed immediately. For larger

orders, measuring execution quality is very di¢ cult. This is in part because large orders

are split up and traded in small chunks, generally over a number of trading venues and a

longer period of time. A �rst step towards coming up with a quality measure lies in the

information that analytic �rms provide to their customers. These �rms allow traders to

enter large orders and then split the orders and route them to various venues. As a measure

of performance of the service they o¤er, analytic �rms that provide these algorithmic trading

tools attempt to calculate measures of how markets moved from the moment that a large

order started to be executed. One of the statistics in this context is volume-weighted average

price (VWAP). The VWAP of a stock over a given time horizon (for example, a day) is

used by traders as a benchmark or target price against which to evaluate the average price

received on the large order that is being executed. Receiving an average execution price

close to the VWAP implies that the market impact of the order was relatively low.35

All of the desirable characteristics and quality measures described above can be applied

to any given trading mechanism. But the next section explores whether, in the market as

a whole, there is intrinsic bene�t to having trading platforms using a variety of trading

mechanisms.

Should there be di¤erent trading mechanisms for di¤erent types of order �ow?There are, of course, already a variety of trading venues in place, and these venues may

have very di¤erent trading mechanisms (some of the speci�c mechanisms are discussed in

Section 2.4). The SEC has stated a belief that competition among venues will improve

execution quality and innovation, and will also reduce the systemic risk from the potential

failure of one central exchange.34See page 37 of SEC (2010).35For a more detailed description of VWAP strategies, see for example Madhavan (2002).

22

What is unclear, however is whether it is desirable for these competing venues to have

di¤erent trading mechanisms in place. This invites the preliminary question whether dif-

ferent investors have di¤erent preferences over mechanisms/trading venues.

The SEC notes that �a signi�cant percentage of the orders of individual investors are

executed at OTC market makers�(that is, in internalization pools, after having been routed

there through retail brokerage �rms), and that �a signi�cant percentage of the orders of

institutional investors are executed in dark pools.�36 Both individual and institutional in-

vestors are considered long-term investors, and this group seems to favor the less-transparent

trading venues, that is, internalization pools and dark pools. In contrast, �a large percent-

age of the trading volume in displayed trading centers is attributable to proprietary �rms

executing short-term trading strategies.�37

At issue is whether there is something about the types of order-�ow/investors that in-

herently makes some trading venues more attractive to them than others. The answer

has implications for many other questions regarding market design. Let us assume for a

moment that less-transparent trading mechanisms (such as dark pools) hinder price discov-

ery. (Whether or not that is the case remains an open question, and is the topic of the

next subsection below.) If less transparency reduces price discovery, but venues with less

transparency were especially bene�cial for long-term investors, then regulators may face a

trade-o¤ when considering, �rst, whether to allow trading venues with less transparency

than exchanges, and second, what degree of opacity to allow.

If less transparency bene�ts long-term investors, regulators should be less concerned

both with the fraction of total volume traded in the dark, and with the degree of frag-

mentation of liquidity (the latter because fragmentation would be a natural consequence of

allowing di¤erent trading venues geared toward di¤erent types of investors).

There remains no clear conclusion about whether di¤erent investor types do really have

preferences or need for di¤erent mechanisms. Not only is it di¢ cult to trace back order-�ow

and identify the trader�s motives (whether or not the trader is a long-term investor), but

it also remains unclear how to de�ne �long-term� investors. In view of these di¢ culties,

the SEC seems to have taken an alternative approach: to distinguish among types of order

�ow, by considering the size of the orders.38

Speci�cally, the SEC has chosen to di¤erentiate between small and large orders (the

latter comprising at least $200,000 in market value). It has proposed three changes that

concern the level of transparency required from dark pools/ATSs. These changes would

demand more disclosure of order- and trade-information (the exact nature of the proposed

changes will be discussed in the next subsection). But the SEC is plans to exempt large

orders (as just de�ned) from these changes, so that they could continue to be executed with

the current low level of pre- and post-trade transparency at the dark pools.

36See page 67 of SEC (2010).37See page 69 of SEC (2010).38See page 43 of SEC (2010) or SEC (2009).

23

There are a number of potential problems when granting exceptions for a portion of the

order �ow. The argument in favor is that, while exempting large orders will come at a cost in

terms of lower price discovery and e¢ ciency, this cost will be outweighed by the bene�ts for

the traders that use this orders, who are believed to be mostly long-term investors. The �rst

potential problem is that the system may be abused in the sense that short-term investors

such as speculative traders may adapt their trading strategies and order submissions to

take advantage of the protection (that is, lower transparency requirements) given to large

orders, once the transparency requirements for smaller orders are toughened. Given that

the exception for orders depends only on the price, it would impossible to prevent this kind

of behavior.

The second potential problem is that, even if large orders are used only by long-term

investors, not all long-term investors are able to use them. For example, while institutional

long-term investors naturally execute large orders, individual retail investors with the same

long time-horizon, but smaller orders, may not be able to take advantage of the protection

granted to large orders, putting �smaller�traders at a disadvantage.

The third potential problem also concerns retail versus institutional order �ow: it has

long been thought that integrating these two order �ows would be bene�cial. Up until

about �ve years ago, large orders were traded in upstairs markets where trade sizes were in

the hundreds of thousands, while the average execution size on the NYSE �oor was in the

hundreds (around 700). In the last few years, exchanges have started to introduce order

types that would allow better integration of large, institutional orders with the ever-smaller

average sizes at the exchanges (currently below 300 shares). This e¤ort on the part of the

exchanges came in response to competition for institutional order �ow from other venues,

like dark pools, that have speci�cally targeted large orders (for example, by introducing a

minimum order size of 100,000 shares). While it is natural for trading venues to compete

for order �ow, the regulators�concern was mostly that integrating the order �ow on a given

platform would lead to more price discovery, especially given that large orders are currently

mostly executed at venues with less price-transparency. If it is true that institutional order

�ow bene�ts from less transparency, then granting exceptions for large orders and allowing

those orders to trade at venues with di¤erent mechanisms should lead to less integration of

retail and institutional order �ow, as institutional order �ow could trade on these venues

while order �ow from individual retail investors would be unable to follow.

Current SEC proposals/discussed options As stated previously, one of the main

concerns expressed by the SEC is that a market microstructure with less regulated trans-

parency will lead to suboptimal price discovery. Much of the theory is inconclusive with

regard to this question, and the actual market structure is more complex than what has

been captured with theoretical models to begin with. Nevertheless, at least at �rst glance,

there is evidence to suggest that less transparency may indeed lead to less price discovery

24

in practice. Many dark pools allow traders to submit orders that consist of quantities only,

and execute any trade using the displayed NYSE prices as reference (usually using the mid-

point). In some sense, these pools are �stealing�the public price from the NYSE, in that

they bene�t from it as a reference price, without having their order �ow contribute to the

discovery of that price.

While it is true that orders executed at the National Bid and Best O¤er (NBBO)39

on other venues do not help form the NBBO, it may very well be that the NBBO re�ects

the correct �calculated� price by incorporating more information than the total demand

and supply that is eventually routed to the NYSE. In practice, the NBBO may re�ect a

more general market demand and supply in that, if it did not, there would be arbitrage

opportunities: for example, a large seller who knows that the NBBO will be used for any

sell order he places on other dark pools may have an interest in submitting a buy order to

the NYSE shortly before submitting his large sell order elsewhere. The buy order would

potentially increase the price at the NYSE and the seller would consequently receive a

better price on a sell order placed elsewhere. If enough traders are aware of this potential

market-manipulation they, would naturally compete in setting the NBBO, so that even if

the quantities submitted to the NYSE were small, the NBBO would represent much more

demand and supply (despite its being executed elsewhere).

Interestingly, there is evidence that traders do in fact try and manipulate the NBBO.

This can be seen by observing some of the rules that dark pools have implemented. Perhaps

the most obvious example is a dark pool created by the NYSE itself, called NYSEMatch-

point. NYSE Matchpoint is a pool for large orders. Traders submit orders consisting only

of quantities, and the orders are then crossed once per hour. The price at which the trades

are executed is the average NBBO of a randomly selected minute within the preceding hour

interval. Choosing the minute randomly makes it prohibitively costly to try to manipulate

the NBBO used in the large hourly cross. Pipeline, a dark pool with a completely di¤erent

mechanism, but which also uses the NBBO as a reference price, has a similar solution,

de�ning a proprietary block price-range (that includes the NBBO, as well as information on

short-term volatility and other factors aimed at reducing the potential for manipulation).

These and other mechanisms are discussed in more detail in Section 2.4 below.

Leaving aside the arguments of the previous two paragraphs, the main concern remains

that the fraction of volume traded on exchanges (under strict transparency requirements)

has dropped continuously and signi�cantly over the past few years.40 As a consequence,

the SEC has brought forward a set of proposals, as well as another option to be considered.

The proposals include de�ning a new, lower, cap on the volume dark pools are allowed to

trade before having to comply with three new requirements, for pre-trade transparency,

IOIs (Indications Of Interest), and post-trade transparency. The other option that is being

39The NBBO is the highest bid and lowest o¤er for a security across all exchanges and market makers. Aformal de�nition is found in Rule 600(b)(42), page 479 of RegNMS, SEC (2005).40See Westbrook and Kisling (2009).

25

discussed is the introduction of a trade-at rule (discussed below). While the proposals are

deemed likely to be implemented in the near future, the trade-at-rule is currently considered

a less practical option.41 But controversy remains around all these proposed measures.42

The current regulation regarding dark pools requires no pre-trade transparency, and

only limited post-trade transparency, for pools that handle less than 5% of the total volume

of a given stock (calculated as the moving average over a time period of a few months).

In particular, no order information, bids, or Indications of Interest have to be displayed.

Indications of interest are sent out by some venues to other traders on the venue when an

order enters their system, informing the other traders of the availability of that order and

requesting a counterparty, without that IOI being a binding bid or ask. Moreover, IOIs

may be sent only to a select group of participants at the given trading venue.

Regarding post-trade transparency, trades currently have to be published in real-time

after execution, but the published information is not required to include the information

about where the trade was executed (that is, the speci�c dark pool that executed the order

remains unknown). Not reporting the execution venue makes it harder for other market

participants to detect the existence and location of a large order, making in harder to engage

in potentially parasitic trading strategies against that order (the exact nature of some of

the strategies will be discussed in Section 2.4). For venues that execute a volume above 5%

of the market total for a stock, stricter transparency requirements hold: best prices have to

be published pre-trade and full reporting has to be done after the trade.

The proposed change consists of lowering the threshold for the current transparency

requirements. A likely new threshold may be 1%, but anywhere down to 0.25% is being

considered. Venues with volumes above the threshold would have to abide by the stricter

pre-trade and post-trade requirements. Moreover, it is proposed that IOIs would have

to be published, and potentially be considered binding o¤ers. If put into practice, these

proposals would signi�cantly and immediately impact trading at the larger pools, while

also immediately and signi�cantly a¤ecting the growth of pools in general. Pools above the

threshold would e¤ectively lose some of the characteristics that made them attractive to

traders to begin with.

There are a number of potential problems with increasing transparency requirements on

the dark pools. Since opacity is what attracts traders, the new rules may lead to a drying-

up of liquidity in dark pools, making it increasingly hard for a submitted orders to �nd

counterparties. Order �ow would go either to exchanges or back to OTC market-makers

(that is, mostly broker/dealer banks). It is interesting to look back at what made dark pools

so popular initially. A recent article, Nelson (2009), makes the observation that, when the

trade-through rule was put in place and all electronic best bids and asks in a given market

were thus protected from being traded through, it lead to an increase in the execution costs

41See for example Mehta (2009a).42For a more detailed discussion, see for example D�Antona (2010).

26

for large orders by requiring �that smaller, better priced orders to be executed before a

large order at an inferior price can be executed as a block�on any exchange. Dark pools,

on the other hand, were free to cross orders at any price, independent of the best bid and

o¤er, meaning that a large order would, in some sense, trade-through after all. This was,

and is, one of the main advantages provided by large pools, and is most bene�cial to traders

who trade large quantities. Currently, 8% of the total 10.8% of dark (non-internalization)

pools are geared speci�cally towards execution of large orders.43

The SEC has found large orders to be submitted mostly by institutional investors, who

are largely part of the group of long-term investors that the SEC is trying to protect. If

the proposals on transparency were put in place, this group would be the one most likely

to su¤er from the implications.

Moreover, as pointed out in the section on theory above, it is unclear whether the price

on a small quantity on an exchange should apply also for the execution of a large quantity.

A trading platform worth mentioning in this context is Pipeline: while over 90% of the

trades executed there execute at the midpoint of the NBBO,44 Pipeline also provides a

block price-range that allows for some variation, and more aggressive pricing of orders, so

that they may still execute in case su¢ cient liquidity is not available on the opposing side

at the NBBO. Finally, upstairs markets historically also sometimes priced block-trades at

prices di¤erent from the NBBO, especially when, say, the broker/dealer had to take the

contra-side of a given trade and was then obliged to unwind the position over time.

Finally, the trade-at rule considered by the SEC in addition to the transparency pro-

posals would likely have even more dramatic e¤ects on dark pools. From the SEC concept

release: the trade-at rule would �prohibit any trading center from executing a trade at

the price of the NBBO unless the trading center was displaying that price at the time it

received the incoming contra-side order.�If the trading venue did not have an order priced

at the NBBO, it could either execute the incoming order at a signi�cant price improve-

ment, or route out part of the order to buy up all volume displayed at the NBBO on other

exchanges, and then execute the rest of the order on its own venue at the NBBO. As a

recent article, Mehta (2009a), put it, �a prohibition on trading at the NBBO would cause

dark pool volume to capsize.�Brokers who internalize orders (executing at the displayed

quotes and making the spread) would lose much of their income by having to send volume

to exchanges (or improving the price for the customer), which would eventually translate

into higher commissions. Institutional investors would be likely to su¤er by having to trade

a large fraction of their orders in markets with higher transparency (where their interests

would also be more likely to be detected). In addition, the trade-at rule may be di¢ cult

to implement for certain venues, especially those whose mechanisms, are unlike exchanges

and geared towards large orders (for which the NBBO may or may not be the correct price

43See Westbrook and Kisling (2009).44See for example page 90 of Traders Magazine (2007).

27

to begin with).

Returning to the proposed changes in transparency requirements, note that exempting

large orders like the SEC envisions,45 addresses some of the problems, but presents nothing

like a full solution. One of the di¢ culties is that, if dark pools are no longer dark for small

orders, these orders may move to exchanges or market-makers, as dark pools would have no

added bene�t from a trading-mechanism perspective, and would o¤er less liquidity than, say,

exchanges. As mentioned before, the lack of small orders likely would make it more di¢ cult

to �nd a counterparty to a large order. A large number of dark pools would be likely to

disappear, concentrating liquidity in the remaining dark pools and thereby mitigating their

problems. Liquidity dry-ups in remaining pools may also be mitigated as order submission

strategies adapt, as follows. Currently, a large part of the small orders routed to a given

dark pool are part of a much larger �parent� order. If post-trade transparency were to

increase, other traders would have an easier time reconstructing the �parent� order from

a collection of possible �child�orders that were executed at various venues (a practice to

which algorithmic traders already devote signi�cant e¤ort). Consequently, traders may

decide to send the entire �parent� order as a block to a dark pool, leading to more large

orders than we currently observe. Either way, though, it is important to note that order

�ow would end up segregated, with large orders remaining on dark pools and small orders

being executed elsewhere.

The above considerations illustrate the remaining uncertainty about what would happen

to trading volume and spreads if much of the volume traded on dark pools today were re-

captured by exchanges and OTC market-makers. If volume and spreads are driven largely

by the market structure, the total volume would be expected to decrease and spreads to

widen as they did under the market structure of about �ve years ago. Unlike execution

times, whose decrease can be explained by advances in electronic trading over the last �ve

years, the increase in volume and decrease in spreads are hard to explain with reference

only to technology and fundamental-value reasons. Market microstructures thus emerge as a

possible driver of the changes by exclusion, a hypothesis strengthened by the close proximity

in time between the changes in regulation and the changes in trade volume and spreads.

(Note that the SEC also considers market microstructure a signi�cant driver of trading

behavior in general: variance ratios, for example, were speci�cally designed to isolate the

part of short-term volatility believed to be driven by market microstructure, as explained

above.)

In terms of other consequences of the proposed changes in regulation, it seems that

OTC market makers (which are broker dealers and thus not categorized ATS), would likely

bene�t, due to an expected increase in volume routed to them (a point which is discussed

in a recent article by Nelson (2009)). From a systemic point of view, though, it may not

be bene�cial to have already large, and systemically important banks take on more market

45See for example Brigagliano (2009).

28

making and order execution responsibilities.46

In any case, if the new changes were to be realized, there would likely be just two main

groups of non-agency based trading venues remaining: exchanges and dark pools (the latter

geared towards large quantities). The following section analyzes these venues in more detail,

starting with a brief discussion of the mechanisms of some of the most prominent dark pools

designed for the execution of large quantities, and then moving on to new order types (that

is, changes to the trading mechanisms) that exchanges have introduced with the intention of

attracting and integrating large orders. It also discusses how these mechanisms are designed

to mitigate the e¤ects of gaming strategies, both by the traders submitting the large orders

and those traders trying to detect them.

2.4 Speci�c venues and mechanisms

Since the recent establishment of multiple trading venues, these venues have been competing

for order �ow in the US equities market in a number of di¤erent ways. Most notably, they

compete through prices (in that some venues have started to pay for order �ow), and

through mechanisms (that is, by o¤ering innovative mechanisms or order types in the hope

to attract more liquidity).

Competition in prices is discussed in Section 2.4.1. A number of di¤erent mechanisms

are described one-by-one in Sections 2.4.2 through 2.4.5. This section �rst describes several

popular ATSs (POSIT, NYSE Matchpoint, Liquidnet, and Pipeline) and then outlines three

new order types: �rst, so-called iceberg orders (where part of the quantity is hidden), second,

discretionary orders (where price is hidden), and third, mechanisms where essentially the

entire order is hidden (including volume orders, discretionary reserve orders, hidden-limit

orders, and dark reserve orders). For Iceberg orders, the only type that has existed long

enough to be studied extensively, academic literature exists and is included in the discussion.

While new order types are not mechanisms in the typical sense, the introduction of a new

order type a¤ects the trading mechanism for the corresponding exchange. Part of the

discussion in these sections focuses on how the mechanisms attempt to reduce the potential

for gaming strategies that may be employed by traders.

The last Section, 2.4.6, revisits some of the theory literature that is relevant for the

theoretical part of the thesis (from Section 4 onward), which analyzes games with either

iceberg, discretionary or volume orders.

There is also a substantial literature focusing on the interaction and competition among

trading venues and mechanisms. In a seminal paper, Glosten (1994) analyzes a market

46 In the recent crisis, the banks�market-making functions in credit derivatives, repurchase agreements,and prime brokerage led to a much faster deterioration of the stability of a bank if word got out that itwas in distress, which in turn led to an increase in systemic risk associated with the consequently morelikely failure of that bank. Speci�cally, customers in these markets (that is hedge funds, and institutionalinvestors) would immediately divert their business to other banks upon hearing that a bank they had beenexecuting trades with was in distress, and the bank�s situation would dramatically worsen with the loss ofthe capital that had been associated with the bank�s market-making business.

29

with a limit-order book and concludes that a di¤erent structure (say, a dealer market)

would not be pro�table competing against the limit order book, and later also considers

competition among limit-order books in Glosten (1998). Pagano (1989) considers the rela-

tionship between trade volume (order size) and liquidity in a given market, when traders

can choose to trade across a number of markets. Chowdhry and Nanda (1991) consider a

model based on seminal papers by Kyle (1985) and Admati and P�eiderer (1988), and ana-

lyze how market-makers at di¤erent markets compete when there are informed traders who

can trade across the di¤erent markets. Parlour and Seppi (2003) analyze the competition

between a limit-order market and a hybrid market (which consists of both a limit-order

book and specialists). Hendershott and Mendelson (2000) consider competition between a

call market and dealers. Finally, Foucault and Parlour (2004) analyze the case of di¤erent

exchanges competing over listing fees.

2.4.1 Transaction Fees, Liquidity Rebates and Access Fees

This section discusses how trading venues have been competing for order �ow by using prices,

namely by paying and/or charging for execution of orders. Traditionally, exchanges �nanced

themselves with transaction fees charged on a per-share basis for trades they executed.

Notably, these fees were the same for both the buyer and the seller in a given transaction.

Here, price competition among exchanges simply meant o¤ering lower fees.

About 75% of the volume currently traded in US equities is displayed volume, meaning

that it is traded at exchange-like venues, in that the venues have order books displaying the

best bids and asks at all times (for a more detailed description, see Section 2.3.1). Traders

considering these venues have two options: they can submit either an order that will be

posted as a standing order in the book, or a marketable order that will execute against

existing orders in the book. It has become common to refer to traders posting visible orders

as �liquidity makers,� and to those who send marketable orders in response as �liquidity

takers.�

Competition for order �ow has recently focused on attracting liquidity makers, who must

be given incentive to post visible orders, since those orders also constitute free options to

other traders and/or may give away information about the value of the asset. In addition,

a plentiful order book is advantageous for the trading venue, since order-routing systems

prefer to send marketable orders to venues with a good execution history (owing to past

depth). As a consequence, many venues have started to introduce �make-or-take�pricing

when executing trades: paying out liquidity rebates for the order that was posted in the

book, and charging access fees (that are larger than the rebates) for the marketable order

that was placed against the standing limit order in the book.

Initially, venues could set rebates and access fees at their discretion and, while always

on the order of 0.25 cents per share, venues varied in the magnitude of their rebates. The

overall e¤ect, though, was an increase in total displayed volume and a reduction in spreads.

30

Still, signi�cant problems associated with make-or-take pricing eventually motivated the

SEC to introduce a access-fee cap of 0.3 cents per share (in RegNMS, which passed in

2005)47. This cap mitigated the problems, but could not eradicate them, as discussed in

Angel, Harris, and Spatt (2010).

The �rst, most obvious e¤ect of rebates and access fees is what the authors refer to

as �obfuscation of prices� from the actual best bid and ask: rebates lead to a reduction

in spreads, but the average spread net-of-costs remains the same o¤setting the bene�t to

liquidity takers by charging them access fees. Liquidity makers� rebate is o¤set by the

reduced spread (that is, the higher buy prices and lower sell prices that they must submit

to remain competitive).

Another problem the paper raises is that the trade-through rule protects quoted prices,

not quoted prices net-of-costs, so rebates end up a¤ecting competition among trading

venues. Every exchange is obliged to rout any marketable order out to any other exchange

o¤ering a better quoted price (implying that the best quoted price is protected, by being

guaranteed to execute ahead of less-competitively priced quotes). Access fees accrued as an

order is routed to a make-or-take venue must be paid either by the routing exchange or by

the trader who sends the marketable order. If the routing exchange agrees to pays the fee,

it loses money and is moreover likely to attract traders wishing to avoid the fee, rather than

those seeking the exchange with the best price. But if the exchange passes fees on, it will

be less attractive to traders, also increasing their uncertainty about the expected trading

cost.

The gravest problem Angel, Harris, and Spatt (2010) discuss with rebates is that they

distort the incentives of traders. Brokers executing orders on behalf of their clients are gen-

erally allowed to keep the rebates they collect, giving them incentive to send non-marketable

orders to venues that pay rebates, while sending marketable orders to venues that do not

charge fees, or to internal crossing networks of dealers. The act of collecting rebates is

itself questionable: these payments would constitute �illegal kickbacks�48 in the context of

common law. Further, distorted routing decisions lead to inferior execution of non- mar-

ketable orders in two ways. First, since venues that pay rebates also charge access fees, an

order at a given quoted price will remain in such a venue�s book longer than in that of a

venue without access fees. Second, the order will interact with less-favorable order �ow (or

more-toxic �ow), meaning that it is more likely that markets will move against the order

whenever it executes. This risk is reduced by routing to, say, the internal-crossing network

of a dealer, since such dealers select which order �ow to accept, and will allow relatively

uninformative retail orders to be executed, while excluding what is perceived as informed

order-�ow from, say, proprietary trading desks and institutional traders. Exchanges, on the

other hand, allow anyone to trade anonymously, so a marketable order faces comparatively

47See page 517 in SEC (2005).48See page 45 of Angel, Harris, and Spatt (2010).

31

more-informed order �ow on the exchange.

Angel, Harris, and Spatt (2010) go on to recommend that the SEC either require all

brokers to pass rebates and access fees on to clients, or eliminate them altogether in favor

of transaction fees.

2.4.2 Sample ATS and Internalization Pools

This section brie�y discusses examples of dark pools belonging to three categories: simple

call markets, negotiation-based pools, and dark pools with other mechanisms.

Call Markets (POSITMatch, NYSE Matchpoint) Call markets collect bids and

execute them at regular intervals (no bids are displayed at any time). Two examples are

POSIT and NYSE Matchpoint.

POSIT Match allows traders to execute blocks of shares at scheduled crosses throughout

the day. Moreover, all trades are executed at the midpoint of the NBBO.49

NYSE Matchpoint, created in 2008, allows traders to submit orders that consist of only

quantities. Orders are crossed at �ve hourly sessions. In each session, demand and supply

are matched, and all trades are executed at a price that is calculated as the average NBBO

of a randomly-selected minute within the preceding hourly interval. Choosing the minute

randomly makes it prohibitively costly for traders to try to manipulate the NBBO used in

the large hourly cross. The NYSE has speci�cally indicated that Matchpoint was created to

allow for less-costly execution of large orders.50 Moreover, traders can submit orders that

consist of baskets (portfolios) of di¤erent securities, as well as imposing other constraints

that are especially useful for portfolio managers.51

From a mechanism-design perspective, call markets are less likely to be manipulated

than markets with sequential trading mechanisms, and have less potential for information

leakage. On the other hand, immediacy is lost, in that traders have to wait for the next

crossing session. This loss in immediacy will be especially unattractive to traders that use

continuous hedging strategies, but it may not cause much concern for investors wishing

simply to buy and hold a large block of a stock.

Negotiation based pools (Liquidnet) Currently, the largest negotiation-based pool is

Liquidnet. Liquidnet links in with traders�order-management systems and, upon submis-

sion of an order, tries to locate potential matches by sending out indications of interest

49Posit Match (by the company ITG) is one of the oldest dark pools, originally created in 1987, when ito¤ered hourly crosses- the number of crosses o¤ered has since increased to 12 per day.An academic discussion of the mechanism is found on page 35 in Angel, Harris, and Spatt (2010).For more detail on the precise execution rules, see for example ITG (2009) and

http://investor.itg.com/phoenix.zhtml?c=100516&p=irol-newsArticle&ID=958312&highlight.Moreover, ITG now o¤ers a number of other solutions for order execution, including continuous crosses,

see http://www.itg.com/.50See for example http://www.nyse.com/equities/nyseequities/11811.07351327.html.51See NYSE Matchpoint (2008) for a more detailed description of the mechanism.

32

to other participants who may be interested in trading against the order. Once matched

through the system, traders bargain in bilateral negotiations on an execution price and size

for the order (which may not always lead to an agreement to trade). To prevent abuse of the

system by traders who, for example, submit orders with the sole purpose of investigating

liquidity on the other side of the market (but with no intention of trading), Liquidnet has

a rating system in place. Speci�cally, a trader�s rating is lowered if he backs out of a trade,

and traders can choose, for example, to be matched only with traders that have su¢ ciently

high ratings (that is, high completion rates on trades). As Angel, Harris, and Spatt (2010)

put is :�Liquidnet thus ensure that only traders who have a high probability of arranging

trades obtain information about future traders.�52

Liquidnet has very low pre-trade transparency. Moreover, while it has a low minimum-

size requirement for orders, bundling exists in practice, in that most trades executed on the

platform are large (with an average execution size of over 50,000 shares). In fact, Liquidnet

regularly reports some of the largest trades in the equities markets, in excess of a million

shares53.

Regarding gaming strategies, it is important to note the following. Since trades are exe-

cuted as a result of bilateral negotiations, traders are naturally discouraged from submitting

gaming strategies consisting of a sequence of orders, hoping for execution at the most favor-

able price. This practice would be encouraged if trades were executed by matching (binding)

electronic orders. The downside of bilateral negotiations, though, is that some e¢ ciency is

lost compared to a central exchange-type mechanism in which sellers would compete over

the price at which to sell to a buyer, or vice-versa.

Other dark pools (Pipeline) Of the remaining dark pools, Pipeline�s mechanism stands

out. Pipeline imposes a minimum order size for every trade: for liquid stocks, the minimum

order size is 100,000 shares, while for less-liquid stocks, the minimum is 25,000 shares. When

an order is entered into the system, Pipeline publicizes only the existence of an order for a

given stock, but does not display the size of the order, the side (whether it is an order to buy

or sell), or the price. Instead, at all times, Pipeline calculates and displays a proprietary

�block price range�for each stock, giving a reference price range within which traders can

expect their block orders to be executed. This block price range is a function of the current

midpoint of the NBBO, but also re�ects parameters that capture more fundamental or

long-term characteristics of the stock (such as price trends and past volatility).

Traders orders are categorized as either passive or aggressive. Less-competitively priced

orders are termed passive, and more-competitively priced orders (de�ned as orders priced

above the NBBO for the stock) are termed aggressive. If a trader submits a passive order,

traders that submitted aggressive orders on the other side will be informed and will have the

option to trade against the passive order at its price. In contrast, the submitter of the passive52See page 36 in Angel, Harris, and Spatt (2010).53See for example Liquidnet (2009).

33

order will not learn any new information. In addition, aggressive orders are considered

binding o¤ers and will execute immediately against existing or incoming aggressive orders

on the other side (at a price equal to the midpoint of the NBBO).54

By providing information to submitters of aggressive orders, the mechanism encourages

submission of competitively-priced orders. But submitting an aggressive order with the

purpose of �shing for liquidity on the other side remains risky, given that the order may be

automatically executed at the midpoint, and that the minimum size of orders is very large.

From a mechanism-design perspective, it is important to note that, since the passive

order sets the price when a passive and an aggressive order are matched (even if the passive

order is submitted after the aggressive order), the mechanism is vulnerable to trading strate-

gies in which traders submit sequences of passive orders, hoping to execute at the lowest

possible buy price (or highest possible sell price). To prevent these strategies, traders who

submitted passive orders are prohibited from increasing the buy limit-price (or decreasing

the sell limit-price) on the order for a random amount of time. On the other hand, pas-

sive orders may be changed to be less-competitively priced at any time, and the prices of

aggressive orders may be increased or decreased at any time (without losing time priority).

2.4.3 Iceberg Orders

Iceberg orders are the oldest of the order types described in this subsection, having existed

in exchanges throughout the world for over 15 years. As a result, a relatively large number of

empirical studies on the use and e¤ects of iceberg orders have been conducted by academics

(some of which are discussed below), while few exist for the more recent order types discussed

in the following subsections.

Iceberg orders derive their name from the small peak visible above water, while the

rest of an iceberg is hidden below. Consistent with this image, iceberg orders consist of a

price and two quantities, a displayed �peak�quantity, which is usually small, and a hidden

quantity, which is usually much larger. The peak quantity is usually required to be above

some minimum size (which varies by exchange). Hidden volume has lower priority than

visible volume at the same price (and, as usual, time priority exists among hidden volume

of di¤erent orders). Moreover, if an incoming order is executed against the iceberg order,

the latter is consequently updated such that its visible part is again equal to the original

peak size (that is, some or all of the remaining hidden volume becomes visible, depending

on the peak size).

Iceberg orders (or, interchangeably, hidden orders) are used by traders to avoid disclosing

the total size of the trade they wish to execute. The most common explanation for traders�

preference for opacity, which is also the one that has been given so far, is that traders�hope

to reduce the market impact of their orders by limiting information leakage. (Leakedinformation can be about the total size of the order, or about the value of the asset). With

54See Pipeline Trading (2009) for a more detailed description of the mechanism.

34

respect to the bene�ts of iceberg orders, a second explanation is also sometimes given,

namely that the iceberg orders reduce adverse selection in the execution of the order.The adverse selection argument is based on a common assumption in the market mi-

crostructure literature, namely that limit orders are submitted by uninformed liquidity

providers, whereas market orders are submitted by both informed and uninformed traders.55

Limit orders that are visible in the book then constitute valuable free-options to informed

traders, who will trade against them by submitting market orders.56 From the viewpoint

of the uninformed submitter of limit orders, trading against a potentially informed trader

induces an adverse-selection cost, and iceberg orders may o¤er uninformed traders protec-

tion from being picked o¤, as uncertainty about the available volume at a given price may

discourage market-order submission by informed traders.57

In evaluating these two explanations (market-impact cost or adverse-selection cost), one

should note the following. If iceberg orders were mainly used by informed traders trying to

reduce market impact, then hidden depth should increase when impact costs are expected to

be high, and hidden depth should be informative, that is, contribute to the price discovery

in the market, and therefore be predictive of future price movements. If, however, iceberg-

order submission is driven by a desire to reduce adverse-selection costs, then one should

�nd more order submission in situations where the option value of a limit order would be

high, such as when volatility or the degree of informational asymmetry is high, or when the

size of the order is large.

The following paragraphs present some general statistics on the use of iceberg orders,

review empirical studies on the motivation for their use, and analyze their impact on order

�ow and prices. Most of the research is from the early 2000s and was conducted on ex-

changes abroad, since iceberg orders and the electronic trading of equities in the US started

relatively late58 and a number of other hidden order types were introduced soon afterwards

(complicating the analysis).

The research shows that iceberg orders are widely used, often by traders believed to

have information, and the data support both explanations for this use: that iceberg orders

reduce price impact, as well as that they reduce adverse-selection costs.

Basic statistics on the use of iceberg ordersA number of relatively early studies found that iceberg orders (or hidden orders) consti-

tute a signi�cant fraction of traded volume on most exchanges that o¤er them (the extend

to which iceberg orders are used varies by exchange).

A study on the Australian Stock Exchange by Aitken, Berkman, and Mak (2001) found

that 6% of the orders were hidden, accounting for 28% of the total volume For the best �ve

55See for example Glosten (1994). Note that later work challenging this assumption also exists, for exampleKaniel and Liu (2006).56See for example COpeland and Galai (1983) and Aitken, Berkman, and Mak (2001).57See for example Aitken, Berkman, and Mak (2001).58Until 2005, the dominant exchange was the NYSE, which was manual- and quote-driven, as opposed to

exchanges like the Paris bourse, that had been order-book driven for years before that.

35

price limits on the Euronext during a sample period in 2000, Winne, D�Hondt, and Heude

(2003) found that hidden depth was on average about the same as displayed depth available

at the best �ve limits (meaning that, on average, only half of the market�s total depth was

visible at the best �ve prices).59 A study on the Spanish Stock Exchange by Pardo and

Pascual (2003) found that 26% of all transactions involved hidden orders. Tuttle (2002)

studies of 100 stocks on NASDAQ shows that the introduction of the SuperSOES, which

allowed for iceberg orders, led to a 42% increase in displayed depth, with hidden depth

making up 22% of depth at the best prices. Note, though, that this study had data only

on the SuperSOES part of the market-maker orders. ECNs, which also allow for iceberg

orders and which contribute 78% to the inside market, were missing, so hidden depth was

understated).

Iceberg orders tend to be competitively (or aggressively) priced in the sense that they

are submitted within or at the best quotes (or among the �ve best limits of the book). For

the Spanish stock exchange, Pardo and Pascual (2003) found that 93% of hidden orders

were placed inside the best quotes. As one would expect, hidden orders are less likely

to be executed, due both to their invisibility and to their lower priority compared with

visible quantities: in the Euronext study mentioned above, the authors found that execution

probabilities during the continuous trading session were 35% for hidden orders, compared

to 50% for displayed orders.

Winne, D�Hondt, and Heude (2003) also investigated the behavior of aggregate depth

displayed, and hidden depth, over the course of the day (at the �ve best limits). Displayed

depth was not only equal on average to about half of total depth, but also fairly constant

over time. While hidden depth, by contrast, both on the bid and the ask sides varied

considerably. Moreover, the authors found negative correlation between the volumes at the

bid and ask sides: hidden depth on the ask side would be high when hidden depth on the

bid side was low, and vice versa. The authors concluded that traders engaged in �depth

management�60 trying to reduce the informativeness of the visible book.

Motivation for the use of iceberg ordersTwo studies seem to validate the market-impact and adverse-selection arguments for

the use of iceberg orders. The �rst study, Harris (1996), tested the market-impact cost

hypothesis, positing that iceberg orders should be more attractive in markets where the

tick size is small, because small tick sizes facilitate front-running. Comparing data from the

Paris Bourse with data from the Toronto Stock Exchange, where minimum price variations

for stocks are on average 12 times larger, the study con�rmed the hypothesis: traders at the

Paris Bourse hid quantities more often, and the di¤erence in trading behavior was more pro-

nounced for larger orders (which have higher associated costs of exposure). Quantitatively,

in Paris, 5% of the smallest orders and 74% of the larger orders were hidden, compared to

59Over all price levels, an average of 35% more depth than displayed was available.60See page 10 of Winne, D�Hondt, and Heude (2003).

36

1% and 13%, respectively, in Toronto.

The second study, Pardo and Pascual (2003) (mentioned above) investigated whether

hidden order submission increases when informational asymmetries are high. Using data

from the Spanish Stock Exchange, the authors found a signi�cant jump in the number

of hidden orders around 15:30pm Spanish time, with a peak during the next hour, and

a reduction until the end of the trading day. In Spain, 15:30pm tends to be a time of

maximal uncertainty, as it is one hour before the opening of the NYSE, where relevant

information (with respect to both prices and daily statistics) will be revealed. In addition,

public announcements in Spain are usually made in the afternoon.

Also relevant are two studies that investigate which groups of traders tend to use iceberg

orders. The �rst, Tuttle (2002)(also mentioned above), was conducted on NASDAQ, and

�nds that iceberg orders are used disproportionately often by �expert�traders. Speci�cally,

professional traders for wirehouses and investment banks each provide about 5% of the

displayed inside market depth, but when hidden depth is included, their share jumps up

to 8.8% and 16.6%, respectively. To the extent that wirehouses and investment banks are

better-informed about asset values than institutional or private investors, hidden volume

would thus also correspond to �informed�volume.

Using data from the Euronext Paris, Declerck (2000) �nds that traders with access to

more levels of the order book use hidden orders disproportionately more often. Traders

on the EuroNext with o¢ cial market-member status are able to see not only the �ve best

limits, but rather the entire book, as well as the identities of the brokers treating the order.

These �dual traders�(trading both on their own accounts and for their clients) have superior

information compared to individual investors, at least about order �ows. The study �nds

that traders with o¢ cial-market member status submit twice as many hidden orders as

traders without this status. In addition, they submit twice as many hidden orders on their

own accounts (22%) compared to on behalf of their clients (9%).

Impact on order �ow and pricesRecent research also investigates the e¤ects of hidden depth on order submission and

price behavior. Winne, D�Hondt, and Heude (2001) conducted a study on the Euronext

and showed that market orders are more aggressive (i.e. larger) when hidden depth is

large on the opposite side of the book. According to the authors, traders are more likely

to submit larger market orders when they have recently discovered hidden depth, because

they then believe that even more depth is hidden, and thus that their market orders are

likely to be executed at favorable prices (that is, due to the expected hidden depth, even

if the displayed quantity is small, the market order is expected to execute at a price near

the best bid or ask rather than walking down the book to increasingly worse prices). The

increase in market-order submission when hidden depth was just discovered suggests that

the traders submitting the market orders do not believe hidden depth is informative; or else

they would not be encouraged to trade against it. This result supports the adverse-selection

37

explanation for the use of hidden orders.

Another study, Hasbrouck and Saar (2001), conducted on the Island ECN, found that a

signi�cant fraction (27.7%) of displayed orders submitted to the ECN were cancelled almost

immediately, within 2 seconds of their submission. (The fraction of immediately cancelled

shares, rather than orders, is 32.5%) These short-lived orders were called ��eeting orders�

by the authors, who also hypothesize that their prevalence can be explained by traders

��shing for hidden depth.�This again suggests that the traders �shing for depth are not

concerned to trade against it, and thus seem not to believe that it is informative, much the

same as suggested by the EuroNext study cited above.

In contrast with these �ndings, though, the NASDAQ study, Tuttle (2002), mentioned

earlier suggests that hidden depth may, on average, be informative. The study regresses

measures of the price impact and information costs of individual trades against various

variables, including displayed depth and non-displayed depth. To measure depth, the author

reconstructs a limit order-book with hidden and displayed liquidity from all the National

Market quotes submitted into the inter-market SuperSOES system (introduced in 2000),

and breaks quotes down according to market-maker types.

The results of the analysis lead the author to conclude that �[...]hidden depth on the

relevant side of the market is a strong indicator that the market will move against the trade�

and that �hidden size is highly predictive of market price-movements� it is, in e¤ect, the

trading of �smart money�, selling before the market moves down, buying before the market

moves up. It is possible that this e¤ect is due to hidden size being used as a vehicle for

certain market participants (investment banks and wirehouses) to work large orders as a

liquidity provider rather than liquidity demander.�61

Finally, in a paper titled �Can order exposure be mandated?,� Anand and Weaver

(2003) investigated the e¤ects on quoted depth and volume, following the Toronto Stock

Exchange�s decision to abolish iceberg orders in 1996, and then to reintroduce them in 2002.

The results of the analysis, which found no signi�cant changes in volume or quoted depth,

seem to suggest that, once hiding quantities with iceberg orders ceased to be an option in

1996, traders in part switched to submitting more market orders instead (rather leaving the

exchange, which would have lead to a reduction in total volume). When iceberg orders were

reintroduced, the market�s total depth (displayed plus hidden depth) increased considerably,

again with no signi�cant e¤ect on volume or quoted depth. Given the result from the data,

the answer to the title of the paper is negative. The authors conclude saying: �This paper

therefore suggest that order exposure cannot be mandated and that attempts to mandate

exposure will cause traders to seek other methods to supply liquidity thus negating the

desired e¤ect of increasing quoted depth. Thus, markets and regulators seeking to increase

liquidity would be best served not to attempting to mandate order exposure.�62

61See page 34 in Tuttle (2002).62See page 425 of Anand and Weaver (2003).

38

2.4.4 Discretionary Orders

Discretionary orders have been introduced into a number of exchanges and ECN�s over the

past two or three years. A discretionary order consists of a quantity and two prices, one of

which is displayed, and one of which is hidden. For a discretionary buy order, the hidden

price is above the visible price (and conversely for a sell orders).

An incoming limit order priced within the discretionary range of an order (that is,

between the displayed and the discretionary price) will be executed against the discretionary

order at the limit order price. By hiding the more aggressive discretionary price at which,

say, a trader is willing to buy, the trader can hope to mitigate the information leakage that

would ensue from displaying the price, and can also hope to get execution at a price below

the discretionary price (if the price was displayed, other traders would raise the price on

the limit orders they submit).

From a mechanism-design perspective, discretionary orders may allow, say, a buyer to

separate more- and less-patient sellers from one another. Relatively impatient sellers may

submit limit orders closer to the displayed price, hoping for execution. Relatively patient

sellers may try to �sh for higher hidden prices before submitting more aggressively priced

(lower) sell orders, thus risking the loss of a sale to another trader who puts in a low-priced

sell order ahead of them.

Some exchanges allow for variations of discretionary orders. The NYSE, for example,

allows for parts of discretionary orders to be routed to other exchanges, and traders can

specify minima for the quantities to be routed out. Orders with that capability are called

�discretion limit orders.�63

There has been some debate around discretionary orders. At �rst, the NYSE only

allowed brokers and specialists to submit discretionary orders, and only later was this order

type made available to all traders. Some traders have pointed out that the bene�ts of

discretionary orders are reduced by the existence of sweep orders. Sweep orders �sweep�

entire price ranges; for example, a sell sweep order would consist of a quantity, a high

starting price, and a lower ending price. The sweep order would automatically generate a

sequence of limit orders beginning with a limit order at the high starting price. This order

would try to execute against any visible or hidden quantity at the starting price. If no

execution, or only partial execution, occurred, an order at the next-lowest tick size would

be sent out for execution, and so forth, until either all quantity was sold, or the end price of

the sweep order was reached. With the widespread use of sweep orders, which o¤er an easy

way to �sh for hidden discretionary prices, traders submitting discretionary orders may be

left with less price improvement (as their orders are mostly executed at the discretionary

price).

Also worth noting is that discretionary orders have been forbidden for Multilateral Trad-

ing Facilities (MTF) in Europe (MTF�s are essentially like ATS). Speci�cally, the European

63See NYSE Arca (2010).

39

Commission has found discretionary orders not to be in compliance with the provisions of

the MiFID (Markets in Financial Instruments Directive), due to lack of transparency.64 As

a consequence, platforms operating as MTFs in Europe, such as BATS, were forced to stop

o¤ering discretionary orders within a week of the Commissions decision (in May of 2009).65

At the same time, the London Stock Exchange has recently introduced orders with even less

pre-trade transparency, hiding both quantity and price. These completely hidden orders

are discussed in the next section.

2.4.5 Volume Orders, Discretionary Reserve Orders, Hidden Limit Orders,Dark Reserve Orders

The four order types in this subsection are all less than 2 years old, and are all very similar

to each other in that they are essentially completely hidden orders. The exact speci�cations

vary slightly, depending on the exchange on which they are o¤ered. This section o¤ers a

brief overview of each order type and a discussion of their common features, as well as the

advantages they o¤er from a mechanism design perspective.

Note that the detailed execution rules for each order type also vary slightly, and will be

omitted in this section. (The detailed rules for volume orders are provided in Sections 4 and

5. Volume orders are the only one of the four types in this group not currently traded on

exchanges. They are, however, among the older types (originally invented and patented by

Gomber, Budimir, and Schweickert (2006)), and also among the most comprehensive (with

respect to the number of parameters that a trader can specify).

Dark reserve orders were introduced to the NYSE about 2 years ago, and are entirely

hidden orders. That is, traders submit a quantity and a price, and no information is

published in the book. In addition, the NYSE allows for bundling by o¤ering minimum

execution-sizes on dark reserve orders: a trader could specify that he wants to buy, say,

10,000 shares at price P, and that at least 7,000 of those have to execute at once. The

minimum size condition is in place to prevent other traders from �shing for the hidden

order.

Hidden limit orders can, as of January of 2010, be submitted to the London Stock

Exchange (LSE). Like dark reserve orders, hidden limit orders are entirely hidden and

allow traders to specify minimum execution sizes. The LSE explicitly denotes minimum

execution size (MES) as an �anti-gaming feature� designed to �protect participants from

small-volume orders (�pinging�) which aim to discover the presence of hidden orders�.66 The

LSE introduced hidden limit orders (with MES) along with a number of other order types

(mostly orders allowing for convenience, like pegging a price of an order, whether it is hidden

or not, to a speci�c distance from the midpoint or best bid). The increase in complexity

64See CESR (2009).65See BATS Europe (2009).66See page 8 of LSE (2009).

40

of order-matching (due largely to MES) forced the LSE to entirely redesign its matching

algorithm and order entry system: a costly endeavor. The LSE�s decision to o¤er these new

order types, thus illustrates just how seriously exchanges are taking traders�demands for

innovative, opaque order types.

Discretionary reserve orders were recently introduced on the Toronto Stock Exchange

(TSX). These orders specify a displayed portion, as well as a hidden portion that consists

of a quantity and a discretionary price range (that is, a visible price and a discretionary

price). Much like the SEC, Canadian regulators are concerned with these opaque orders�

possible e¤ects on price discovery. In October 2009, the Securities Administrators and

the Investment Industry Regulatory Organization of Canada published a joint consultation

paper, CSA and IIROC (2009), asking for comments on market design and raising questions

very similar to those raised by the SEC in its recent concept release. Moreover, as mentioned

in the section on discretionary orders, regulators in Europe are discussing similar questions,

even though the market structure and regulatory framework is less well-de�ned there than

in the US, with respect to exchanges and what, in the US, are called ATSs.

Unlike the other three types just mentioned, volume orders, as de�ned by their inventors,

have never been introduced in exchanges. The closest thing to a volume order on an

exchange is the discretionary reserve order on the TSX. In practice, however, volume orders

can be arbitrarily close to dark reserve orders and hidden limit orders, as will be evident

from their de�nition. Volume orders consist of two parts. First, there is a displayed portion,

which can be arbitrarily small (so that the order would look essentially like a completely

hidden order once placed in the book). Second, there is a hidden portion of the order,

which consists of a quantity, together with a price and an optional minimum execution size.

Volume orders derive their name from the minimum execution size on the hidden quantity,

which is referred to as a �volume condition.�The price on the hidden quantity can be above

or below the visible price, whether the order is to buy or sell.

The di¤erence between volume orders and hidden orders should be small in practice,

since the volume order requires an arbitrarily small hidden part. Some practitioners, though,

claim that requiring a minimum of as low as 100 shares to be displayed for any order

constitutes an imposition for most traders.67 The validity of this statement is hard to

verify. If volume orders were used in exchanges, traders would have an incentive to set the

displayed part of the order to be equal to the typical size of a small order (rather than

submitting a visible part of, say, 1 share, which may look �suspicious�). Thus, as long

as the obligation to display some quantity does not discourage the submission of volume

orders altogether, they may retain an advantage over completely hidden orders, because the

displayed part of the order would presumably contribute to more to price discovery, while

the hidden part would still o¤er traders the advantages of completely hidden orders.

Finally, it remains to mention the advantages of volume orders in discouraging gaming

67See Rosenblatt and Gawronski (2007).

41

strategies. As noted above, minimum execution sizes (or volume conditions) on the hidden

part protect volume orders from traders who try to inexpensively ��sh�for hidden volume.

In addition, execution rules for volume orders specify �passive price setting�: that, say, a buy

limit order priced below the hidden price of a volume order is executed at the hidden price of

the volume (rather than its own price). This contrasts with the rules for discretionary orders,

in which a buy limit order submitted to the book, and falling within the discretionary price

range of a discretionary order resting in the book, will be executed at its own price (not the

discretionary price of the discretionary order). The advantage of this passive price setting

is that it naturally removes the incentive for other traders to submit sequences of orders

trying to edge against execution (by submitting sequences of sell orders with decreasing

prices, or sequences of buy orders with increasing prices).

2.4.6 Theoretical Literature

The theory part of this thesis, presented in Sections 4 and 5, analyzes optimal order sub-

mission in a one-shot game. Speci�cally, a buyer and seller, each of whom can have either

of two types, can trade up to two units through a limit order book. I assume that the

buyer submits the initial order to the empty book and the seller responds to it. In Section

4, valuations for the units are private, and in Section 5 they are interdependent. In both

cases, players know only their own valuations. Each section considers four games: in the

�rst, the buyer can submit only a limit order to the book, and the seller responds with

either a limit or a market order. In the following three games, the buyer and seller can also

submit iceberg, discretionary, or volume orders. The resulting equilibria are analyzed from

a mechanism design perspective, with respect to what strategies each order type o¤ers to

buyers. The e¤ect of the introduction of di¤erent order types is analyzed with respect to

volume and transparency. Each section ends by analyzing the buyer-optimal mechanism

and its relation to equilibria of games involving orders.

A crucial topic of this analysis is how to set prices for each mechanism or order type

when there is asymmetric information. There is a large amount of literature in this area.

One seminal paper by Myerson (1981) considers the case in which one seller sells one unit of

a good to one buyer, whose type (that is, valuation) is privately known only to the buyer,

and the seller�s and buyer�s valuations are private both private. Myerson and Satterthwaite

(1983) consider the case in which there is again one seller, one buyer, and one unit of a

good, but there is also uncertainty about both the buyer�s and the seller�s types (that is,

the types are privately known). Moreover, all valuations are again private. Both papers

demonstrate the impossibility of implementing ex-post e¢ cient outcomes in these games.

The two mechanism-design papers most closely related to the model presented in the

thesis are by Maskin and Tirole. The authors analyze a principal-agent game in which the

principal (the buyer) proposes a contract to the agent (the seller). Exchanged quantities are

continuous. The �rst paper, Maskin and Tirole (1990), considers the case of private values,

42

in which there is uncertainty about both the buyer�s and the seller�s types. For the case of

quasilinear utilities, the case of two-sided uncertainty simpli�es, in that the principal does

not lose from revealing his type, and the optimal contract is the same as when the buyer�s

type is known. In the model considered in this thesis, the analogous result holds for private

values. The second paper, Maskin and Tirole (1992), establishes equilibrium concepts for

the principal-agent game when values are interdependent and there is uncertainty about the

buyer only (the principal). For the case of quasilinear utilities, the authors also describe the

case of two-sided uncertainly, that is, when the seller�s type is also unknown. The model in

the present thesis solves for the optimal contract with interdependent values and two-sided

uncertainty, and compares the optimal allocation with equilibrium allocations of speci�c

sequential games in which players submit orders. Although the model setup in the thesis is

slightly di¤erent than that of Maskin and Tirole, it retains most of their notation.

The present thesis also analyzes the buyer�s choice of optimal order-type. In partic-

ular, buyers have the option to submit limit orders or, depending on the game, iceberg,

discretionary or volume orders. There are a number of papers analyzing a trader�s choice

between submitting market orders or limit orders. Note that this setup di¤ers from the

present model, which implicitly assumes that the buyer has already decided to submit an

order to the book, rather than submitting a market order to trade against any existing

orders in the book. Moreover, most of the previous literature assumes that there are both

informed and uninformed traders in the market, whereas in the present model all players

are informed in the sense of knowing their own valuation on each unit.

The �rst seminal paper in the context analyzes the buyer�s choice of optimal order-type

is by Glosten and Milgrom (1985). Here, it is the informed trader that submits market

orders, wishing to trade immediately on the information that he possesses about the value

of the asset. Fouccault (1999) considers a dynamic model, and concludes that traders

with information may submit limit orders instead of market orders if the value of their

information is su¢ ciently long-lived. Kaniel and Liu (2006) �nd, with similar arguments,

that limit orders may end up conveying more information than market orders. (In addition,

as mentioned in Section 2.4.3, there is empirical evidence that informed traders also submit

orders to the book, rather than only taking liquidity from the book).

The theoretical literature on iceberg orders suggests two di¤erent rationales for the use

of iceberg orders. Harris (1998) argues that uninformed traders use iceberg orders if they

need to trade large quantities. In doing so, they avoid being front-run by other �parasitic�

traders. Note that the uninformed traders lack information about the true value of the

asset, but that the information about their demand is still valuable because it is information

about future order �ow. Aitken, Berkman, and Mak (2001) argue that uninformed traders

use iceberg orders to reduce the adverse selection cost of their orders being picked o¤ by

informed traders (who consider the limit orders as standing free options). By hiding units,

informed traders will trade less aggressively against the uninformed traders�orders. The

43

other argument for the utility of iceberg orders is that informed traders use them to trade

on their information, while reducing the information leakage that would occur if the order

was displayed.

Moinas (2006) explicitly models the submission of iceberg orders in a multi-period model,

assuming that an informed trader may be present and choose to submit the iceberg order at

the beginning of the game (otherwise, an uninformed trader submits the �rst order). The

author compares the informed trader�s payo¤s in the model where iceberg orders can be

submitted to that trader�s payo¤s when only limit orders can be submitted. In the former

case, the informed trader has a strictly higher payo¤ if he chooses to submit an iceberg

order. In order not to discourage order submission by the uninformed sellers, who must

trade against what could be an informed iceberg order, the informed trader has to trade

less often (and thus hide) units less often, leading to an overall reduction in the informed

trader�s payo¤s, as compared to the case with only limit orders.

I am not aware of theoretical literature on discretionary orders or volume orders, nor

on completely hidden orders/ dark reserve orders, which are the closest to volume orders in

practice. This lack is due, presumably, to the novelty of these order types, all which have

been implemented only in recent years and months, or, in the case of volume orders, have

never yet been implemented.

44

3 Overview of the Theory Sections

This section provides an overview of the theoretical analysis in this thesis. Sections 4 and

5 present a number of models for a market with a central order book. The questions

motivating all models is how di¤erent order types a¤ect trade. Taking a mechanism design

perspective, three order types �iceberg orders, discretionary orders, and volume orders�are

analyzed, as well as their e¤ect on expected trade volume and transparency.

Section 3.1 provides a very brief description of the three order types and some context;

Section 3.2 presents an overview of the model setup and the main results.

3.1 Background for the Di¤erent Order Types

Limit orders specify a price and a quantity to be bought at that price, and they aredisplayed in full in the order book. As is standard in order books, there is price priority

(higher-priced buy orders and lower-priced sell orders are listed and executed �rst). Con-

sequently, traders essentially submit demand schedules with buy limit orders (and supply

schedules with sell limit orders).

Market orders specify a quantity to be bought from the book. When a trader submitsa market buy order of a given quantity to the book, the order is executed against the lowest-

priced eligible sell orders/o¤ers in the book (where eligibility depends on the execution rule

for the sell order type). Analogously, market sell orders execute against the highest eligible

bids �rst.

Iceberg orders consist of a displayed price and quantity, and an additional quantityat the displayed price, which remains hidden (that is, not displayed in the book). The term

�iceberg order�derives from the fact that only part of the order, the tip (or peak), is visible.

Iceberg orders are also sometimes referred to as hidden orders or reserve orders.

Q

P

Hp

Lp

Twounit limit orderpriced at ),( LH pp .

Q

Hp

Lp

P

Iceberg order :)),(( LHH pppvisible part ),( LH pp ,one unit hidden at Hp .

Figure 1: Example Limit and Iceberg Order.

Discretionary orders consist of a displayed price and quantity, and hidden discre-tionary prices on part or all of the visible quantity. For buy orders, the hidden discretionary

prices are higher than the displayed ones; for sell orders, they are lower.

45

Volume orders consist of a displayed price and quantity, and a hidden quantity ata hidden price. The hidden price on the hidden quantity may be above or below the

displayed price. Moreover, the order may specify a minimum execution size on the hidden

quantity. The minimum execution size is called a �volume restriction� from which the

order also derives its name. Speci�cally, the submitter can specify that the hidden part

may execute against incoming orders only if at least a minimum fraction of the hidden

quantity (for example, 70%) executes. Note that while volume orders are not currently

used on exchanges, similar types of orders, namely entirely hidden orders, do exist.

P

Q

Lp

Hp

Discretionary order )),(( LHL ppp :Visible part ),( LL pp , and hiddenprice of Hp the first unit. P

Q

Lp

Hp

Volume order ]),[,( HHL ppp :visible unit at Lp , and a bundle oftwo units at Hp .

Figure 2: Example Discretionary and Volume Order.

The execution rules for the di¤erent order types are discussed in detail in the cor-responding Sections 4 and 5. These rules specify how precisely each order type is crossed

against incoming orders.

Iceberg orders, discretionary orders and volume orders are variations of the standard

limit order, aimed at mitigating the problem of information leakage associated with limit

orders being visible to everyone in the market. The leaked information can be either infor-

mation about future order �ow, or information about the value of the traded asset, implying

that it is valuable information in a setting of both interdependent values (or common values)

or a pure private values setting.

Consider an example in which the book suggests that there is a substantial buy interest

from some trader A (who is anonymous) at some price P. First, seeing this interest, other

traders might conclude that the asset is undervalued at P, updating their belief about

the value upwards and reducing or withholding their supply at P (this would be true in a

setting with interdependent or common values). Second, some traders might engage in front

running: submitting a buy order priced slightly above P, hoping to buy up all the liquidity

around P, and forcing trader A to buy back from them at a higher price shortly afterwards.

This strategy would potentially be pro�table in both private values and interdependent

values settings.

Information leakage especially a¤ects agents trading large quantities, inherently, have

to weigh immediacy against price: trading the entire order at once or in large chunks would

46

be faster, but allows for less price discrimination. With information leakage, trading faster

by displaying larger quantities becomes even more costly. In turn, hiding part of the order

information, as would be possible with the three order types discussed here, may allow for

these traders to trade larger sizes (and/or faster) with comparatively less price impact.

For completeness, it remains to mention that, the potential bene�ts for traders notwith-

standing, regulators are concerned that the reduction in transparency may come at the

cost of reduced price discovery. Another concern is that too little transparency may reduce

investors�con�dence in prices, discouraging order submission and thus trading volume68.

3.2 Overview of the Model Setup

A simple game theory model of an exchange is considered. Speci�cally, a buyer and a seller,

both of which can be either of two (privately known) types, can trade up to two units of

an asset through an order book, which is empty at the beginning of the game. For ease of

exposition, I assume that the trader submitting the initial order to the empty order book is

a buyer (as is assumed without loss of generality, throughout the rest of the thesis). The

seller then responds to the displayed buy order, and trade (if any) happens as the buy and

sell order are crossed.

It is important to note that this model analyzes a buyer�s problem of submitting an

optimal order (bid) to a limit order book; in practice, a buyer would �rst have to choose

between submitting an order to be posted in the book, or responding to any sell orders

already contained in the book by submitting a marketable buy order.

Next, the respective probabilities for the types of the buyer and seller are common

knowledge, but the types themselves are only private. Each buyer type and each seller type

assigns a value to the �rst and second unit of the asset. For the buyer types, the value

on the �rst unit is higher than that on the second, while sellers would be willing to sell

the �rst unit for less than the second unit. The types for the buyer and seller are denoted

�large�and �small�. By assumption, the values for the large buyer type are above that of

the small buyer type on each unit, and the values of the large seller type are below that of

the small seller type on each unit. Consequently, large types �as their name suggests�have

a relatively stronger desire to enter a given trade and are likely to trade relatively larger

quantities in equilibrium. (Note that there are eight parameters for the values, since there

is a buyer and seller, each having two types, and each type has valuations on two units.)

Two basic settings are considered. First, in Section 4, examines a setting in which

buyer and seller valuations of the asset are private (and equal to the privately known values

described above). Second, in Section 5, an interdependent values setting is considered, in

which each buyer type�s valuation for a given unit at any point in the game depends on

his privately known value as well as on his estimate of the expected seller value for that

68This section only provides a sketch of the nature of the questions; an extensive analysis of the currentmarket structure and regulatory concerns is found in Section 2.

47

unit. (Similarly for the valuations of the seller types, which now depend on the seller type�s

privately known values, as well as on the expected buyer value.) Moreover, the degree to

which the expected seller value enters each buyer type�s valuation is parametrized.

As will be seen later, for both private and interdependent values, the buyer and seller

valuations will induce supermodularity in the payo¤s of the game (supermodularity in val-

uations and execution probabilities of orders), inducing a �monotonicity�structure in equi-

libria (both points are discussed in detail where relevant).

For each setting, the model setup includes four games, which di¤er in the set of ad-

missible orders. In the basic game, traders can submit only limit orders to the book, and

either limit orders or market orders to respond to the book. In the other three games, the

set of admissible orders additionally includes either iceberg orders, discretionary orders, or

volume orders.

Players have initial beliefs, and they form Bayesian updates over the course of the game

on the equilibrium path. Two additional (non-standard) assumptions about beliefs have to

be made, because some of the orders contain hidden parts, implying that certain deviations

are invisible. The �rst assumption is that nothing is hidden, given a visible buy order

that constitutes an o¤-the-equilibrium-path deviation (in games where the set of admissible

orders includes orders that have invisible parts, such as discretionary orders). Second,

only equilibria in which all orders have positive execution probabilities are considered (thus

excluding non-serious o¤ers).

Expected payo¤s in this model are the determined in a standard way, calculating the

expectation is over the value of the units traded and the prices paid/received.

With respect to transparency, pre-trade and post-trade transparency are distinguished.

Pre-trade transparency refers to what can be inferred about the buyer and seller types by

looking at the limit order book once the initial buy order has been posted (for a given set

of parameters, types can be known or unknown). Post-trade transparency refers to what

can be inferred about the buyer and seller types once the responding sell order has been

submitted and crossed with the buy order.

For each setting and each of three order types, characteristics of the equilibria of the

basic game (with limit orders only) are compared to those of the games that also include one

of the three other order types. All comparisons of equilibria are done for given parameter

combinations, which allows one to analyze, for example, what e¤ect the introduction of

discretionary orders may have on volume or transparency. All equilibria calculated are pure

strategy equilibria. Many of the existence proofs involve �nding numerical examples, which

are included in the appendix, along with the theoretical proofs.

The analysis also includes a mechanism design section that analyzes the mechanism

design analogue of the optimal buy order for a buyer, namely the buyer-optimal mechanism

in a principal-agent game in which the buyer is the principal and the seller is the agent. Only

non-randomizing contracts are considered (much like only pure strategies are considered in

48

the games involving order types). It is investigated whether equilibria of games with one

of the order types considered can replicate the allocation of the buyer-optimal mechanism

(by implementing the same trades and prices/transfers that an optimal mechanism would

prescribe), or why it may not be possible to do so.

3.3 Aside: Perfect Information

This section provides an example for the simpler case of perfect information, when buyer

and seller types are known (and values are private or interdependent). The thesis considers

two-sided uncertainty throughout; the purpose of this section is to provide some intuition

for the di¤erent order types.

Let vBik; vSjk, be the values for buyers and sellers, where i; j 2 fL; Sg refers to the type,which can be large or small, and k 2 f1; 2g refers to the �rst and second unit respectively.

Example 1 (Perfect Information) Assume that the buyer type is large, with valuations

of (vBL1; vBL2) = (32; 31) on the �rst and second unit, and that the seller type is small with

valuations of (vSS1; vSS2) = (25; 30).

The e¢ cient outcome would be to trade both units, as the buyer�s valuation on both unitsis above that of the seller.

In the game with only limit orders, the optimal order for the buyer is to submit a one-unit limit order at 25, in response to which the seller would submit a one-unit market order,

so that the unit would be traded, and the buyer would be left with a payo¤ of 32 � 25 =7. (The buyer�s payo¤ from the optimal two-unit buy order, (p1; p2) = (30; 30), would be

(32 + 31) � (30 + 30) = 3). Consequently, when limit orders are used, there is less tradingthan would be e¢ cient.

Next, consider the game in which the buyer may also submit iceberg orders. Iceberg ordersallow the buyer to hide a unit, but with perfect information there is no reason to do so, since

hiding a unit would, if anything, reduce the execution probability of the unit.

Similarly, there is no reason to hide a price with discretionary orders. If anything, hidinga price would reduce the execution probability of the unit, compared to displaying it.

There are, on the other hand, equilibria involving volume orders. For example, the buyermay submit a volume with a visible one-unit order priced at 8 (or any price below 25), and

a hidden bundle of two units priced at (27:5; 27:5). If the beliefs of the seller are that the

bundle is hidden at 27:5 in equilibrium, then his best response is to sell the two units by,

say, submitting a two-unit limit order priced at 27:5. As a consequence, the payo¤ for the

buyer would be 63�55 = 8, and the e¢ cient trade would be implemented. Note though, thatthere are multiple equilibria in this game, and that all equilibria with volume orders depend

on the seller believing that a unit is indeed hidden.

Finally, consider the (buyer-) optimal mechanism that the buyer could propose to the

seller. The optimal contract would be to o¤er to buy a bundle of two units at a price of 55

49

(again, 27:5 for each unit). The seller would accept the contract (which would leave him

with no rent), and the buyer�s payo¤ would be 63�55 = 8. This payo¤ is equal to the payo¤in the equilibrium with volume orders above, and it does also implement the e¢ cient trade.

If, instead of values being private, they were interdependent, for example with � = 0:2, the

results in this example would not qualitatively change. Speci�cally, the buyer�s valuation

on the units would be (32 + 0:2 � 25; 31 + 0:2 � 20) = (37; 37): calculated as the sum of

the buyer�s private valuations on the units, (vBL1; vBL2) = (32; 31), and the product of the

interdependence parameter � = 0:2 and the (buyer�s expectation of ) the seller�s private

valuations, (vSS1; vSS2) = (25; 30). Similarly, the seller�s valuation of the units would be

(30; 36:2). Thus, it would be e¢ cient to trade both units, the optimal limit order would

be a one-unit order (now priced at 30), neither iceberg nor discretionary orders would be

bene�cial, the optimal volume order would have a bundle hidden (at 33:1), and the optimal

mechanism would also propose to buy a bundle of two units (at 33:1).

3.4 Overview of the Results for Private Values

The following is an overview of the results for the private values analysis.

3.4.1 Limit Orders and Market Orders Only (Basic Game)

Buyers submitting an order have no incentive to hide their type, because sellers sell a given

unit if and only if it is priced above their reservation value for that unit. As a consequence,

buyers submit their (individually) optimal limit order, trading o¤ price with execution

probability. (Lower-priced buy orders might not execute when the seller is of the small type

and therefore has a higher reservation value.)

Trading volume may not be increasing in the probability that the seller type is large,

because buyers may be tempted to gamble and submit lower-priced orders if they think

that the order is likely to execute. Equilibria do display a kind of monotonicity, though,

in that large types trade larger quantities in equilibrium, and trade volume increases with

higher buyer valuations and lower seller valuations.

3.4.2 Game with Iceberg Orders

Iceberg orders are weakly dominated and thus not used in the private value case. The

intuition is that hidden units have a weakly lower execution probability than visible units

at any given price.

3.4.3 Game with Discretionary Orders

First, note that limit order equilibria remain equilibria once discretionary orders are intro-

duced. This result is a consequence of the assumptions about o¤-equilibrium-path beliefs.

50

Next, there exist equilibria in which discretionary orders are used and buyers achieve

strictly higher payo¤s than in the corresponding limit order equilibria. In all discretionary

order equilibria, buyers pool on the visible part of the order, and large buyers addition-

ally submit a discretionary price. Visible and discretionary prices are set such that small

sellers (who have high valuations) �gamble�by o¤ering to sell at the higher, discretionary

prices and large sellers (who have low valuations) sell at the lower, visible prices. That is,

discretionary orders allow buyers to do screening of the sellers with respect to their (ab-solute) valuations on a given unit: in equilibrium, seller types can choose to sell a given unit

at a lower, visible price with certainty (receiving a lower payo¤ with certainty), or at the

higher, hidden price with a lower execution probability (thus entering a lottery consisting

of a higher payo¤ if the unit executes at the hidden price, and a payo¤ of zero if it does

not).

As mentioned in more detail in Section 2.4.4, exchanges have advertised discretionary

orders as a way for traders to increase execution probability of an order, while reducing

information leakage and obtaining price improvement with respect to the hidden price (that

is, whenever the order executes against a limit order with a price within the discretionary

range between the visible and hidden price). This advantage of discretionary orders likely

exists due to the fast-pace trading environment, which implies that a traders who submit

sequences of, say, sell orders trying to �sh for hidden discretionary buy prices, risks losing

execution against another market participants that submits a lower-priced order within the

discretionary range of a discretionary order before the trader�s sequence of decreasing price

has reached the hidden discretionary price.

Let � be the probability that the seller assigns to the buyer type being large, and � the

probability the buyer assigns to the seller type being large. That is, �; � de�ne the seller�s

and buyer�s initial beliefs about the other type.

Example 2 (Discretionary order equilibrium, Screening) Letf�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 6; 5; 9; 8; 4:5; 7; 1; 3gA discretionary order equilibrium exists in which both buyer types submit a visible limit order

at (pL; pL) = (3; 3) and the large buyer type additionally hides two prices (pH ; pH) = (7; 7).

The large seller type submits a two-unit limit order selling two units with certainty at pL,

and the small seller type submits a two-unit limit order priced at (pH ; pH), which only

executes when the buyer type is large. Payo¤s are 3:5 and 8:6 for the large and small buyer

type respectively.

Note that in the corresponding limit order equilibrium, the small buyer would have submitted

the same limit order (and received the same payo¤s). The large buyer�s optimal strategy in

the limit order equilibrium would have been to submit a two-unit order with an execution

probability of 1+ � (meaning that the large buyer only trades one unit with the small seller

type). Thus, volume in this discretionary order equilibrium is increased compared to the limit

51

order equilibrium. Moreover, the payo¤s for the large buyer are higher in the discretionary

order equilibrium and the e¢ cient trade for the large buyer is implemented (since he trades

two units with each seller type).

Below is an illustration of the discretionary order equilibrium.

Equilibrium orders:Small buyer submits a twounit order at Lp .Large buyer submits a discretionary order ))(),(( HLHL ppppthat is, two units visible at Lp and a hidden price Hp on each unit.Small seller submits a twounit limit order at Hp ,which executes if the buyer is large.Large seller submits a two unit limit order at Lp ,which always executes at ),( LL pp .

P

Hp

Lp

Q

Figure 3: Example Discretionary Order Equilibrium.

Generally, the introduction of discretionary orders can lead to both an increase and a

decrease in trading volume, compared to the corresponding limit order equilibrium (which

one of the two possibilities holds depends on the speci�c parameter values). A decrease in

volume happens if in the absence of discretionary orders, large buyers are pricing their orders

aggressively by bidding high in order to guarantee execution. In practice, this situation

seems to not be very common, as traders tend to engage in order splitting (submitting

many small, not aggressively priced orders).

Next, transparency and volume do not always move together: it is possible that pre-

trade and post-trade transparency is lower for the discretionary order equilibrium, and yet

trading volume is higher than for the corresponding limit order equilibrium.

Moreover, discretionary order equilibria may lead to an increase in trading volume while

leaving the type of the buyer (who submitted the initial order) unknown at the end of the

game. In practice, this result would make discretionary orders even more attractive to

buyers who want to trade more quantity later, and are thus concerned with information

leakage over time, as executed trades are successively posted to the book.

Example 2, continued (Discretionary order equilibrium, Transparency) In the

discretionary order equilibrium considered so far, there is no transparency pre-trade, in that

buyer types pool onto the same visible order. In contrast, pre-trade transparency is higher

with limit orders only, as the buyer types separate: the small buyer submits an order with

expected trade volume of 2� (thus buying two units from the large seller only), and the

large buyer submits an order with expected trade volume of 1+� (buying one and two units

from the small and large seller type, respectively). The matrices below illustrate post-trade

transparency for the equilibrium with limit orders and the discretionary equilibrium. Rows

and columns indicate the buyer and seller types (L=large, S=small), question marks mean

52

that the type is unknown to the market, and numbers in brackets refer to units traded.

post-trade transparency

with limit orders

seller S seller L

buyer SS; S

(0)

S;L

(2)

buyer LL; S

(1)

L;L

(2)

post-trade transparency

discretionary equilibrium

seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 1: Post-trade transparency with LO and discretionary equilibrium

3.4.4 Game with Volume Orders

First, limit order equilibria remain equilibria when volume orders are introduced (much like

when discretionary orders are introduced).

There exist a number of equilibrium classes in which volume orders are used. In those

equilibria, volume orders allow buyers to submit buy orders that resemble a supply sched-ule and they also allow for bundling (two features that may be optimal from a mechanismdesign perspective). In addition, as with discretionary orders, volume orders allow the buyer

types to do screening of the seller types with respect to the seller types�valuations.There are two scenarios for equilibria. The �rst scenario comprises equilibrium classes

in which buyers either separate on the visible part of the order that they submit, or pool

onto both the visible and the hidden part of the order. Thus, upon seeing the (visible) part

of the order posted to the book, sellers correctly identify the buyer�s type and anticipate

any hidden volume. These equilibria are less interesting, in that the buyers would be

indi¤erent between hiding the units and either displaying them or submitting a buy order

that is a supply schedule (if either of these two options were possible in a limit order book).

The important thing to note is simply that volume orders provide a way to submit supply

schedules in these equilibria.

In the second scenario, buyers pool onto orders whose visible parts are identical and

either a) only the large buyer submits a hidden price or b) each buyer submits a di¤erent

hidden price. Thus, sellers cannot perfectly predict what (if any) volume may be hidden

at a given price. Instead, sellers form beliefs about the probability that a unit is hidden at

any given price in equilibrium.

Equilibria in the second scenario can be categorized into two main groups. In the �rst

group, both sellers pool onto a visible part of the order and the large buyer hides one or

two units, but would be indi¤erent to displaying them. These equilibria can have demand

features (if the hidden price is below the visible price) or supply features (if the hidden price

or prices are above the visible price), and they may also have involve bundling.

53

Example 3 (Volume order equilibrium, Supply and Bundling) Letf�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:5; 0:5; 7; 3; 9; 8; 5; 7; 4; 6gA volume order equilibrium exists in which both buyers types submit a visible limit order

at pL = 5 and the large buyer type additionally hides a bundle of two units at pH = 5:5

(that is, the large buyer submits a volume order of (pL; [pH ; pH ])). The large seller types

submits a volume order (pL; [pH ; pH ]) selling one unit at 5 whenever the buyer type is small

and a bundle of two units at 5:5 whenever the buyer type is large. The small seller type

submit a on-unit limit order at pL = 5 which executes (with certainty) at this price.

In contrast, in a limit order equilibrium, the small buyer would have submitted the same

order, while large buyer would have submitted a two-unit order priced at (6; 6), with the

same execution probability of 1 + � as in the volume order equilibrium. The payo¤ of the

small buyer is thus the same in both equilibria, but that of the large buyer increases from 4

to 0:5 � (9� 5) + 0:5 � (17� 11) = 5 in the volume order equilibrium.

Q

Equilibrium orders:Small buyer submits a visible order at Lp .Large buyer submits a volume order

]),[,( HHL pppvol , thus hiding a bundle at Hp .Small seller submits a limit order at Lp ,which always executes at Lp .Large seller submits a volume order ]),[,( HHL pppvol ,which executes at Lp if the buyer is small and at

),( HH pp if the buyer is large.

Hp

Lp

P

Figure 4: Example Volume Order Equilibrium.

In practice, buyers cannot submit buy limit orders that resemble supply schedules to any

limit order book. At the same time, buyers want to and do engage in price discrimination,

which is what a supply schedule would accomplish. Buyers currently try to replicate a

supply schedule dynamically by submitting a sequence of buy orders with increasing prices

at the inside of the book (that is, at the best price), buying up liquidity at each price point

in the upward sequence. By hiding a bundle of units at a price higher than the visible/best

prices and thus generating a supply schedule of buy orders, volume orders would thus

provide buyers with a �static�alternative to the dynamic strategy currently used to price

discriminate. In addition, the fact that the bundle would be hidden, has advantages in

practice. Buyers may be reluctant to display a larger quantity as part of a supply schedule

for fear of being front run. Posted best prices would have to be conditional on the size of a

responding order (that is, quantity the seller plans to sell), implying that the book would be

54

more di¢ cult to interpret for traders. Moreover, calculating those best prices would involve

solving a combinatorial problem. In contrast, there is little information leakage with hidden

bundles, and a simple price and time priority rule for any hidden quantities in general ,

which makes it easy to determine the order of execution against any eligible incoming sell

orders.

The second group consists of volume order equilibrium classes for which an analogue

the discretionary order equilibrium class exists, having the same expected trades for each

pair of buyer and seller types, the same number of visible and hidden units, and the same

structure of prices (that is, magnitude of hidden versus visible prices).

3.4.5 Optimal Mechanism

As suggested by Maskin and Tirole (1990), who consider buyer-optimal mechanisms when

values are private, the buyer-optimal mechanism will be shown to not involve pooling of

buyer types. That is, the problem simpli�es to the full information program, in which the

optimal mechanism for each buyer can be calculated independently.

The transfers associated with the optimal mechanisms for a given buyer type are equiv-

alent to either a demand schedule or a supply schedule,or a schedule that involves bundling.

Demand schedules can be implemented with simple limit orders; supply schedules cannot.

On the other hand, it is possible to implement supply schedules and bundling with volume

orders.









discriminate.

Below are two examples illustrating how the discretionary order and volume order equi-

libria compare to the buyer-optimal mechanism.

Example 2 continued (Optimal mechanism versus Discretionary order equilib-rium) For the same parameters as in the volume order equilibrium example, the optimal

contract for the large buyer type and for the small buyer type is to propose to buy two units at

a price of 4 = 1+3, implying that the large seller type will sell two units in the equilibrium

of the principal-agent game (and the small seller type will not sell any units). Associated

payo¤s are 9:1 and 4:9 for the large and small buyer type respectively.

As can be seen, the large buyer type would do better in this case than in the discretionary

55

order equilibrium (while trading less in equilibrium) and the small buyer type would do better

(while trading the same amount in equilibrium, namely, two units if the seller type is large).

Example 3 continued (Optimal mechanism versus Volume order equilibrium)For the same parameters as in the volume order equilibrium example, the optimal contract

for the buyers is exactly as in the volume order equilibrium. That is, the small buyer type

optimally proposes to buy one unit at 5 and the large buyer type optimally proposes the

seller types with a menu of two choices, to sell one unit for 5 or a bundle of two units for

11 (that is, 5:5 per unit).

3.5 Overview of the Results for Interdependent Values

The following is an overview of the results for the interdependent values analysis, which is

structured like the corresponding section for private values, presenting four games involving

the di¤erent order types and concluding with a mechanism design section.

3.5.1 Limit and Market Orders Only (basic game)

In contrast to the private values setting, the buyer type�s optimal strategies cannot be

calculated independent from each other, because the seller�s response to a given buy order

depends on his belief about the buyer�s type. The large buyer type may thus �nd it bene�cial

to pool onto submitting the same order as the small buyer type, because revealing his type

to be large would cause the seller to adjust upwards his expected valuations of the asset.

3.5.2 Game with Iceberg Orders

With interdependent values as opposed to private values, iceberg order equilibria exist.

(Limit order equilibria also remain equilibria.) In iceberg order equilibria, buyer types pool

onto a visible order, and the large buyer type also hides a unit. Intuitively, hiding a second

unit and pooling with the small buyer onto a visible order is bene�cial to the large buyer,because it allows him to receive a lower price on the visible unit compared to what he would

have had to o¤er in a separating equilibrium. The hidden unit, though, executes only when

the buyer type is large, so its price has to be high enough for the seller types to want to sell

the unit conditional on the buyer type being large.

Volume may increase or decrease following the introduction of iceberg orders. With

respect to transparency, iceberg orders have a similar bene�t as discretionary orders (dis-

cussed above, in Section 3.4.3), in that the type of the large buyer is revealed only if he buys

two units (as opposed to always, if he submits an order with same execution probability

of 2 units in a separating equilibrium; if the equilibrium was pooling and two units were

bought by the buyer types, no iceberg order would be needed.).

56

Example 4 (Iceberg order equilibrium, Pooling) Letf�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:3; 0:1; 0:3; 8; 3; 11; 9; 5; 9; 3; 4gFor these parameter values an iceberg equilibrium exist in which the small and large buyer

type submit a visible one-unit order at 7:49 and the large buyer type additionally hides a

unit.

In the absence of iceberg orders, a separating limit order equilibrium would exist in which

the large buyer submits an order priced at (p1; p2) = (8:3; 6:7), with expected trade volume

of 1+� (as in the iceberg order equilibrium). The small buyer type would submit a one-unit

order at 7:4. Total volume in the iceberg order equilibrium is thus the same as with limit

orders, but the large buyer type does better (spending 7:49 � (1 + 0:1) = 8:239 instead of

8:3 + 0:1 � 6:7 = 8:97), and the small buyer type does worse. (Both seller types also do

worse.)

Q

Lp

PEquilibrium orders:Small buyer submits a visible order at Lp .Large buyer submits an iceberg order ))(( LL pp ,thus hiding a unit at Lp .Small seller submits a limit order at Lp ,selling the unit at Lp with certainty.Large seller submits a two unit limit order at Lp ,selling one unit at Lp , if the buyer is small and two unitsat Lp if the buyer is large.

Figure 5: Example Iceberg Order Equilibrium.

3.5.3 Game with Discretionary Orders

When values are interdependent, the structure of the equilibria and the conclusions regard-

ing transparency and volume are the same as in the private values setting. There are now

more equilibrium classes, though, and intuition for the equilibria is also di¤erent.

Large buyers hide prices for two reasons. First, as in the private values case, volume

orders allow buyers to do screening of the sellers. Second, pooling allows the large buyersto receive lower prices on the visible units than they would have if their type was known.

Moreover, since both buyers pool onto the visible prices and buy at the visible prices when

the seller type is large, the small buyer type is at a disadvantage: he ends up paying more

than he would need to pay on an order with the same execution probability if he was able

to credibly signal his type.

57

3.5.4 Game with Volume Orders

There exist a number of equilibrium classes in which volume orders are used. Features

included in these equilibria are pooling to the advantage of the large buyer, bundling,supply schedules, and screening (as de�ned before) as well as screening with lotteriesacross units (which is de�ned below).

Much like in the private values setting, there are equilibria in which buyers separate on

the visible part of the volume order, or pool on the entirety of the order. These equilibria

are mentioned for completeness: they are less interesting in the sense that buyers may as

well display the entirety of their order (if it were possible to do so).

As for the equilibria in which buyers pool on the visible part and either only one buyer

hides units or both buyers hide units, one can distinguish three main groups of equilibria.

First, there are equilibria that are a combination of iceberg equilibria and either demand

or supply schedules. While this does not hold for all equilibrium classes in this group, the

main idea here is that the large buyer pools onto a visible part of the order for the �rst

unit, and then hides another unit at a higher or lower price. While the price of the hidden

unit re�ects the fact that the type of buyer hiding it is large, pooling is advantageous for

the large buyer type because it allows him to achieve a lower price on the �rst unit.

An example of this kind of equilibrium class would be the one in Example 3 above. With

interdependent values, though, the large buyer would have the added bene�t that the price

pL would be lower than what he would have to o¤er on that unit if his type was known.

The second group, as in the private values setting, contains equilibrium classes that

closely resemble discretionary order equilibria. Submitting hidden units allows the large

buyer to screen the seller types. In addition, the visible prices onto which the buyer types

pool are lower than what the large buyer type would have to pay on an order with the same

execution probability if his type was known to be large.

The third group contains equilibrium classes of two kinds. The �rst kind are equilibria

in which buyers separate the seller types with respect to their valuation on the second unit:

both sellers sell one unit with certainty at the �rst visible price, while on the second unit,

the small seller prefers to gamble for a higher (hidden) price and the large seller prefers to

sell the unit with certainty at the lower (visible) price. Note that these equilibria do not

exist in the case of private values, where the optimality of the small buyer�s order implies

that the large seller receives no rent on the second unit. In addition, equilibrium classes of

this kind exist that also have supply features (in that the hidden price on the second unit

is above that of the visible unit).

Example 5 (Volume order equilibrium, Supply, Screening and Pooling) Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f 0:1; 0:1; 0:1; 8; 7:8; 12; 11; 7:5; 7:6; 2:7; 5g

58

For these parameter values, an equilibrium exist in which the buyer types pool onto a visible

order (pL; pL) = (8:34; 6:072), while the large seller type hides a unit at pH = 8:7 . In

particular, the large buyer submits a volume order vol((pM ; pL); pH). The small seller type

submits a volume order vol(pM ; pH) = vol(8:34; 8:7), selling the �rst unit at 8:34 with

certainty and the second unit at 8:7 if the buyer type is large. The large seller type submits

a two-unit limit order priced at pL, thus always selling two units at (pM ; pL) = (8:34; 6:072).

(Note that the prices are set such that the large seller type is just indi¤erent between selling

the second unit for sure and gambling for the higher price; he makes a rent of 0:26 on the

second unit with either strategies. )

Q

Equilibrium orders:Small buyer submits a visible order at LM pp , .Large buyer submits a volume order

)),,(( HLM pppvol , thus hiding a price Hp .Small seller submits a volume order ),( HM ppvol ,receiving Mp on the first unit,and selling a second at Hp if the buyer is large.Large seller submits a two unit limit order at Lp ,which executes at Mp , Lp .

Hp

Lp

P

Mp

Figure 6: Example Volume Order Equilibrium.

The second kind of equilibria screens sellers by o¤ering a menu of lotteries across units.

The example class illustrated below resembles a �ctitious discretionary order equilibrium

which buyers submit a visible supply schedule (rather than a demand schedule), and another

unit is hidden above the visible prices. Note that the hidden price is sometimes paid for the

�rst and sometimes for the second unit, something that can only be achieved with volume

orders. In equilibrium, seller types are faced essentially choosing between two options: a

lottery on the �rst unit (which may or may not execute at the hidden price), or a �rst unit

sold with certainty in addition to a lottery on a second unit (which may execute at two

di¤erent prices). Since the choices involve lotteries across units, the seller types�relative

magnitude of the valuations on the �rst and second unit, or the �slope�of the supply line

de�ned by the seller types�valuation, enters the choice. Intuitively, the slope of their supply

line, can also be thought of as the market�s depth on the supply side. Thus buyer types

using volume orders in these equilibria are essentially screening the opposing side of the

market (the supply side), with respect to depth.

Example 6 (Volume Order Equilibrium, Supply, Pooling and Screening withLotteries across Units) Letf�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:4; 0:2; 0:8; 14; 11; 17; 16; 6; 11; 2; 4gFor these parameter values a volume order equilibrium exist in which the small and large

59

buyer type submit a visible one-unit order at pL = 8:8 and the large buyer type additionally

hides a unit at pH = 12:8 while the small buyer hides a unit at pM = 11:15 (that is, the

large buyer submits a volume order vol(8:8; 12:8) and the small buyer type submits a volume

order of vol(8:8; 11:15).).

The small seller type then submits a one-unit limit order at pH = 12:8, which executes

whenever the buyer type is large. The large seller type submits a volume order (pL; pM )

which executes at pL; pH when the buyer type is large and at pL; pM when the buyer is

small.

Equilibrium orders:Small buyer submits a volume order ),( ML ppvol .Large buyer submits a volume order ),( HL ppvol .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a volume order ),( ML ppvol ,selling one unit at Lp , and a second unit either at Hpif the buyer is large, or at Mp if the buyer is small.

Q

Hp

Lp

P

Mp

Example Volume Order Equilibrium.

As for the e¤ect of volume orders with respect to transparency, the results are the same

as with private values.

3.5.5 Optimal Mechanisms

Following Maskin and Tirole (1992), who analyze buyer-optimal mechanism in the common

values setting, buyer optimal non-randomizing mechanisms are calculated by �rst identifying

the Rothschild-Stiglitz-Wilson allocation (that is, the least cost separating allocation) for

the given set of parameters.

The optimal mechanism may sometimes involve separation for the buyers. If so, optimal

separating contracts may look like supply schedules, as was found in the private values

case. Moreover,as in the private values setting, each of the separating contracts can be

implemented with volume orders.

In contrast to the private values case, though, the optimal mechanism generally does not

simplify to the full information program, so that the optimal mechanism sometimes involves

buyer types pooling. Pooling contracts may sometimes prescribe the same allocation for

both buyer types, in which case the optimal pooling contrast looks like a supply schedule

as well (formally, there would be two identical supply schedules).

It is investigated whether the allocation of any optimal mechanism can be replicated

(that is, implemented) with orders. Speci�cally, for any given set of model parameters, an

optimal mechanism is an allocation, that is, trade quantities and transfers to be exchanged

60

for each state (that is, buyer and seller combination). Thus, it is analyzed whether, for any

given model parameters, it is possible to �nd a set of admissible orders such that, state by

state, the same number of units and the same payments are exchanged in the equilibrium

of the associated game as in the optimal mechanism.

Some optimal mechanisms can be directly implemented with limit orders. Some mech-

anisms cannot be implemented for the relatively simple reason that they, for example,

prescribe a zero transfer when a buyer-seller pair trades two units, which is something

that could not happen in an equilibrium with orders. Since all payo¤s are linear in trans-

fers/prices, though, a given buyer or seller type�s strategy depends only on the expected

transfers he will receive. Thus, equivalence classes of mechanisms are de�ned: by iden-

tifying mechanisms that have the same traded quantities and expected transfers for each

type of buyer and seller (but di¤erent transfers for a given pair of buyer and seller type).

The equivalence classes are non-empty because the matrix that maps transfers (for each

buyer-type and seller-type pair) into expected transfers for each buyer or seller type, does

not have full rank.

With the equivalence classes in place, a given mechanism is considered replicable if

there is any mechanism in its equivalence class that is replicable. In the example above, the

mechanism that had a zero transfer when a given pair of buyer and seller types traded two

units may still be implementable if the equivalence class contains a transformation of the

mechanism whose allocation is replicable (as equilibrium prices and traded quantities in a

game involving orders).

Not all allocations/optimal mechanisms (that is, not all equivalence classes) can be

replicated as equilibria, implying that buyers may achieve higher payo¤s with the optimal

mechanism than in equilibria with any of the order types. The reason for this result is that

the incentive and rationality constraints imposed in the principal-agent game are �fewer�

than those imposed on equilibrium strategies and outcomes (because sellers have more

available deviations in the sequential game).

61

3.6 Overview Table for the Results

The following table provides an overview of most of the results for the private values and

the interdependent values settings in Sections 4 and 5 respectively. (Note that the case of

perfect information is not part of the analysis in those sections, and is added here merely

for illustration purposes.) For iceberg, discretionary and volume orders the entries in the

table describe the respective order type�s advantage over limit orders. All three order type

have ambiguous e¤ect on volume and advantages with respect to transparency described

earlier in Section 3.

PerfectInformation

Private Values(buyer-seller-uncertainty)

Interdependent Values(buyer-seller-uncertainty)

IcebergOrders

Not used Not used Pooling (Large buyer pays

less on visible unit )

DiscretionaryOrders

Not used. Screening (Separate sellers by

valuation on a given unit)

Screening (Separate sellers by


Pooling (Large buyer pays


VolumeOrders

Same as Mechanism

Design, but multiple

equilibria.

Same as Mechanism Design,

but multiple equilibria.



Supply schedules

and bundling


valuation on a given unit,

and with lotteries across

units)

Supply schedules

and bundling

Pooling (Large buyer pays


MechanismDesign

Optimal contract

reaches e¢ ciency,

full rent extraction.

Optimal contracts same as

Full Information contracts;

do not always reach e¢ ciency.

All (optimal) contracts are

demand schedules or supply

schedules (which may feature

bundling).

All (optimal) contracts can be

replicated with volume orders

(though multiple equilibria).

Optimal contracts with pool-

ing exist.

Optimal contracts do not al-

ways e¢ ciency.

Not all optimal contracts can

be replicated (even ex-post

implementable conracts).

Table 2: Overview table

62

4 Private Values

The theoretical part of this thesis presents a simple game theory model of an exchange. A

buyer and a seller, both of which can be either of two (privately known) types, can trade

up to two units of an asset through an order book. This section contains the theoretical

analysis of the thesis for the case in which the buyer and seller values for the two units are

private (Section 5 contains the analysis for the case in which values are interdependent).

Within each of the sections, set of four games which di¤er in the set of admissible orders is

considered.

This section has six subsections. Section 4.1 describes the physical setup. Section 4.2

solves a basic game, in which the set of admissible orders consists of only limit orders and

market orders. Sections 4.3, 4.4 and 4.5, analyze games in which the set of admissible

order types additionally includes iceberg orders, discretionary orders, and volume orders,

respectively. The last Section, 4.6, contains a mechanism design analysis.

4.1 Model Setup

This section de�nes the physical environment of the game, that is, the players and the time

line of the game. It also speci�es what would be considered e¢ cient trade.

4.1.1 Players

A buyer and a seller can trade up to two units of an asset. Units will be indexed by

k 2 f1; 2g. Buyers and sellers can be either �large�or �small,�and are labeled Bi and Sj,respectively, with i; j 2 fL; Sg denoting the types as large (L) or small (S).

The types refer to the players�(relative) valuations: for any one of the two units, the

valuation of the large buyer type is above that of the small buyer type; the valuation of

the large seller type is above that of the regular seller type. I assume that the buyer�s type

is large with probability �, and small otherwise. Similarly, the seller type is large with

probability �, and small otherwise. The probabilities are common knowledge at the outset

of the game; individual players, though, only know their own type at that point.

Buyer and seller valuations are denoted, respectively, vBik and vSjk. Here, vBik refers to

buyer Bi�s valuation for the kth unit, analogously for sellers. The valuations are common

knowledge, and assumed to have the following structure:

vBi1 > vBi2 i = L; S

vBLk > vBSk k = 1; 2

vSj1 < vSj2 j = L; S

vSLk < vSSk k = 1; 2

Thus, buyer valuations are decreasing, and seller valuations are increasing. And, as

stated earlier, the large buyer�s valuation is above that of the small buyer�s, and the large

seller�s valuation is below that of the small seller�s.

63

Below is an example of admissible buyer and seller valuations:

Large buyer

Large sellerSmall buyer

Small seller

1BLv

2BLv

1BSv2BSv

1SLv

2SLv1SSv

2SSv

QQ

PP

Figure 7: Example Buyer and Seller Valuations.

Individual players only know their own type, and have initial beliefs about the other

player�s type. Initial beliefs are denoted �i(0) = (�ij(0))j=fL;Sg for buyers, and �j(0) =

(�ij(0))i=fL;Sg for sellers (where �ij(0) denotes the probability that buyer i assigns to seller

j�s type, and analogously for sellers). Initial beliefs are assumed to be consistent in that

�i(0) = � and �j(0) = �. Thus, all games considered in the following are Bayesian games

of imperfect information, where the uncertainty stems from players�uncertainty about the

other player�s type.

4.1.2 Game

In all games (both in Sections 4 and 5), a buyer and a seller trade with each other by

submitting orders to an order book. The time line for the games is as follows:

t = 0� The order book is empty.

t = 0 The buyer submits a buy order to the book,

t = 0+ The book is updated to include on the initial order;

t = 1 The seller submits a sell order upon seeing the updated order book,

t = 1+ The game ends: trade (if any) happens;

and payo¤s are realized.

Regarding the time line, note the following. First, the game starts out with an empty

order book in order to focus on the problem of optimal order submission for the initial player,

thus abstracting from the trade-o¤ between submitting a new order versus responding to

orders that might already be posted in the book. Consequently, it is possible to consider

the bid side and the ask side of the book independently, so that the assumption of the initial

64

order being a buy order can be made without loss of generality (the game is analogous for

the case of an initial sell order).

As an aside, with respect to the trade-o¤ between submitting a new limit order (that

is, a simple buy or sell order) versus a market order (thus buying/selling from the book by

responding to existing limit orders): it is often argued that traders might prefer limit orders

over market orders if a) they are su¢ ciently patient and would rather wait for execution than

incur the spread associated with responding to the other side of the book, or b) they have

a su¢ cient preference for execution uncertainty versus price uncertainty, thus preferring

the �xed limit order price over an uncertain average execution price on the market order

quantity.

Throughout this dissertation, attention will be restricted to pure strategy equilibria;

some of the notations in the model setup have been adapted to this restriction.

4.1.3 E¢ cient Trade

A trade in this model is e¢ cient if for every unit traded (independently), or every non-

separable combination of two units traded, the reservation value of the buyer is above that

of the seller (reservation values, in turn, depend on the trader�s valuations).

4.2 Limit Orders Only

This section presents the complete model for the basic game, as well as a brief characteri-

zation of the equilibria.

4.2.1 Actions

In this version of the game, the set of admissible orders to be submitted to the book consists

of limit and market orders.

Limit orders specify a price and a quantity to be bought at that price, and they aredisplayed in full in the order book. As is standard in order books, there is price priority

(higher-priced buy orders and lower-priced sell orders are listed and executed �rst). Con-

sequently, traders essentially submit demand schedules with buy limit orders (and supply

schedules with sell limit orders).

In this model (since at most two units are traded), a buy limit order speci�es a price p1on the �rst unit and possibly a price p2 on the second, with p1 � p2.

Market orders specify a quantity to be bought from the book. When a trader submitsa market buy order of a given quantity to the book, the order is executed against the lowest-

priced sell orders in the book, that is, the best o¤ers (analogously, market sell orders would

execute against the highest bids �rst).

The execution rules are as follows: if a market order is submitted in response to alimit order, the quantity speci�ed in the market order executes at the price of the limit

65

order. If a limit order is submitted in response to the initial limit order, then any trade

is executed at the price of the initial limit order (even if the price of the sell limit order is

below that of the limit order in the book). In the basic game, the execution rules imply

that it would be su¢ cient to allow sellers to submit only market orders.

4.2.2 Payo¤s

This section de�nes strategies and payo¤s. Players�strategies are conditional on all infor-

mation available to them at their decision nodes.

The buyer moves �rst, at time t = 0, thus, the buyer�s strategies are mappings fromthe buyer type into the set of admissible buy orders, sbuyer 2 Sbuyer, where Sbuyer : i 7�!buy order. The seller chooses a responding sell order at time t = 1, once the book has been

updated to re�ect the initial buy order, thus the seller�s strategies are mappings from theseller type and the initial buy order into the set of admissible sell orders, sseller 2 Sseller,where Sseller : j � buy order 7�! sell order.

Given a pair of buyer and seller strategies (that is, buy and sell orders), it is possible to

calculate the payo¤s associated with the resulting trade. In particular, the execution rules

will de�ne how many units, if any, are traded, and at what prices. For ease of exposition,

payo¤s will be written directly as a function of the resulting trades, which are de�ned next.

Let xijk and pijk denote, respectively, the execution probability and price for unit k,

which is to be exchanged in the trade between the buyer of type i and the seller of type

j. Individual trades are denoted (xij ; pij), where xij = (xijk )k2f1;2g and pij = (pijk )k2f1;2g.

Given that attention is restricted to pure strategy equilibria, only integer quantities are

traded, so xijk 2 f0; 1g for all (i; j; k). Moreover the second unit is only traded if the �rstunit was traded, implying xij1 � x

ij2 for all (i; j).

Next, expected trades for the buyer are denoted as (xBi; pBi) = (xij ; pij)j2fL;Sg.

Speci�cally, these are the trades that buyer Bi anticipates when he does not know the

seller�s type, but knows the individual trade xij that will occur conditional on the seller�s

type being equal to j.

Payo¤s for the seller, for a trade xij ,are:

V ij (xij ; pij) =

2Pk=1

�pijk � vSjk

�xijk

Expected payo¤s for the buyer, for a trade xBi are:

U i(0; xBi; pBi) =P

j2fL;Sg

Pk2f1;2g

�vBik � pijk

�xijk

!�ij(0)

Above, the �rst parameter in U i, refers to the time at which the expectation is formed,

in this case t = 0, that is, the beginning of the game.

66

4.2.3 Solving for Equilibria

Calculating the subgame perfect equilibria of this Bayesian game involves solving the game

backwards. At time t = 1, when the initial buy order has been published in the book,

seller types have to decide how many, if any, units to sell. (Sellers may sell the units by

submitting market orders with the quantity they wish to sell, or by submitting a sell limit

order with any prices that are less than the displayed prices; in either of these cases, any

trades will execute at the prices of the initial buy limit order.) Due to the private values

setting, the seller types are indi¤erent with respect to the type of the buyer who submitted

the order. A seller�s best response is thus to accept to sell any given unit as long asthe price on the unit is above his reservation value. Seller j�s reservation value on unit k

is given by vSJk. In particular, given the structure of the valuation, if a small seller sells a

given unit at a given price, so will the large seller, which implies xiLk � xiSk 8i; k.Given the seller�s best responses at time t = 1, buyer types have no incentive to hide

their type (there is no downside to submitting a buy limit order that reveals their type). A

buyer�s optimal strategy is independent from the other buyer�s type, and essentially involves

trading o¤ price with expected execution probability for each possible limit order.

Since buy limit orders, are demand schedules, pij1 � pij2 has to hold for any two-unit

limit order. That is, a second unit only executes if the �rst unit executed, which implies

xij1 � xij2 8i; j in any equilibrium.

It is optimal for buyers to set the limit order prices as low as possible given a desired

execution probability, meaning that prices are generally equal to the reservation values of

the buyers trading the unit, with the caveat that the price schedule may have to be modi�ed

to resemble a demand schedule.

The following table illustrates the six potentially optimal limit orders the buyer may

submit. These orders are the lowest price orders for the six possible execution probabilities,

from no trade to trading both units with certainty. For each of the six options, the table

includes the expected trade volume, vol(xBi), the execution probabilities per unit depending

on the seller types, xBi, and the prices per unit (pij1 ; pij2 ).

Here, expected trade volume, of a given expected trade xBi is

vol(xBi) =P

j2fL;Sg

Pk2f1;2g

xijk

!�j .

(A given unit k will be traded with probability vol(xBik ; k) =P

j2fL;Sgxijk �j).

67

vol(xBi) xBi = (xiL1 ; xiL2 ; x

iR1 ; x

iR2 ) (pij1 ; p

ij2 )

2 (1; 1; 1; 1) (vSR2; vSR2)

1 + � (1; 1; 1; 0) (max(vSR1; vSL2); vSL2)

1 (1; 0; 1; 0) (vSR1; 0)

2� (1; 1; 0; 0) (vSL2; vSL2)

� (1; 0; 0; 0) (vSL1; 0)

0 (no trade) (0; 0; 0; 0) (0; 0)

Table 3: Potentially optimal limit orders in private values setting.

As can be seen in the table, to guarantee, for example, an execution probability of 2,

buyers have to assure that the small seller sells his second unit, which implies that p2 � vSS2.At the same time, limit orders have to be demand schedules, so p1 � p2.

Regarding the optimal buy order with execution probability 1+�, prices have to be set

such that both types of sellers sell their �rst unit (which implies p1 � vSS1), and only thelarge seller sells his second unit, so that p2 � vSL2 has to hold (as well as p2 � vSS2). Inaddition, the order has to be a demand schedule, implying p1 � max(vSS1; vSL2) (note thatthe assumptions on the valuations do not exclude that vSS1 < vSL2).

The optimal limit order for buyer i at the outset of the game is found by calculating

the associated expected payo¤ for each of the six options and choosing the one that has the

highest payo¤.

4.2.4 Characteristics of the Equilibria

In this section, I describe characteristics of the subgame perfect equilibria with respect to

trade volume and transparency.

First, the structure of the valuations for buyers and sellers implies that large types trade

more volume in equilibrium. For sellers, this is simply because their strategy involves selling

units as long as the price is higher than their reservation value, implying that large sellers

will sell at least as many units as small sellers.

The following paragraphs illustrate why large buyers trade more volume than small

buyers in equilibrium. First, note that there can be no equilibria in which the buyer types�

expected trade volume is 1 and 2�, or 1 and 2�. Consider the case when the equilibrium

volumes traded are vol(xBR) = 2� for the small buyer and vol(xBL) = 1 for the large.

In particular, vol(xBR) = 2� means that the small buyer buys two units at pBR2 = vSL2

if the seller type is large, which is only possible if vSL2 < vSS1 (otherwise, the execution

probability on that order would be 1 + �). This in turn implies that the large buyer�s

optimal strategy cannot be such that vol(xBL) = 1, because he could do strictly better

submitting an order with expected volume of 1 + � by adding a second unit priced at pBR2(since the small buyer �nds it bene�cial to buy a second unit at that price, so must the

68

large type). Similarly, it cannot be that the trading volume of the small buyer is 1 and that

of the large buyer is 2�, because the large buyer would �nd it optimal to submit an order

with expected trade volume 1 + � instead.

Next, consider two expected equilibrium trades xBRand xBL for the large buyer and the

small buyer (with equilibrium prices pBRand pBL). Buyer i will prefer the strategy of the

small buyer type over that of the large buyer type whenever:

U i(0; (xBR; pBR)) � U i((xBL; pBL))()Pk2fL;Sg

Pj2fL;Sg

vBik(xSjk � xLjk )�j

�P

k2f1;2g

Pj2fL;Sg

(pSjk xSjk � pLjk x

Ljk )�j

Note that the right hand side is independent of the buyer type. Next, note that for

xBRand xBL, and any xBi; exBi that do not have associated volumes of 1 and 2� (or 1 and2�), vol(xBi) > vol(exBi) implies xijk > exijk 8j; k. That is, strategies that lead to higher totalvolume generally also have higher execution probabilities for each unit and for each type of

seller.

As a consequence, it cannot be that the small buyer trades more in equilibrium, because

vol(xBR) > vol(xBL) would imply that the right hand side above would be supermodular

in vBik and (xSjk � xLjk ), meaning that if the small buyer preferred xBR, so would the large

(which cannot be).

The following statement summarizes the analysis above, and also includes comments

about trade volume as a function of the seller type being large.

Proposition 1 i) In any equilibrium of the game, large buyers (sellers) always trade weaklymore volume than small buyers (sellers). Consequently, trading volume is increasing in the

probability � that a buyer is large.

ii) On the other hand, expected trade volume may not always be increasing in the probability

� that the seller is large.

Part i) has been shown above. For part ii), a numerical example can be found in the

appendix. In terms of the intuition, note the following. For a given buy limit order, expected

trading volume is increasing in the probability � that a seller is large (as large sellers are

more likely to sell). Trade volume will not always be increasing in �, though, since buyers

may �nd it optimal to gamble and submit a low-priced order aimed at the large seller type,

risking non-execution (for � = 1 the seller type is known to be large and buyers will submit

accordingly low buy limit orders).

This section concludes with a remark on transparency, which is included for complete-

ness, and in order to de�ne a benchmark for the games to follow (in which other order types

may be used). First, though, pre-trade and post-trade transparency for all the games to be

considered in this thesis will be de�ned.

69

Transparency, both pre-trade and post-trade, refers to what can be inferred aboutthe buyer and seller type. Speci�cally, types might be known or unknown at any given

point in time. Pre-trade transparency refers to before the trade (but after the initial order

is posted), and post- trade refers to after the game has ended.

Corollary 1 Pre-trade and post trade transparency.i) Whether the buyer type is revealed pre-trade depends on whether there happens to be

pooling or separation in equilibrium.

ii) Transparency may not increase post-trade. If it does, then trades reveal at most the seller

type.

A given buyer type�s optimal strategy is independent of the other buyer type, which

implies i). Moreover, for a given posted buy order, seller strategies depend only on the

seller�s (reservation) values on the units, which implies ii), as no information about the

buyer is revealed by the fact that trade happens. Transparency may not increase if, for

example, the large buyer type submitted an aggressively priced buy order with an execution

probability of 2, so that both seller types would sell two units at the limit order prices.

Finally, note that Section 4.6 considers the buyer-optimal mechanism. Anticipating

some of the results: it is not always possible to implement the e¢ cient allocation, nor the

�rst best with limit orders, since the optimal mechanism may include submitting a buy

limit order that looks like a supply schedule (that is, p1 < p2). In contrast, some of the

other order types discussed in the next subsections can implement any optimal mechanism.

4.3 Iceberg Orders

This section presents a game in which buyers can use limit orders as well as iceberg orders

to post to the book, and sellers may use limit or market orders to respond (as in Section

4.2, market orders are su¢ cient here, though).

Iceberg orders specify a price that is displayed together with both a visible quantityat that price and a second quantity that is hidden. Hidden quantities are not displayed in

the book, though execute against incoming market orders. In this model, iceberg orders

consist of a visible one-unit or two-unit order, and include a hidden unit at the �rst visible

price.

Let p1 be that price of the �rst visible unit in the iceberg order (and p2 < p1 the price

of the second visible unit, if there is one). A seller submitting a two-unit market sell order

against the iceberg order would sell two units at the price of the �rst visible unit, p1. If

there had been no hidden unit, the seller would either not have sold a second unit, or (if

the visible limit order consisted of two units) he would have sold a second unit at p2.

In order to de�ne an equilibria involving iceberg orders, it is now necessary to introduce

interim beliefs for the sellers. In particular, the seller updates the initial beliefs �ij(0)upon seeing the initial buy order in the book. Let �ij(1) denote the seller�s interim beliefs.

70

With this, one can introduce expected payo¤s for the seller types, when they expect trade

xSj ; pSj :

Vj(1; xSj ; pSj) =

Pi2fL;Sg

Pk2f1;2g

�pijk � vSjk

�xijk

!�ij(1)

Regarding beliefs, there are three points to note. First, in any equilibrium, beliefs on

the equilibrium path are updated via Bayes�rule. Second, for deviations o¤ the equilibrium

path, I assume that sellers believe there is no additional hidden part to any visibledeviation. It is necessary to make an assumption regarding deviations that are not visible,because iceberg orders allow for part of the order to be hidden (as do discretionary orders

and volume orders, which are introduced later). The motivation for the speci�c assumption

here is as follows. Since seller types are already endowed with beliefs about the deviating

buyer�s type (namely, that the buyer type is large, unless he can credibly show he is not),

it seems intuitive not to endow them with yet another layer of beliefs, this time about the

full description of the order (any hidden parts that might exist in addition to the visible

parts of the deviation), especially considering that o¤-equilibrium-path beliefs cannot be

veri�ed, unlike, in some sense, beliefs about (invisible parts of) orders played in equilibrium.

Nevertheless, the assumption is not without loss of generality.

Third, only equilibria that satisfy the intuitive criterion will be considered. That is,the seller assumes that a deviating buyer�s type is large, unless the deviation credibly shows

he is not. By �credibly,� I mean that a large buyer would prefer to be identi�ed as large

and choose his optimal action accordingly, rather than choosing the deviation in question

and being thought of as a small buyer.

Before analyzing the equilibria with iceberg orders, consider the following remark.

Observation 1 All limit order equilibria from Section 4.2 remain equilibria once iceberg

orders are introduced.

Limit order equilibria remain equilibria when iceberg orders are introduced, if the asso-

ciated equilibrium beliefs are that no units are hidden (and the o¤-equilibrium-path beliefs

are as speci�ed above, namely, that no unit is hidden).

Regarding equilibria with iceberg orders, I now make the following re�nement. I consider

only �proper� iceberg order equilibria, de�ned as equilibria in which iceberg ordersare submitted with positive probability such that at least one player does strictly better

than in the corresponding limit order equilibrium.

With the re�nement in place, we can now consider candidate equilibria. In any candidate

equilibrium involving iceberg orders, sellers have to believe that at least one buyer type hides

an additional unit at p1, and at least one of the seller types has to submit an order aimed

at executing against that hidden order.

Next, given the private values setting, seller payo¤s from trading against a given order

are independent of the type of the buyer who submitted the order. As a consequence, seller

71

types base their responses solely on the prices they receive. Together with the argument in

the previous paragraph, this implies that hidden units have the same execution probability

at any given price than visible units at that price.

Moreover, buyer valuations are also independent of the seller type. Thus, if a unit was

hidden and executed with positive probability in equilibrium, it must be that a buyer would

be (strictly) better o¤ if the execution probability (strictly) increased. If the buyer would

not be better o¤with a higher execution probability, he would not have hidden an additional

unit at that price to begin with.

As a consequence of the arguments in the preceding paragraphs, iceberg orders are

weakly dominated, that is, there are no equilibria involving iceberg orders in which buyers

do strictly better than when only submitting limit orders. This result is captured in the

next proposition.

Proposition 2 There are no equilibria in which iceberg orders are used.

Note that in Section 5, on common values, iceberg orders will be used. Moreover,

concealing one�s type as a buyer may still be advantageous in a private values setting; in

fact, this is the case for discretionary orders, which are presented next.

4.4 Discretionary Orders

In this section, the set of admissible orders for buyers consists of both simple limit orders

and discretionary orders; sellers may use limit or market orders to respond.

Discretionary orders specify a quantity and a visible price for that quantity, as well asa hidden price on any part of that quantity, where the hidden price is above the displayed

price for buy orders, and below the displayed price for sell orders. If a market order is

submitted against a discretionary order, the order executes at displayed prices. If a limit

order (discretionary or not) is submitted, execution happens at the price of the responding

limit order if the price is in the discretionary range (between the discretionary price and

the visible price). If a limit order is submitted that crosses the (visible) limit order part

of the discretionary order, then execution happens at the visible price of the discretionary

order. Note that given the execution rules, it is su¢ cient if sellers can submit limit orders,

as any equilibrium in which market orders are used by sellers could be replicated with only

limit orders.

Note that the assumption about beliefs introduced in Section 4.3 is kept in place, so

that sellers expect that there is no hidden part to a visible deviation (all deviations are thus

simple limit orders).

The section is divided into two parts. The �rst part, 4.4.1, describes the equilibria, the

second, 4.4.2, analyzes the equilibria with respect to volume and transparency.

72

4.4.1 Equilibria

First, note that since discretionary orders hide part of the order information, it is necessary

to consider interim beliefs for the sellers. I assume that these beliefs are as speci�ed in

Section 4.3 on iceberg orders.

Next, much like was the case in the section on iceberg orders, limit order equilibria

remain equilibria when discretionary orders are also allowed. The corresponding belief for

these equilibria is that nothing is hidden in equilibrium. This result is captured in the

following remark.

Observation 2 All limit order equilibria from Section 4.2 remain equilibria once discre-

tionary orders are introduced.

Regarding equilibria involving discretionary orders, I will make the same re�nement

introduced for candidate equilibria involving iceberg orders, namely that I consider only

�proper� discretionary order equilibria. In these proper discretionary order equilib-ria, discretionary orders are submitted with positive probability, and at least one player

does strictly better than in the corresponding limit order equilibrium (the corresponding

equilibrium means the limit order equilibrium from Section 4.2 for the same parameters).

In general, there could be three kinds of equilibria involving discretionary orders. First

are equilibria in which buyer types separate on the visible part of the orders. In this case,

hidden prices will be correctly anticipated by the sellers, so buyers may as well display

them. I exclude these equilibria.

In the second kind of equilibria, the buyer types pool on the visible part of the order

and submit di¤erent hidden prices. In this case, if for any given unit both buyer types have

submitted a hidden price, then both buyer types may as well increase the visible price on

the unit until it is equal to the lower of the two hidden prices. Thus, this kind of equilibria

can also be excluded.

This leaves the third group of equilibria, where buyer types pool on the visible part of

the order, and only one buyer type hides at least one unit. I will consider only equilibria of

this group.

The following paragraphs explain the construction of equilibria with discretionary orders,

which involves a number of steps. First, the visible order has to be the optimal limit order

from Section 4.2 for the buyer who only submits that order (and does not hide any prices).

Second, the buyer who hides one or two prices will do so only if it increases the execution

probability on the corresponding units (otherwise he may as well submit only the lower

visible price on that unit). The two previous arguments allow the visible orders that may

be part of a discretionary order equilibrium to be narrowed down to one of the three following

orders: the optimal one-unit order with execution probability � (one price would be hidden

on that unit by the large buyer), the optimal two-unit order with execution probability 2�

73

(one or two prices would be hidden by the large buyer), or the optimal two-unit order with

1 + � (a price would be hidden on the second unit by the large buyer).

Next, the structure of valuations for the buyers implies that it must be the large buyer

who hides units in equilibrium. If instead the small buyer preferred to hide a price rather

than only submitting the visible part of the order, so would the large buyer, which cannot

be in equilibrium (where only one of the two buyers hides prices).

The picture below illustrates the buyer orders in the candidate discretionary order equi-

libria, which can be divided into classes A through D. Thick lines are visible units (onto

which both buyer types pool), dotted lines refer to hidden units (which are submitted by

the large buyer type). It is possible to narrow all candidate equilibria down to these four

based on the arguments in the previous two paragraphs and three more comments, which

will be discussed next.

Buyer orders in candidate discretionary equilibria A through D.

class A

Q

P

LpHp

class Cclass B

Q Q

Lp LpHp HpP P class D

Q

LpHpP

Figure 8: Buyer orders in candidate discretionary equilibria A through D.

As for the execution probabilities on a given unit in each of these equilibrium classes, it

must be that:

A B C Dorder/units 1 2 1 2 1 2 1 2

visible � 0 � � � � 1 �

discretionary 1 0 1 1 1 1 1 1

Table 4: Execution probabilities for orders in equilibrium classes A through D.

First, it is important to note that in the case of private values, the optimal order with

execution probability 2� is priced at p1 = p2 = vSL2, that is, the prices on both visible

units in this case are identical, as illustrated for classes B and C.

Second, for class C note that given execution rules and incentive constraints, the hidden

price on the �rst and second unit are identical. If prices were di¤erent, the higher-priced

unit would be entered into the order book as the �rst unit (implying pH1 > pH2 ). The only

way the second hidden price would execute is if the seller submitted a two-unit limit order

74

priced at pH2 on both units, and as a consequence both hidden units would execute at pH2 .

Thus, the �rst hidden price pH1 would not enter the equilibrium payo¤s, but it would tighten

equilibrium constraints by making the sellers�deviation to submitting a one-unit limit order

priced at pH1 more attractive. As a consequence, buyers are better o¤ submitting pH1 = p

H2 ,

as illustrated.

Third, regarding class D, which has execution probability of 1+�, note that discretionary

equilibria can only exist if that optimal limit order has p1 > p2. If the optimal order of

1 + � execution probability was �at, that is, priced at p1 = p2 = vSL2 (which would hold

if vSS1 < vSL2), then the hidden price would have to be higher than the visible prices.

As a consequence, the only way for the hidden unit to execute in a discretionary order

equilibrium would be if the small seller submitted a one-unit limit order at the hidden price

and the large seller submitted a two-unit market order (or a two-unit limit order at the

visible prices). As a result, the execution probability on the visible order would be 2�,

instead of 1 + �, which cannot be.

Next, given p1 > p2, the hidden price on the second unit has to be equal to the visible

price on the �rst unit, p1. If the hidden price was on the �rst unit, it would not increase

the �rst unit�s execution probability. If there is a hidden price pH2 on the second unit, the

only time both the �rst and second unit execute jointly is when a seller submits a two-unit

limit order priced at p1, and in that case the seller receives p1 on both units. As a result,

pH2 does not enter equilibrium payo¤s. A pH2 higher than p, though, would tighten sellers

incentive constraints by making it more attractive for the sellers to deviate to submitting a

one-unit limit order at pH2 . Consequently, buyers are better o¤ submitting pH2 = p1.

In order to verify whether any of the candidate equilibria can in fact be equilibria for

some parameter combination, I will now turn to the seller�s responses for each of the four

candidate classes. At this point, I would like to forestall that the structure of the valuations

for sellers will imply that large sellers sell more units (and at lower prices) than small sellers.

Equilibrium A From the optimality of the small buyer�s strategy, which is to submit a

one-unit order with probability of execution equal to �, we know that p1 = vSL1. Thus the

equilibrium payo¤ of the large seller would be 0. Given a discretionary price pH1 , the small

seller�s individual rationality (IR) constraint will be satis�ed if �(pH1 � vSS1) � 0. Given

vSL1 < vSS1, though, any pH1 that satis�es the small seller�s IR constraint has to violate

the large seller�s incentive constraint (as deviating to submitting a limit order at pH1 yields

�(pH1 � vSL1) > 0 to the large seller). Therefore, no equilibrium of this class exists.

Equilibrium D From the optimality of the small buyer�s order (which must be the opti-

mal limit order with expected execution probability of 1 and � on the �rst and second unit

respectively), we have p1 = vSS1; p2 = vSL2. Given this, the large seller�s equilibrium pay-

o¤ from trading the second unit is 0 (he submits a two-unit market order in this candidate

equilibrium). For the small seller to want to sell a second unit at pH2 = p1 (rather than

75

deviating to selling only one unit at p1), it must be that �(pH2 � vSS2) � 0. If that equationheld, though, then the large seller would �nd it pro�table to deviate to the small seller�s

strategy (thus selling the second unit at pH2 = p1 with probability �, rather than selling it

at p2 with probability 1). Therefore, no equilibrium of this class exists.

Equilibria B and C In equilibria of class B, one of the two types of sellers submits a two-

unit market order (selling two units at the visible prices with probability 1), and one type of

seller submits a one-unit limit order priced at the discretionary price, which executes with

probability �. In equilibria of class C, one seller again sells two units with certainty at the

visible price, while the other now submits a two-unit limit order priced at the discretionary

price, which executes in full with probability � (so that the expected trade volume for that

seller is 2�). In both equilibrium classes, the structure of the valuations implies that it

must be the small seller who sells at higher prices with lower execution probability (if the

small seller preferred to submit a market order, and sell two units at the low visible prices,

so would the large, implying that no units would ever execute at the hidden prices, which

cannot be since we are considering only equilibria in which all submitted order parts have

positive execution probability).

The following proposition states that equilibria of both classes B and C may exist for

certain execution probabilities in the private values setting. Intuition for the results will be

provided subsequently; the appendix contains numerical examples.

Proposition 3 There are two classes of equilibria, named B and C. For a given set of

parameters �, an equilibrium of both classes B and C may exist. As for the equilibrium

strategies/orders:

In both classes B and C, buyers pool onto a visible two-unit limit order priced at p1 = p2 =

vSL2.

The large buyer hides a price on the �rst unit at pH1 (in class B), and on both units at

pH1 = pH2 (in class C).

Large sellers submit a two-unit limit order at the visible prices p1 = p2 in both classes B

and C.

Small sellers submit a one-unit sell limit order at the hidden price pH1 in class B and a

two-unit sell limit order at pH1 = pH2 in class C.

Large types always trade more in equilibrium than the corresponding small types.

Compared to the corresponding limit order equilibrium, payo¤s for large buyer types are

strictly better and payo¤s for small buyers are the same.

Expected trade volume for each player type is described below.

76

type/expected volume B C

large buyer 1 + � 2

small buyer 2� 2�

large seller 2 2

small seller � 2�

Table 5: Player�s expected trade volume in equilibrium classes B and C.

Note that in spite of the private values setting, large buyers bene�t from the ability

to pool with small buyer types in discretionary order equilibria, in that the large buyer

achieves higher payo¤s than in equilibria with simple limit orders. Intuitively, discretionary

orders provide buyers with a potentially less costly strategy for achieving an expected trade

volume of 1 + � and 2, compared to the optimal limit orders with the same trade volume.

The reason that submitting discretionary orders can be less costly is that discretionary

orders allow large buyers to separate the seller types. Given the equilibrium buy orders,

the seller types have two choices, and will auto-select by type: sellers can either trade with

certainty by selling at the lower, visible prices, or sellers can submit a limit order at the

higher hidden price, but that order will have a lower execution probability. Presented with

this menu of choices in equilibrium (trading o¤ prices and execution probability), large

sellers who have lower valuations (and thus more to gain from a trade) choose to trade with

probability 1. Small sellers will gamble for the hidden prices.

Analyzing the equilibria in more detail, �rst note that the large seller�s expected payo¤

from selling two units at p1 = p2 is positive. Given the optimality of the small buyer�s limit

order, it must be that p1 = p2 = vSL2. Thus the large seller type makes zero expected

payo¤ from trading the second unit, and strictly positive payo¤ on the �rst.

If the large seller were to deviate to selling one or two units at pH1 , he would make a

higher per-unit payo¤ if the unit executed, but the probability of execution is lower than 1,

namely �.

To sum up, in deciding whether to adhere to his equilibrium strategy and sell at the

visible prices, or deviate and gamble for the higher hidden prices, the large seller is trading

o¤ his positive payo¤ from selling the �rst unit atp1, with that of selling one or two units

at pH1 (in equilibria of classes B and C respectively).

Note that there are no one-unit discretionary equilibria in this private values section

(unlike for Section 5). If they existed, supermodularity would imply that buyers pool onto

a visible unit with execution probability � and price p1, and the large buyer would hide a

pricepH1 > p1. Small sellers would sell at pH1 (with execution probability �) and large sellers

would sell at p1 with probability 1. This is because the execution probability on the visible

part would have to be � (so that hiding strictly increases the unit�s execution probability

to 1). The visible price would have to be such that the large seller is indi¤erent between

trading and not trading, due to the optimality of the small buyer�s equilibrium strategy.

77

Now, though, if the individual rationality constraint for the small seller were satis�ed given

the hidden price pH1 > p1, then deviating to the small seller�s strategy would be pro�table

for the large seller.

Finally, note that there are generallymultiple equilibria involving discretionary ordersfor given parameters �. For a given � there is exactly one limit order equilibrium, and thus

one optimal limit order for the small buyer. That optimal order remains the small buyer�s

order in the discretionary equilibrium, and it is also the visible part of the discretionary

order for the large buyer. Multiplicity arises in two ways. First, a given visible order onto

which buyers pool might be part of an equilibrium of two di¤erent classes, say B and C (a

numerical example is given in the appendix). Second, if an equilibrium of a certain class,

say B, exists for a given displayed order, then there is generally a continuum of equilibria,

because a continuum of hidden prices will �work� as well. This result is due to the fact

that payo¤s are linear in the hidden prices, so that individual rationality and incentive

constraints simply impose lower and upper bounds on what the hidden prices may be.

4.4.2 Volume and Transparency

We can look at transparency and volume in all equilibria.

Corollary 2 TransparencyIn all equilibria, any remaining post-trade uncertainty is about the buyer type.

Post-trade transparency refers to whether the type of the buyer and seller playing the

game can be inferred after the trade (if any). Any inference will be based on the knowledge of

the parameters of the game, the posted orders, and traded quantities and prices. To see the

result of the corollary, consider the following matrices, illustrating post-trade transparency

for the discretionary equilibrium classes B and C. In the matrices, rows and columns indicate

the buyer and seller types respectively (L=large, S=small), question marks mean the type

is unknown to the market, and numbers in brackets refer to units traded.

post-trade transparency equilibrium B

seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(1)

?; L

(2)

post-trade transparency equilibrium C

seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 6: Post-trade transparency in equilibrium classes B and C.

For any discretionary order equilibrium, it is possible to identify the case of a small

buyer type and a small seller type, because traded volume is lowest (or zero) in this case.

78

The case of a large buyer type and a small seller type can also be inferred uniquely, because

the traded volume is highest and the execution price for at least one unit is higher than the

posted price. The only two cases that cannot be distinguished from each other are those of

a large seller type trading with either a large or small buyer, because traded quantities and

prices are the same for both buyer types.

This model suggests that discretionary orders might allow large buyers to trade larger

quantities (two units in this model) without having their type revealed after the trade, as

long as the market is two sided (that is, as long as large sellers are present).

In practice, such a characteristic could make discretionary orders more attractive to

large traders. For one, the literature already argues that information leakage associated

with published orders is what drives traders to choose market orders over limit orders.

Next, it seems intuitive that a buyer wishing to trade more units over time would prefer to

not have his type revealed (inferred) so that the price he receives on the following trades is

not a¤ected. Discretionary orders would allow the buyer to do just that, provided markets

are two sided. So far, though, discretionary orders are mostly advertised as a way to receive

price improvement over the hidden prices, and as a way to quickly take advantage of price

movements (if, say, asks move down unexpectedly, a hidden unit with a discretionary price

higher than the new best ask would immediately execute).

The following corollary concerns the e¤ect of volume orders on expected trading volume.

Corollary 3 VolumeDepending on the parameter combinations, it is possible that the introduction of discre-

tionary orders leads to an increase or a decrease in expected trade volume.

See appendix for a numerical example. As mentioned, intuitively, the introduction of

discretionary orders provides buyers with potentially less costly alternatives to attain either

an execution probability of 1+� (by hiding one price, as in class B earlier), or an execution

probability of 2 (by hiding two prices, as in class C earlier).

Volume may be increased compared to the limit order equilibrium if, for example, both

buyers in the limit order equilibrium submitted orders with execution probability 2�. Vol-

ume may be reduced following the introduction of discretionary orders if the equilibrium of

class B is played when previously the optimal strategy for the large buyer was to submit

a limit order with expected trade volume of 2. Note, though, that one generally does not

observe very aggressive pricing in practice (that is, large, aggressively priced orders that

are displayed). Instead, traders engage in order splitting, and/or submit market orders

against best asks (as was discussed in Section 2). Thus, in practice, the introduction of

discretionary orders is likely to increase the number of buy orders submitted to the book.

The following corollary concerns the relationship between volume and transparency.

79

post-trade transparency with limit orderswhen vol(xBL) = 1 + �; vol(xBS) = 2�

seller S seller L

buyer SS; S(0)

S;L(2)

buyer LL; S(1)

L;L(2)

post-trade transparency equilibrium Cseller S seller L

buyer SS; S(0)

?; L(2)

buyer LL; S(2)

?; L(2)

Table 7: Post-trade transparency in limit order and class C equilibrium.

Corollary 4 Volume and transparencyi) It is possible that pre- and post-trade transparency is lower for the discretionary order

equilibrium compared to the corresponding limit order equilibrium, and yet trading volume

is higher.

ii) It is possible to have the same trading volume in the discretionary order equilibrium as

in the corresponding limit order equilibrium, with any remaining uncertainty being about the

buyer in the discretionary equilibrium and about the seller in the limit order equilibrium.

See the appendix for numerical examples. Below is an illustration for part i).

The matrices above illustrate the case in which given some parameter �, equilibrium of

class C is played, and the optimal strategy of the large buyer in the corresponding limit

order equilibrium would be to submit an order (pBL1 ; pBL2 ) with pBL2 > pBL1 and an expected

trade volume of 1 + �. The optimal strategy for the small buyer is to submit (pBS1 ; pBS2 )

with pBS1 = pBS2 and an expected trade volume of 2� in both equilibrium C and the limit

order equilibrium. Thus, trade volume would be increased to �(2)+ (1��)(2�) in C (from�(1 + �) + (1� �)(2�) in the limit order case).

As can be seen in the tables, pre-trade, no types are known in equilibrium C. Instead,

buyer types separate in the limit order case. Post-trade, there is also less transparency in

equilibrium C than with limit orders, because the buyer type remains unknown when the

seller type is large.

For part ii), consider the case where, given some parameter �, equilibrium C is played,

and the optimal strategy of the large buyer in the corresponding limit order equilibrium

would be to submit an order with the same expected trade volume (of 2). Trade volume

would be the same in C and in the limit order case. Moreover, in regard to post-trade

transparency, any remaining uncertainty in C is about the buyer type (see above), whereas

for the limit order equilibrium the remaining uncertainty post-trade would be about the

seller type (see below).

80

post-trade transparency with limit orders

when vol(xBL) = 2; vol(xBS) = 2�

seller S seller L

buyer SS; S

(0)

S;L

(2)

buyer LL; ?

(2)

L; ?

(2)


seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 8: Post-trade transparency in limit order and class C equilibrium.

Part i) of the corollary above states that trade volume and transparency (both pre- and

post-trade) do not necessarily move together; that is, it might be possible to achieve more

volume with less transparency. This point was somewhat controversial among regulators.

The general opinion was that (increased) transparency in �nancial markets was desirable,

and, more speci�cally, that low pre-trade transparency would be detrimental to trade vol-

ume, because the increased uncertainty about execution probability and execution prices

when hidden volume is present may discourage the submission of responding orders (as was

mentioned in Section 2). As a consequence, regulators in some countries (such as the U.S.)

were opposed to the introduction of iceberg and discretionary orders, or slow to allow it.

However, in the last �ve years, opinions have changed. Notably, the SEC approved the

creation of dark pools, Alternative Trading Systems, and, most recently, the Matchpoint

platform of the NYSE. All of these platforms use trading mechanisms with little or no

pre-trade transparency, and sometimes no post-trade transparency either (dark pools, for

example, do not even publish executed trades).

Part ii) of the corollary also concerns to transparency. Post-trade transparency usually

refers to what information is published after the trade; in opaque markets, venues may

not publish any information post-trade, in more transparent markets prices and quantities

of each trade may be published, as well as information about which venue executed the

trade. In this model, post-trade information is simply the quantity and price associated

with a given trade, independent of whether limit order or discretionary order equilibria

are concerned. Limit order and discretionary order equilibria di¤er though, with respect to

what information the trade price and quantity allow one to infer, that is, which player�s type

can be identi�ed post-trade. In particular, it is possible to achieve the same trade volume

but have the uncertainty about types be �switched�. In discretionary order equilibria, the

buyer�s type remains unidenti�ed if the market is two-sided (that is, when liquidity exists

on the other side of the market or, in this model, when the seller type is large). In contrast,

in the corresponding limit order equilibrium, seller types remain unknown if the buyer type

is large. If such a characteristic of discretionary orders were to hold in practice, this would

encourage the use of these orders, as buyers who plan on trading a large quantity (and

81

who have to decide how to optimally split the order) would bene�t from not having their

type revealed after a larger-than-usual trade in a sequence of trades (as revealing their type

would likely increase the cost at which the remainder of the total quantity could be �lled).

4.5 Volume Orders


and volume orders; sellers may use limit orders, market orders, or volume orders to respond.

Volume ordersVolume orders consist of a visible quantity and price as well as a hidden quantity.

The (hidden) price on the hidden quantity may be higher or lower than the visible price.

Moreover, the submitter can include a volume restriction on the hidden quantity, such that

an incoming order only executes against the hidden quantity if at least a minimum fraction

(say 70%) of the hidden quantity is traded. This option essentially allows for buyers to

specify bundling, which can be very bene�cial.

In this model, a volume order consists of a displayed one- or two-unit order (at p1 or

(p1; p2)), together with either one hidden price pvol1 , or two identical hidden prices (pvol1 =

pvol2 ). The volume condition, or minimum size condition, in this model is that in orders

with two hidden prices, buyers can specify that if there is execution at the hidden price,

then both units have to be sold at once.

Regarding the execution rules of volume orders: A seller submitting a sell market order

(of one or two units) against the volume order would sell at the displayed prices. If the seller

submits a limit order or a volume order against the initial volume order, then any executable

displayed unit trades �rst, and any hidden unit trades at its hidden price (examples for such

cases will be provided along with the equilibrium discussion). For volume orders with two

hidden units, note the following: Consider volume order vol(pL; (pM ; pM )), that is a visible

price of pL and two hidden units at pM , without a volume condition. If a seller submits a

two-unit limit order priced higher than pL and up to pM , both units execute at pM . If the

seller submits a volume order at vol(pL; (pM ; pM )), then only two units execute (since that

is the most that can be traded in this model), with one unit at pL and the other unit at pM .

If the buy order has a volume condition instead, then it is assumed that two units execute

at pM . In a model with up to three units traded, one unit would trade at pL and two units

at pM in this case. The assumption is necessary so that the idea of bundling can still be

captured even if the seller�s response was a volume order of the kind described, which is

a valid response for sellers. Moreover, since hidden volume always executes at the hidden

price which is set by the buyer, volume orders imply passive price setting (as de�ned in

Section 2).

Finally, as before, only pure strategy equilibria are considered, and the assumption

about beliefs introduced in Section 4.3 is kept in place, so that sellers expect that there is

no hidden part to a visible deviation (all deviations are thus simple limit orders).

82

This section is divided into two parts. The �rst part, 4.5.1, describes the equilibria; the


4.5.1 Equilibria

First, note that since volume orders hide part of the order information, it is necessary to

consider interim beliefs for the sellers. It is assumed that these beliefs are as speci�ed in

Section 4.3 on iceberg orders.

Next, analogously to what was the case with iceberg orders and discretionary orders,

limit order equilibria remain equilibria when volume orders are also allowed. The corre-

sponding belief for these equilibria is that nothing is hidden in equilibrium nor for deviations

o¤ the equilibrium path. This result is captured in the following remark.

Observation 3 All limit order equilibria from Section 4.2 remain equilibria once volume


Regarding equilibria involving volume orders, the same re�nement is made as was in-

troduced for candidate equilibria involving iceberg orders and discretionary orders, namely

that only �proper�volume order equilibria are considered. In these proper volume or-der equilibria, volume orders are submitted with positive probability, and at least one player

does strictly better than in the corresponding limit order equilibrium (the corresponding

equilibrium means the limit order equilibrium from Section 4.2 for the same parameters).

There are two general scenarios. The �rst scenario comprises cases in which buyerseither separate on the visible part of the order that they submit, or pool onto both the

visible and the hidden part of the order. In these cases, upon seeing the (visible) part of

the order posted to the book, sellers correctly identify the buyers�type and anticipate any

hidden volume. As a consequence, the fact that volume is hidden is not relevant: buyers

could instead have displayed the hidden volume without further consequences, if it were

possible to do this in the limit order book. In fact, volume orders with a hidden price of

pvoli < p1 could be expressed with a simple demand schedule, volume orders with a hidden

price of pvoli > p1 would have to be expressed with a supply schedule (which is not something

that can currently be done in exchanges).

In the second scenario, buyers pool onto orders whose visible parts are identical, and

either only one buyer submits a hidden price, or each buyer submits a di¤erent hidden price.

If so, sellers cannot perfectly predict what (if any) volume may be hidden at a given price.

Instead, sellers form beliefs about the probability that a unit is hidden at any given price

in equilibrium. These beliefs are consistent with the buyers�equilibrium strategies and the

sellers�prior beliefs about the buyer types.

Let us �rst consider the equilibria in the �rst scenario above. The following proposition

holds.

83

Proposition 4 Equilibria exist in which volume orders are used and buyers either separateon the visible order or pool onto the entirety of the order (that is, its visible and hidden

parts).

The appendix contains numerical examples. These equilibria are mentioned mainly for

completeness; they are not very interesting in that the buyers may as well display the hidden

units, that is, submit demand schedules with simple limit orders (if the hidden prices are

lower than the visible prices) or supply schedules (if the hidden prices are higher than the

visible prices and it were possible to submit supply schedules on the exchange).

What is relevant about these equilibria, though, is that they illustrate that it may

sometimes be optimal for buyers to submit something resembling a supply schedule. As

discussed in Section 4.6, which explores the mechanism design approach to this game, the

optimal buyer contract does in fact sometimes look like a supply schedule. Note that given

the current private values setting and the fact that there are only two units, the ability

to bundle units does not give the buyer more abilities than the use of a supply schedule

would. Bundling will be more important later, though, in Section 5.5, when values are

interdependent.

The following proposition characterizes all equilibria in the second scenario.

Proposition 5 Equilibria involving volume orders in which buyers pool onto the visible partof the order and submit di¤erent (if any) hidden parts can be categorized into three groups:

i) Equilibria in which buyers would be indi¤erent to displaying the entire order by submitting

a demand schedule (class G), or a supply schedule (classes I, J, Kb) if it were possible to

submit a supply schedule to the limit order book.

ii) Equilibria that replicate the discretionary equilibria of class B or C (in that they have

almost identical orders and identical expected trades, and thus lead to identical allocations).

iii) Equilibria that display some non-monotonicity.

The appendix contains a detailed proof and numerical examples. An outline of the proof

is provided here, as well as the intuition for the di¤erent equilibria in each of the groups.

In order to �nd all the possible equilibria, it is �rst useful to distinguish between two cases:

equilibria in which buyers pool onto a visible part, but only one buyer submits a volume

order and hides at least one price/unit, and equilibria in which buyers pool onto a visible

order and both buyers submit volume orders whose hidden parts di¤er.

In the �rst case, one can next distinguish equilibria in which there are one or two visible

units and either one unit is hidden at a price above the �rst visible unit, one unit is hidden at

a price below the �rst visible unit, two units are hidden with a price above the visible units

(if two units are hidden at a price below the �rst visible unit, the second unit would never

execute), or, �nally, two units are hidden at a price above the visible units and a volume

condition is imposed (specifying that the hidden units have to execute as a bundle). In all

84

of these possible equilibria, valuations for the buyer imply that the large buyer must be the

one hiding units and thus achieving a higher execution probability than the small buyer

who only submits the displayed order. As for seller responses, for a given set of visible buyer

orders, there may be a number of equilibria that di¤er depending on the seller�s responses

(and, consequently, by the speci�c equilibrium value of the hidden price or prices). Note

that seller valuations also imply that in any candidate equilibrium, it must be the large

seller that sells a higher quantity and at a lower price, rather than the small seller.

In the second case, both buyers submit volume orders (that is, orders that have hidden

prices). Here, one can distinguish equilibria in which there is one visible unit and either

both buyers each hide one unit (which for each buyer may be priced higher or lower than

the visible unit), or one buyer hides one unit and the other hides two.

The proof provided in the appendix looks at each candidate equilibrium class, excludes

all those that cannot exist given the private values setting, and provides numerical examples

for those equilibrium classes that do exist.

The paragraphs to follow present the intuition for the proposition. Volume orders are a

generalization or combination of iceberg and discretionary orders, but provide more freedom

as to how units can be hidden (for example with bundling conditions imposed), and at what

price. In the private values setting, there are no iceberg equilibria, and, consistent with that,

there are no equilibria involving volume orders that resemble an iceberg order equilibrium.

In contrast, when values are interdependent, volume order equilibria are able to replicate

iceberg order equilibria (see Section 5.5).

In equilibria of group i), only the large buyer hides units. There are four classes of

equilibria here; three of those classes, G, J and Kb, are illustrated below.

class G

Q

Equilibrium orders:Small buyer submits a visible order at Hp .Large buyer submits a volume order

),( LH ppvol , thus hiding a price Lp .Small seller submits a limit order at Hp .Large sellers submits a two unit limit order at Lp ,which executes at Hp , Lp .

Hp

Lp

P

Figure 9: Volume Order Equilibrium Class G.

In all equilibria of group i), the large buyer would be indi¤erent between hiding and

�e¤ectively displaying�the units (thus revealing his type), if it were possible to do so in the

limit order book. Speci�cally, in class G, the large buyer would be indi¤erent to submitting

a buy limit order priced at (pH ; pL), in class J , he would be indi¤erent to submitting a buy

order that is a supply schedule, (pL; pH), if the book allowed this. In class Kb, he would

85

class J

Q


),( HL ppvol , thus hiding a price Hp .Both seller types submit a volume order ),( HL ppvol ,which executes at Lp , Hp .

Hp

Lp

P

Figure 10: Volume Order Equilibrium Class J.

class Kb

Q


]),[,( HHL pppvol , thus hiding two prices at Hp .Small seller submits a limit order at Lp .Large seller submits a volume order ]),[,( HHL pppvol ,which executes at Lp or Hp , Hp .

Hp

Lp

P

Figure 11: Volume Order Equilibrium Class Kb.

be indi¤erent to submitting two orders, one at pL and a bundle at (pH ; pH)�again, if this

were possible in a limit order book. (Class I is similar to class Kb and is in the appendix.)









discriminate. In addition, the fact that the bundle would be hidden, has advantages in

practice. Buyers may be reluctant to display a larger quantity as part of a supply schedule

for fear of being front run. Posted best prices would have to be conditional on the size of a

responding order (that is, quantity the seller plans to sell), implying that the book would be

more di¢ cult to interpret for traders. Moreover, calculating those best prices would involve

solving a combinatorial problem. In contrast, there is little information leakage with hidden

bundles, and a simple price and time priority rule for any hidden quantities in general ,

which makes it easy to determine the order of execution against any eligible incoming sell

86

orders.

It remains to mention that even in the private values case, the large buyer may not

always be indi¤erent between displaying and hiding units, as, for example, in equilibrium

classes of the next group.

For the equilibria in group ii), again only the large buyer hides units. In this case, though,

the large buyer would not want to display any hidden units. Instead, the equilibria resemble

the discretionary order equilibria of classes B and C in Section 4.4. While execution rules

are di¤erent for volume orders than for discretionary orders, the equilibria in this group

have the same expected trades (type by type) and thus the same intuition: hiding units

allows the large buyer to separate seller types. In doing so, the large buyer can achieve

lower average prices, paying relatively high prices only when the seller type is small, and

relatively lower prices otherwise. Below is an illustration for equilibria B and C, including

the seller responses.

Equilibrium orders:Small buyer submits a two unit order at Lp .Large buyer submits a volume order )),,(( HLL pppvol ,that is, two units visible at Lp and one unit hidden at Hp .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a two unit limit order at Lp ,which executes at Lp , Lp .

class B

LpHp

Q

P

Figure 12: Volume Order Equilibrium Class B.

Equilibrium orders:Small buyer submits a two unit order at Lp .Large buyer submits a volume order ]),[),,(( HHLL ppppvol ,that is, two units visible at Lp and a hidden bundle of two units at Hp .Small seller submit a two unit limit order at Hp .Large seller submits a two unit limit order at Lp ,which executes at Lp , Lp .

class CHp

Lp

P

Q

Figure 13: Volume Order Equilibrium Class C.

For the equilibria that replicate the discretionary equilibria of class C, note that with

volume orders there would be a larger set of parameter combinations � for which these

equilibria exist. To see this point, recall that in class C buyers pool onto a visible two-unit

order and the large buyer also hides two units at a hidden price, which is higher than the

87

visible prices. The small seller then sells two units at the hidden price and the large seller

sells two units at the (lower) visible prices. When prices are hidden with volume orders, the

large buyer has the ability to introduce a volume condition (that is, bundling) and specify

that execution at the hidden price may happen only if the two hidden units are traded at

once. The ability to bundle weakens the incentive constraint for sellers by excluding one

possibly attractive o¤-the-equilibrium-path deviation available with discretionary orders:

to sell one unit at the hidden price. Because this deviation is excluded, the large buyer

can reduce the hidden price at which he buys two units from the small seller, compared to

what he would have to o¤er in a discretionary order equilibrium of class C (as discretionary

orders do not allow for bundling).

Group iii) contains one equilibrium class that displays non-monotonicities in that the

large buyer type does not trade more units than the small buyer type with both the small

and the large seller (and similarly, the large seller type does not trade more than the small

seller type with both buyer types). Large types (buyer or seller) trade more than small

types in expectation, though. Moreover, one buyer type trades one unit with, say, the small

seller type and one unit with the large seller type, and the other buyer type trades one unit

with the large seller and two units with the small seller. The details of the equilibrium class

description are found in the appendix; the class is mentioned here mainly for completeness.

Finally, note that much as was the case for discretionary orders in Section 4.4, there

generally are multiple equilibria involving volume orders for given parameters �. Theargument for this is that, as in the previous section, within each possible volume order

equilibrium class there generally is a continuum of hidden prices that can be part of an

equilibrium for a given set of visible equilibrium prices (again, this is because payo¤s are

linear in the hidden prices, so that individual rationality and incentive constraints simply

impose lower and upper bounds on what the hidden prices may be). In addition, as was

the case for the discretionary order section, for a given � there may be equilibria in more

than one equilibrium class (so, for example, an equilibrium of class B and one of class C

may exist for a given �).


This section will discuss the e¤ects that the introduction of volume orders has on trading

volume and transparency.


tionary orders leads to an increase or a decrease in expected trading volume.

The appendix contains a numerical example. Intuitively, much as in the case of discre-

tionary orders, the introduction of volume orders provides buyers with a potentially less

costly alternative to attain a given execution probability. Volume may be reduced following

88

the introduction of volume orders if, for example, the equilibrium that is analogue to the

discretionary equilibrium of class B (group ii above) is played when previously the optimal

strategy for the large buyer was to submit a limit order with an expected trade volume of

2.

Regarding transparency, a corollary similar to that for discretionary orders holds:

Corollary 6 Volume and transparencyi) It is possible that post- (and pre-) trade transparency is lower for the volume order equi-

librium than for the corresponding limit order equilibrium, and yet trading volume is higher.

ii) ) It is possible to have the same trading volume in the volume order equilibrium as in

the corresponding limit order equilibrium, with any remaining uncertainty being about the

buyer in the volume order equilibrium and about the seller in the limit order equilibrium.

The appendix contains numerical examples. The corollary holds because the equilibria

of group ii) involving volume orders are analogous to the discretionary order equilibria of

classes B and C in Section 4.4. The intuition and conclusion about the potential bene�ts of

volume orders in practice is also the same as for discretionary orders (presented in Section

4.4.2).

4.6 Buyer-Optimal Mechanisms

This section presents the mechanism design problem corresponding to the games in Sections

4.2 through 4.5. In particular, a principal-agent game is considered in which the buyer is

the principal and the seller is the agent. That is, instead of looking at the optimal orders

buyers would submit to an order book, this section looks at the buyer-optimal mechanism

to propose to the sellers.

The physical setup for the game, as well as the formal setup, has been described in

Section 4.1.. The current section consists of two parts. The �rst part, 4.6.1, presents

the setup for the principal-agent game; the second part, 4.6.2, includes an analysis of the

optimal contracts. In particular, the latter section investigates whether it is possible to

replicate the allocation of the buyer-optimal mechanism as equilibrium of games in which

buyers and sellers submit appropriate orders (that is, orders such that the resulting trades

�prices and quantities� lead to an equilibrium allocation that is identical to that of the

optimal mechanism).

4.6.1 Principal-Agent-Game Setup

The buyer and seller play a principal-agent game similar to the one in Maskin and Tirole�s

paper on private values69. In particular, the buyer is the principal proposing a contract

(that is, mechanism) to the seller, who is the agent (the words contract and mechanism

69See Maskin and Tirole (1990).

89

will be used interchangeably). Note that at the outset, all parameters in � are common

knowledge; buyer and seller types, however, are unknown. As a result, the general setup

for the mechanism design analysis is one of two-sided hidden information.

The following paragraphs explain how the contracts work, as well as describe the timing

of the principal-agent game.

Principal-Agent-Game timing In the �rst stage of the principal-agent game, thebuyer proposes a contract/mechanism. A contract speci�es: a) a set of possible actions for

the buyers and sellers, and b) for each pair of actions by the buyer and the seller, a pair

(i; j), a measure on a set of outcomes. An �outcome�here is a quantity of units to be traded

as well as the price(s) paid by the buyer to the seller (it also includes a possible transfer

whenever zero units are traded).

Attention is restricted to non-random mechanisms, that is, non-random outcomes for a

given pair of actions (thus, the measures on the set of outcomes are degenerate, with weight

1 on some outcome).

In the second stage of the game, the seller can decide whether to reject or accept thecontract. If the contract is rejected, the game ends, and both players obtain a reservation

utility.

If the contract is accepted in the second stage, it is implemented in the third stage. Inparticular, the buyer and the seller simultaneously choose from the set of actions speci�ed

in the contract (part a above), and the corresponding outcome is realized (part b above).

From now on, only direct revelation mechanisms (DRM�s) will be considered. In adirect-revelation mechanism, the action the buyer and seller can take is simply to announce

their types. If announcing the type truthfully is optimal, the DRM is said to be incentive

compatible.

Next, note that the buyer not only announces a type in the third stage of the game,

but he may also choose to reveal information about his type at the contract proposal stage

through his choice of mechanism (that is, contract proposed). If so, one would say that

buyer types separate (and there is no need for the buyer to additionally announce a type

in the third stage of the game). Otherwise, no information is revealed, and buyers are said

to pool at the contract proposal stage.

For the case of buyer types separating at the contract proposal stage, let�i = (xi; ti) denote the contract proposed by a buyer who revealed his type to be i: Here,

xi and ti are, respectively, a trade and transfers conditional on the buyer type being i.

Speci�cally, xi = (xij)j2fL;Sg = (xijk )j2fL;Sg;k2f1;2g, with the same notation introduced

in Section 4.2.2 (xijk equal to 0 or 1, depending on whether the kth unit is traded). As

for transfers, ti = (tij)j2fL;Sg, where a given tij may be positive even if the buyer-seller

pair (i; j) does not exchange any units. Note that contracts proposed by buyer i in a full

information setting will also be denoted �i, where full information means that buyer type

90

is known at the outset.

Next, consider the general case of buyer types pooling at the contract proposalstage. For the DRMs being considered, the notation is as follows: both buyer types poolonto announcing a contract � = (�i)i2fL;Sg, where �S and �L are contracts as in the

separating case (described in the previous paragraph). Buyers announce their type in the

third stage of the game, e¤ectively choosing from the menu f�S ; �Lg. Note that the outcome(xij ; tij) that is implemented still depends on the pair of buyer i and seller type j.

Optimal mechanisms, speci�cally mechanisms that are optimal for the large buyertype in that they maximize his expected payo¤, will be considered.

Payo¤s, Incentive Constraints and Equilibrium Allocations First, a notation de-

scribing players�expected payo¤s from the DRM proposed in stage one of the game must

be introduced. The notation, which is the same as that in the Maskin and Tirole papers,

applies both to the interdependent and private values setting (in the latter, the parameter

� below would be set equal to 0).

Let U ij(�nm) and V

ij (�

nm) be, respectively, the buyer�s and seller�s payo¤ when the

buyer type is i and the seller type is j, and the contract �nm is implemented because the

buyer and seller reported n and m in the third stage of the game. Speci�cally:

U ij(�nm) =

2Pk=1

vBikxnmk � tnm

V ij (�nm) = t

nm �2Pk=1

vSjkxnmk

Here, �nm = (xnm; tnm) with the notation from the previous section.

Next, beliefs were previously denoted �i(t) for buyers and �j(t) for sellers. In thissection, the time index will be dropped for convenience (it is to be deduced from the

context). Thus, let �ij now denote the probability buyer Bi assigns to the seller being of

type Sj; and let �ji be the probability seller Sj assigns to the buyer being of type Bi.

Reservation allocations are generally de�ned as � (they are the allocations implemented

in case the principal�s contract is rejected). In this game, the reservation allocation isthe no-trade allocation �0, which is assumed to yield a payo¤ of 0 for all players.

The relevant incentive constraints for the principal-agent game will next be introduced.

Incentive constraints for the principalAn allocation is incentive compatible for the principal (buyer), given beliefs �i if and

only if:PICi

Pj�ijV

ij (�

ij) �

Pj�ijV

ij (�

nj ) 8n

Note that buyers average over the unknown seller type.

Incentive constraints for the agentFor the agent/seller, there are two groups of incentive constraints: �type by type�-

constraints (which have to hold for a given buyer type) and �on average�constraints (which

have to hold as sellers average over buyer types based on their beliefs �j).

91

Given a reservation allocation �, an allocation is incentive compatible for a given buyer

type if and only if:AICij U ij(�

ij) � U ij(�im) 8j;m

IRij(�ij) U ij(�

ij) � U ij(�ij) 8j

For some beliefs �j over the buyer type distribution, the sellers �averaged�constraints

are:AICj(�j)

Pi�jiU

ij(�

ij) �

Pi�jiU

ij(�

im) 8j;m

IRij(�;�j)Pi�jiU

ij(�

ij) �

Pi�jiU

ij(�

ij) 8j

In order to describe the equilibrium contracts, we need to �rst calculate what the buyer�s

payo¤ is from his best deviation. Clearly, any equilibrium payo¤ for a buyer must be higher

than the payo¤ from that best deviation. Unlike for the seller, a buyer�s outside option is not

what he would receive in the no-trade allocation. Instead, buyers could separate, proposing

contracts that get them positive payo¤s while revealing their types. This is captured in the

following de�nition of the Rothschild-Stiglitz-Wilson (RSW) allocation:

De�nition 1 An allocation b�(�0) is RSW relative to �0 if and only if for all i, b�i(�0) = �i,where �i solves:

max�

Pj�ijV

ij (�

ij)

subject to PICn 8n; AICnj and IRnj (�0) 8n; j.

The RSW allocation is the least-cost separating allocation. One needs to �rst solve for

the full information optimal contract �L for the large buyer given the seller constraints.

Second, one can calculate the optimal contract for the small buyer, by �nding the full

information contract and adjusting the transfers such that the large buyer would not like to

deviate to it (it might be that small buyers prefer not to trade as a result). As an aside, note

the similarity between this concept and that of the intuitive criterion imposed on equilibria

in Bayesian games.

With this, the equilibrium allocations can now be de�ned.

De�nition 2 Let b�(�0) be the RSW allocation. Then the set of equilibrium allocationsfor the contract proposal game is the set of allocations � that satisfy PICi 8i; AICj(�j)and IRj(�j; �0) and that weakly Pareto dominate the RSW allocation, that is:Pj�ijV

ij (�

ij) �

Pj�ijV

ij (b�ij(�0)) 8i

Note that as buyers pool onto the equilibrium contract �, the appropriate seller incentive

constraints are those that average over the unknown buyer types. Note that for a given

RSW allocation there is likely to be a multiplicity of equilibria. As stated above, attention

will be restricted to the equilibria (that is, optimal mechanisms) that are optimal for the

large buyer type; in particular, all the numerical examples given fall into that category.

92

4.6.2 Characterization of Optimal Mechanisms

This section analyzes the buyer-optimal mechanisms for the model setup. Since there is

uncertainty about both the buyer and the seller type, the game is one of two-sided imperfect

information. As a result, the general structure of mechanisms proposed by the buyers

consists of a contract � = (�S ; �L), that includes the mechanisms �S and �L proposed by

each buyer type. The next proposition, though, essentially states that the calculation of the

buyer-optimal mechanisms simpli�es due to the private values framework in this section, so

that it su¢ ces to calculate the optimal separating contracts for each buyer type.

Proposition 6 In the private values setting, there is no value to pooling. Speci�cally,

buyers can never achieve a strictly higher payo¤ by pooling onto a contract � = (�S ; �L)

than they would by submitting the optimal separating contract �i = (�iS ; �iL) i 2 fL; Sg, asin the full information case.

First, note that the result of the proposition is not surprising; Maskin and Tirole70 show

in their paper on private values that, for a similar model setup, the optimal mechanism

with two-sided private information about types and quasi-linear utilities simpli�es to the

optimal mechanism with full information.

As for the proof of the proposition here, note that it has a number of steps, is construc-

tive, and that, as expected, it has similarities to the proof presented by Maskin and Tirole71.

The proof shows that, for any given optimal pooling contract � = (�S ; �L), one can

construct a set of separating contracts such that one of two cases holds. Either the payo¤s

of both buyers submitting the separating contracts are equal to the payo¤s in the pooling

equilibrium, or at least one buyer does strictly better submitting a separating contract, im-

plying that the pooling contract cannot have been optimal (that is, part of an equilibrium).

More precisely, for any optimal pooling contract the proof �rst shows that the small

seller�s individual rationality constraint and the large seller�s incentive constraint must be

binding. Using this result, it is possible to de�ne a set of separating contracts. The separat-

ing contracts are such that they implement the same allocation as under pooling, but with

di¤erent transfers. The transfers can be chosen in a way that most seller constraints will

be automatically satis�ed. Note that seller constraints now have to be satis�ed conditional

on the buyer type, not in expectation (so there are, for example, two individual rationality

constraints for the small seller, one per separating contract). It is important to note that

�nding the separating contracts is possible for two reasons: one is the linearity of the payo¤

functions; the other is the private values assumption in this section. In the case of inter-

dependent values, the seller�s payo¤s depend not only on the allocation and transfers, but

70See Maskin and Tirole (1990).71See page 403-403 of Maskin and Tirole (1990).

93

also on the seller�s belief about the buyer�s type. Section 5 shows that, in contrast to the

private values case here, there is value to pooling in the case of interdependent values.

Next, an analysis is provided of whether it is possible to implement the optimal allocation

from the mechanism design problem with limit orders, that is, whether it is possible to �nd

equilibria in the game of Section 4.2, in which the orders submitted lead to traded quantities

and prices that replicate the allocation of the buyer-optimal mechanism.

Proposition 7 Payo¤s in the optimal (separating) mechanism submitted by the principals

in the private values case cannot always be implemented with limit orders. Moreover, this

will always be true for the case in which the transfers in the optimal mechanism resemble

a supply schedule (such that the implied price on the second unit is higher than that on the

�rst).

The appendix contains numerical examples. Intuitively, in this proposition, one of two

things may happen. First, the optimal mechanism may have a payment schedule that can

only be implemented with supply schedules. This would be the case if, for example, the

buyer wanted to buy one unit from the small seller and two units from the large seller and

the transfers tiS and tiL were such that tiL=2 > tiS .

Second, it could be that the buyer does not want to buy anything from the small seller

(thus, tiS = 0) and wants to buy two units from the large seller. In order to implement

the optimal mechanism with limit orders, the buyer would have to submit a two-unit limit

order, priced at p1, p2 (with p1 + p2 = tLL). In the mechanism design framework, the

seller�s options are to reject the mechanism or to accept it and report one of two types. In

this speci�c case, the seller�s options are either to not trade or to sell both units at p1+ p2.

In the principal-agent game, though, the seller can also decide to deviate and sell only one

unit, at p1. In order to make this deviation least attractive, the buyer will choose to set

p1 = p2 = tLL=2. As is shown in the appendix, it may be that even with this choice of

prices, sellers will prefer to deviate to selling at p1, so that the optimal mechanism cannot

be implemented with limit orders.

4.6.3 Optimal Mechanisms Allocations as Equilibrium Outcomes

This section brie�y states what order types may be needed to replicate the allocation of

any optimal mechanism as an equilibrium of the game in which buyers and sellers submit

orders as in Sections 4.2 through 4.5.

Proposition 8 Optimal (separating) mechanisms in the private values case are either sup-ply or demand schedules of buy orders.

See the appendix for a proof. The conclusion of this proposition is twofold. First,

whenever the optimal mechanism entails the submission of a demand schedule, then limit

94

orders allow the buyers to replicate the optimal mechanism. On the other hand, when

the mechanism includes the submission of a supply schedule, then limit orders will not be

su¢ cient, as was mentioned in the previous proposition. Volume orders, in turn, can be used

to replicate a supply schedule, namely by submitting a displayed one-unit order together

with two hidden units at a higher price and a bundling condition, so that the hidden units

can only execute together. There is a problem in using volume orders to submit supply

schedules, though, namely that sellers would have to know or believe in equilibrium that

some units are hidden. That is, depending on the sellers�beliefs, there may be multiple

equilibria for a given visible order. To solve that problem, buyers would have to announce

that some units are hidden, both if the buyers were pooling on the visible part of the

volume order, or if the buyers separated on the visible part of the volume order. On the

other hand, note that in the interdependent values framework of Section 5, hiding units will

have advantages (because it allows the large buyer to conceal his type).

Order books do not currently allow for displaying bundles or blocks of shares, likely

because it would be di¢ cult to �nd a way to display these bundled shares together with

the other orders in the book (on which no bundling condition has been imposed by the

trader submitting the orders). One way to allow for displaying of bundled shares would

be to show market participants an order book that is conditional on what size order they

are considering entering into the book. That is, sellers would face a family of demand

schedules, depending on what size sell order they are considering submitting to the book.

Computationally, this seems unfeasible in practice, though, especially given the prevalence

of high-frequency trading. In addition, it is unlikely that traders would like to display large

bundles or blocks of shares, given the degree of information leakage that would likely be

associated with doing so. That is, allowing for bundling of hidden units may be optimal,

considering that it simpli�es the exposition of the best bids and asks in the book, and that

traders may prefer hiding blocks of shares over displaying them.

95

5 Interdependent Values

This section contains the theoretical analysis for the case in which values are interdependent.

The section has six parts. Section 5.1 presents the amendments to the model setup from

the private values case that need to be made to capture interdependent values. Section 5.2

solves the basic game, in which buyers can submit only limit orders to the book, and sellers

can respond with limit and market orders.

Sections 5.3, 5.4 and 5.5, analyze the games in which the set of admissible order types

additionally contains iceberg orders, discretionary orders and volume orders, respectively.

The last Section, 5.6 contains the analysis of the buyer-optimal mechanism for the

interdependent values case.

5.1 Model Setup

The model setup for the case when values are interdependent is very similar to the setup

for private values, the di¤erence being with respect to the buyer�s and seller�s valuations

for each unit. Speci�cally, buyer (seller) valuations for a given unit of the asset are denotedevBik and ~vBjk (again, indexes i; j refer to the type and k refers to the unit). For any pairof buyer Bi and seller Sj, valuations are interdependent in that the following holds:evBik;j = vBik + � � vSjkevSjk;i = vSjk + � � vBik

The vBik and vSjk are common knowledge and have the same structure as in the private

values setting; the parameter � � (0; 1] is also common knowledge and represents the strength

of the interdependence.

By linking buyer and seller valuations, the interdependent values setting tries to capture

a common values characteristic of �nancial assets. In the case of pure common values, the

possibly unknown value of the asset would be the same for all traders. This assumption

could be motivated by the idea of a resale value that is identical for all traders. Or, it could

be motivated by the belief that the correct value of an asset can be found by aggregating the

information traders jointly have about the value. Interdependent values capture a common

value e¤ect, but still allow for di¤erences in valuations among buyers and sellers that could

arise from, say, di¤erences in inventory, hedging needs, or liquidity constraints.

Next, the de�nitions and notations from the private values case carry over for the players�

beliefs, strategies, and individual and expected trades. What need to be amended are the

players�payo¤s.

Let U ij(xij ; pij) and U ij(x

ij ; pij) denote the payo¤s for the buyer Bi and seller Sj whenthey engage in the trade (xij ; pij). For the general case of interdependent values, these read

as follows (for private values, set � = 0):

U ij(xij ; pij) =

2Pk=1

�vBik + � � vSjk � pijk

�xijk

96

V ij (xij ; pij) =

2Pk=1

�pijk � (vSjk + � � vBik)

�xijk

Next, expected payo¤s at time t are the payo¤s a buyer (seller) anticipates at timet if he does not know the seller�s (buyer�s) type, but knows the individual trade xij that

will occur conditional on his and the other player�s type. Buyers would thus anticipate

(xBi; pBi) = (xij ; pij)j2fL;Sg and sellers (xSj ; pSj) = (xij ; pij)i2fL;Sg. Using the buyer�s

(seller�s) time t beliefs, expected payo¤s at time t are then calculated in the standard

way:72

U i(t; xBi; pBi) =P

j2fL;Sg

Pk2f1;2g

�vBik + � � vSjk � pijk

�xijk

!�ij(t)

Vj(t; xSj ; pSj) =

Pi2fL;Sg

Pk2f1;2g

�pijk � (vSjk + � � vBik)

�xijk

!�ij(t)

5.2 Limit Orders Only

In this variation of the buyer-seller game, buyers submit a buy limit order at time t = 0, and

sellers respond with a sell limit order or a market order at t = 1. The proposition and obser-

vations below concern the main characteristics of the equilibria; the detailed construction

of equilibria can be found in the appendix.

First, note that with interdependent values, large buyers have an incentive to hide their

types. From Section 4.2, we know that this is not the case with private values. As a result,

the optimal strategy for a buyer of a given type will depend on the strategy of the other

buyer�s type and the sellers�beliefs about the buyers. For a given parameter combination,

there may be a multiplicity of equilibria in the case of interdependent values (in contrast

to the case of private values), as speci�ed in the following observation, which is illustrated

with a numerical example in the appendix.

In this game, separating equilibria are unique. Note that whenever there is a multi-

plicity of equilibria, equilibria that are dominated for both buyers are not considered (an

equilibrium is referred to as dominated here if both buyers prefer a di¤erent equilibrium in

the set of multiple equilibria, though both buyer types do not have to prefer the same other

equilibrium). Nevertheless, it may be that there is more than one pooling equilibrium for a

given parameter combination �, as the following observation states.

Observation 4 For a given parameter combination �, there may exist more than one pool-ing equilibrium (that is, an equilibrium in which buyer types pool onto a visible buy order).

72Alternatively, one could write U ij(xij ; pij) =

2Pk=1

�fvalBik;j � pijk �xijk and U i(t; (xij ; pij)j2fL;Sg) =

Pj2fL;Sg

Pk2f1;2g

�fvalBik;j � pijk �xijk!�ij(t). (And similarly for buyers.)

The notation above was chosen so that the connection to private values would be clearer.

97

Second, as in the private values case, the expected payo¤ functions for the buyer and

the seller are supermodular; speci�cally, U i(t; (xij ; pij)j2fL;Sg) and Vj(t; (xij ; pij)i2fL;Sg)

are supermodular in the expected equilibrium quantities and the valuations. As a result,

all equilibria exhibit the same monotonicity with regard to trade volume as in the private

values case: large types trade more in equilibrium than small types do. More generally, the

exact analogue to Proposition 1 holds, as stated below:

Proposition 9 i) In any equilibrium of the game, large buyers (sellers) always trade weaklymore volume than small buyers (sellers). Consequently, trade volume is increasing in the

probability � that a buyer is large.

ii) On the other hand, expected trade volume may not always be increasing in the probability

� that the seller is large.

See the appendix for a proof. The intuition for this proposition is similar to that for

the analogous proposition in the private values case. Regarding transparency, the exact

analogue of observation 1 from the private values case again holds, and is stated here

simply to serve as a benchmark.

Observation 5 Pre-trade and post-trade transparency.i) Whether the buyer type is revealed pre-trade depends on whether there happens to be

pooling or separation in equilibrium.

ii) Transparency may not increase post-trade. If it does, then trades reveal at most the seller

type.

A given buyer type�s optimal strategy is independent of the other buyer, which implies

i). Moreover, for a given posted buy order, seller strategies depend only on the seller�s

(reservation) values on the units, which implies ii).

5.3 Iceberg orders

In this game, buyers submit limit or iceberg orders and sellers respond with either limit or

market orders (though, as in Section 4.3, market orders would be su¢ cient). Iceberg orders

consist of a visible order for one or two units and a hidden unit at the �rst visible price.

The section is divided into two parts. The �rst part, 5.3.1, describes the equilibria; the


5.3.1 Equilibria

First note that as in the private values case, the exact analogue of Observation 2 holds, due

to the assumptions about beliefs o¤ the equilibrium path that were introduced in 4.3. The

Observation is stated below, for completeness.

98

Observation 6 All limit order equilibria from Section 4.2 remain equilibria once iceberg


The following proposition provides a simple description of the equilibria in which iceberg

orders are used, by focusing on the orders submitted by each type (see the appendix for a

detailed equilibrium strategy construction, along with numerical examples):

Proposition 10 There are two classes of equilibria (named E and F) in which iceberg

orders are used. For a given set of parameters �, an equilibrium of both classes E and F

may exist. As for the equilibrium strategies/orders:

In class E, both buyer types pool onto a visible buy order priced at p1, and the large buyer

type hides a second unit at this price. The small seller type submits a one-unit sell limit

order at p1, and large seller type submits a two-unit sell limit order at p1.

In class F, buyers pool onto a visible order priced at (p1; p2), and the large buyer type hides

a unit at p1. The small seller type submits a two-unit sell limit order at (p1; p1), and large

seller type submits a two-unit market order.

Large types always trade more in equilibrium than the corresponding small type.

Payo¤s for the large buyer type are strictly higher than they would be in the corresponding

limit order equilibrium, and payo¤s for the small buyer type can be increased or decreased.

The expected trade volume for each player type is as follows:type/expected volume E F

large buyer 1 + � 2

small buyer 1 1 + �

large seller 1 + � 2

small seller 1 1 + a

The pictures below illustrate equilibria of classes E and F.

class E

Q

Lp

PEquilibrium orders:Small buyer submits a visible order at Lp .Large buyer submits an iceberg order ))(( LL pp ,thus hiding a unit at Lp .Small seller submits a limit order at Lp ,selling the unit at Lp with certainty.Large seller submits a two unit limit order at Lp ,selling one unit at Lp , if the buyer is small and two unitsat Lp if the buyer is large.

Figure 14: Iceberg Equilibrium Class E.

99

class F

Q

Hp

Lp

PEquilibrium orders:Small buyer submits a visible order at ),( LM pp .Large buyer submits an iceberg order )),(( LMM ppp .Small seller submits a twounit limit order at Mp ,selling the unit at Mp if the buyer is largeand two units if the buyer is large.Large seller submits a two unit limit order at Lp ,receiving Mp on the first unit and either Mp or Lp on thesecond, when the buyer is large or small, respectively.

Figure 15: Iceberg Equilibrium Class F.

In the following paragraphs, the intuition for the equilibria of both classes is presented.

Regarding the �rst (visible) unit priced at p1, two conditions must hold. First, p1 has

to be higher than the sellers�reservation values, that is, the sellers�valuations. A seller�s

valuation in turn re�ects his beliefs about the buyer type (a seller�s valuation on the �rst

unit is vSj1 + � � (� � vBL1 + (1 � �) � vBS1) ). Since both buyers submit the same visibleorder, sellers will average over the buyer type according to their initial belief. Second, in

order for the small seller to submit a two unit limit order priced at p1 for both units (or

a two unit market order), p1 must exceed the small seller�s reservation value on (the �rst

and) the second unit. The seller�s valuation on the second unit is calculated conditional on

the seller type being large, and equals vSS2+� �vBL2. (This is because the second unit onlyexecutes when a unit is hidden at that price, which is true if and only if the buyer type is

large.)

Regarding a second unit that is always traded when the buyer type is large, note that

the large buyer always pays a price of p1, no matter what the seller type. Instead of pooling

onto the visible order submitted by the small buyer, the large buyer could separate and

submit a di¤erent, visible two unit order. In order to guarantee execution of the second

unit, its price would have to be p01 = vSS2 + � � vBL2 (which is the same lower bound asfor p1). If a large buyer were to submit a separating two unit order, though, the price he

would have to pay on the �rst unit would be higher than the price p1 in the iceberg order

equilibrium. This is because the seller�s reservation values would increase, re�ecting the

seller�s knowledge that the buyer type is large.

Thus, intuitively, in an iceberg order equilibrium hiding a unit is pro�table for the large

buyer because it allows him to pool onto the �rst (visible) unit with the small buyer, so

that the price for that unit is lower than if the large buyer had separated and revealed his

type.

100

As for the constraints on the valuations, iceberg order equilibria only exist when the

small seller valuations are essentially �at. In order to see how valuations may be �at, �rst

note that the small seller valuation on the �rst unit is calculated averaging over both buyer

types, while the valuation on the second unit is conditional on the buyer type being large.

Given this formal de�nition of the small seller�s valuations, it is theoretically possible that

they are not only �at, but decreasing. In particular, decreasing valuations for the small

seller would result if the vBik were su¢ ciently decreasing in k, the vSjk were not strongly

increasing in k; and the strength of the interdependence, measured by �; were su¢ ciently

strong. While it seems unrealistic in practice to think of decreasing seller valuations, another

interpretation is that sellers are not necessarily �natural� sellers, but simply traders with

some initial valuations (parametrized here by vSJk and vBik) in a market for an asset with

common value characteristics that are su¢ ciently strong. In this case, a trader�s valuations

would largely depend on what he believed (or discovered) about the valuations of the other

market participants.

5.3.2 Volume and transparency

The following two corollaries concern volume and transparency.

Corollary 7 Volume and transparency(i) The introduction of iceberg orders may lead to a reduction in the expected trading volume

and a reduction in payo¤s for the sellers and the small buyer.

(ii) It is possible to have the same amount of volume in the iceberg order equilibrium as in

the limit order equilibrium, but have more pre-trade and post-trade uncertainty about buyer

and seller types.

See the appendix for numerical examples. Intuitively, iceberg orders provide large buyers

with a potentially cheaper way to achieve a trading volume of 1 + � or 2. As a result, it

might be possible for an iceberg order equilibrium to exist when there would only have

been a separating limit order equilibrium in which the large buyer traded, say, two units

rather than 1 + �. Since buy order prices in the separating equilibrium would likely have

been higher as in the limit order equilibrium (at least when the buyer type is large), sellers�

payo¤s are reduced in the iceberg order equilibrium. Moreover, the price(s) of visible orders

submitted when there is pooling will likely be higher than with separation; thus small buyers

are also negatively a¤ected (sellers�reservation values in the iceberg order equilibrium are

higher because they are computed by averaging over both buyer types). These arguments

are explained part i) of Corollary 7.

Regarding part ii), transparency (that is, uncertainty about the buyer�s type), let�s

consider the case of an equilibrium of class E. The table below illustrates the post trade

transparency and the traded quantities (where the same notation as in the private values

analysis applies).

101


when vol(xBL) = 1 + �; vol(xBS) = 1

seller S seller L

buyer SS; S

(1)

S;L

(1)

buyer LL; S

(1)

L;L

(2)

post-trade transparency equilibrium E

seller S seller L

buyer S?; ?

(1)

?; ?

(1)

buyer L?; ?

(1)

L;L

(2)

Table 9: Post-trade transparency in limit order and iceberg order equilibrium of class E.

As can be seen above, iceberg orders allow large buyers to have their type revealed only

when two units are indeed traded. Should this characteristic carry over in practice, it would

provide another argument in favor of using iceberg orders. (In addition to the fact that

iceberg orders may allow large buyer types/traders to submit orders with higher execution

probabilities while still receiving a lower price on the visible units onto which all buyer types

pool.)

Note that in practice, ��shing� for hidden liquidity is common: traders submitting

orders with the goal of discovering hidden liquidity (that is, orders) at a given price, and

then cancelling the order after 1-2 seconds if it does not execute (that is, when no hidden

volume is present).

5.4 Discretionary Orders


and discretionary orders, while sellers may use limit orders or market orders to respond.



5.4.1 Equilibria

Most of the statements from the corresponding Section, 4.4., in the private values setting

carry over to interdependent values; such as the following observation.



The next proposition is similar to Proposition 3, but there are some di¤erences, discussed

below.

Proposition 11 There are four classes of equilibria, named A through D. For a given setof parameters �; equilibria of more than one class may exist. As for the equilibrium strate-

gies (orders):

102

In class A, both buyers pool onto a visible buy order priced at p1, and the large buyer hides

a higher discretionary price pD1 . Small sellers submit a one unit sell limit order at pD1 , and

large sellers submit a one unit market order.

In classes B and C, buyers pool onto a visible two-unit limit order priced at p1 = p2.

The large buyer hides a price on the �rst unit at pH1 (in class B); and on both units at

pH1 = pH2 (in class C).

Large sellers submit a two-unit market order in both case B and C.

Small sellers submit a one unit sell limit order at the hidden price pH1 in case B, and a two

unit sell limit order at pH1 = pH2 in case C.

In class D, both buyers pool onto a visible two-unit limit order priced at (p1; p2); and the

large buyer hides a higher discretionary price pD2 (� p1) on the second unit. Small sellerssubmit a two-unit sell limit order at pD2 , and large sellers submit a two-unit market order.

Large types always trade more in equilibrium than the corresponding small types.

Payo¤s for the large buyer type are strictly higher than they would be in the corresponding

limit order equilibrium; payo¤s for the small buyer and the sellers may be decreased.

Expected trade volume for each player type is as follows:

type/expected volume A B C D

large buyer 1 1 + � 2 2

small buyer � 2� 2� 1 + �

large seller 1 2 2 2

small seller � � 2� 1 + �

Table 10: Player�s Expected Trade Volume in Discretionary Order Equilibrium Classes A,

B, C and D.

See the appendix for a proof including numerical examples. The picture below illustrate

the buyer orders in the equilibria of classes A through D. Equilibrium class D is illustrated

later in more detail.

class A

Q

P

LpHp

class Cclass B

Q Q

Lp LpHp HpP P class D

Q

LpHpP

Figure 16: Buyer Orders in Discretionary Equilibrium Classes A, B, C, and D.

As for the intuition for the equilibria, there are two e¤ects to note. First, as in the private

values case, discretionary orders allow buyers to separate seller types. In equilibrium, large

103

sellers sell (at least one unit) at the high, discretionary price. The discretionary price pD1exceeds the reservation value for the small seller, where the reservation value is calculated

conditional on the buyer type being large, so pDk > vSSk+� � vBLk. As in the private valuescase, buyers may �nd it worthwhile to submit a discretionary order (paying a relatively

high price when trading with small sellers), because it allows them to separate sellers and

receive a lower price on the displayed units (when trading against large sellers).

Second, prices on the displayed units are lower than they would be if the buyer were

known to be large. This is because in equilibrium, small and large buyers pool, submitting

orders with identical visible parts. Compared to the private values case, pooling thus

provides an additional bene�t of discretionary orders (for the large buyer types) when

valuations are interdependent.

Equilibrium A exists here and not in the private values scenario because the small buyer

cannot simply submit his optimal limit order and pay a price that is equal to the large

seller�s reservation value conditional on the buyer being small. Submitting such an order is

no longer possible, because it might also be preferred by the large buyer. As a consequence,

the displayed price p1 may leave the large seller some rent (unlike in the private values case),

so that it may be possible to separate seller types with a combination of prices p1 and pD1(where pD1 is above the reservation value of the small seller, conditional on the buyer type

being large).

Regarding equilibrium D, the same argument that applied to the �rst and only displayed

unit in equilibrium A can be applied to the second unit. That is, in equilibrium, the

displayed price p2 leaves the large seller with some rent (because the small buyer cannot

submit an order at a lower price without the large buyer preferring that order as well). As

a consequence, it may be possible to �nd a discretionary price pD22 such that the small seller

is willing to sell at pD2 , while the large seller prefers selling at a lower price p2, though with

a higher execution probability (which would be 1 rather than �).

class D

Q

LpHpP

Equilibrium orders:Small buyer submits visible order at ),( LM pp .Large buyer submits discretionary order ))(,( MLM ppp .Small seller submits a twounit limit order at Mp ,receiving Mp on the first unit andselling a second at Mp if the buyer is large.Large seller submits a twounit limit order at Lp ,selling the two units at ),( LM pp .

Figure 17: Discretionary Order Equilibrium Class D.

104

It is important to note that while equilibria in class D have the same expected trade

volume per buyer and seller type as equilibria in class F with iceberg orders, there are

signi�cant di¤erences between these equilibrium classes, both regarding the prices paid in

equilibrium as well as regarding the intuition for why these equilibria work. The di¤erences

stem from di¤erences in execution rules between iceberg and discretionary orders. In par-

ticular, incoming market orders do execute against hidden units in iceberg orders, whereas

they do not execute against discretionary orders. Thus, discretionary orders only execute

against limit orders with prices above the visible price and below the higher discretionary

price. Moreover, any execution happens at the price of the incoming limit order. Thus,

intuitively, units that have discretionary prices only execute against incoming (limit) orders

if they would not execute at visible prices. Consequently, in an equilibrium of class F, the

large buyer always pays the price p1 on the second unit, no matter what the seller type.

In an equilibrium of class D, the large buyer pays the lower price p2 whenever the seller

type is large, and the higher price pD2 when the seller type is small. In conclusion, in the

iceberg equilibrium F, hiding a second unit is pro�table because it reduces the price a large

buyer has to pay on the �rst unit. In the equilibria of class D, hiding the discretionary price

allows the large buyer to separate sellers (and is possible because large sellers receive a rent

at visible prices because of the interdependent values framework).


Regarding transparency, volume, and the relation between the two, the results from the

private values section hold, as illustrated in the following Corollaries 8, 9, and 10, which

are, respectively, the analogues to Corollaries 2, 3, and 4 in the private values section. (The

conclusion about the potential bene�ts of discretionary orders in practice is also the same

as in the private values setting in Section 4.4.2.)

Corollary 8 In all equilibria, any remaining post-trade uncertainty is about the buyer type.

Corollary 9 It is possible that the introduction of discretionary orders leads to a reductionin expected trade volume.

Corollary 10 i) It is possible that compared to the corresponding limit order equilibrium,post- (and pre-) trade transparency is lower in the discretionary order equilibrium and yet

trading volume is higher.

ii) It is possible to have the same amount of volume in the discretionary order equilibrium as

in the corresponding limit order equilibrium, with any remaining uncertainty being about the

buyer in the discretionary equilibrium and about the seller for the limit order equilibrium.

Generally, the intuition from the corollaries in the private values section carries over.

With interdependent values, there are two new classes of equilibria that did not exist with

105

private values, namely the equilibria of classes A and D. The intuition for these classes is

similar to B and C: pooling allows the buyer to obtain a better price on the �rst visible

unit. The reason that the equilibria of class A and D did not previously exist was that then

the large seller did not receive a rent when trading the �last�unit at visible prices. (Here

�last�mean the �rst unit if only one unit is traded in total, and the second unit if two units

are traded in total.) With interdependent values, though, the large seller can receive a rent

on the visible units (onto which buyers pool) as well, which is to the detriment of the small

seller who has no way of deviating and signaling his type in equilibrium.

5.5 Volume Orders



5.5.1 Equilibria

In this game, buyers can use limit orders as well as volume orders to post to the book, and

sellers can use limit, market, or volume orders to respond (though, as in Section 4.2, market

orders and volume orders are su¢ cient). Volume orders consist of visible prices, labeled pi,

and hidden prices, labeled pvoli , which may be above or below the visible prices. Moreover,

in the case where there are two hidden prices, it must be that pvol1 = pvol2 . It is important

to note that buyers additionally have the option to explicitly specify bundling: on orders

with two hidden prices, buyers can specify that if there is execution at the hidden price,

then two units have to be sold at once.

Just as in the private values Section, 4.5, there are two scenarios. For completeness, the

description of the scenarios provided there will be restated. The �rst scenario comprises

cases in which buyers either separate on the visible part of the order that they submit, or

pool onto both the visible and the hidden part of the order. Thus, upon seeing the (visible)

part of the order posted to the book, sellers correctly identify the buyer�s type and anticipate

any hidden volume. As a consequence, the fact that volume is hidden is not relevant: buyers

could instead have displayed the hidden volume without further consequences (if it were

possible to do this given the rules for volume orders). In fact, volume orders with a hidden

price of pvoli < p1 could be expressed with a simple demand schedule, and volume orders

with a hidden price of pvoli > p1 would have to be expressed with a supply schedule (which

is not something that can currently be done in exchanges).

In the second scenario, buyers pool onto orders whose visible parts are identical and

either a) only one buyer submits a hidden price, or b) each buyer submits a di¤erent hidden

price. Thus, sellers cannot perfectly predict what (if any) volume may be hidden at a given

price. Instead, sellers form beliefs about the probability that a unit is hidden at any given

price in equilibrium. These beliefs are consistent with the buyers� equilibrium strategies

and the sellers�prior beliefs about the buyers�types.

106

Turning to the equilibria, it is �rst noted for completeness that, as in previous sections

(for example, Section 5.4), the following observation holds:



Regarding equilibria that involve volume orders, let�s begin with those that �t in the

�rst scenario.

Proposition 12 Equilibria exist in which volume orders are used and buyers either separateon the visible order or pool onto the entirety of the order (that is, its visible and hidden

parts).

The appendix contains numerical examples. As in Section 4.5, these equilibria are

mentioned mainly for completeness; they are not very interesting in that the buyers may as

well submit demand schedules with simple limit orders (if the hidden prices are below the

visible prices) or supply schedules (if the hidden prices are above the visible prices and if

it were possible to submit supply schedules on the exchange). What is relevant about this

proposition is that it illustrates the following: it may sometimes be optimal for buyers to

submit supply schedules when choosing among limit and volume orders. This is true in the

case of private values (as explained in Section 4.6) and also holds for interdependent values.

In fact, Section 5.6, which presents the mechanism design approach to this game when

values are interdependent, shows that buyer optimal-contracts may sometimes be a supply

schedule (more precisely, buyers may want to pool onto a contract or submit a separating

contract that resembles a supply schedule).

The following proposition characterizes all equilibria in the second scenario (when buyer

types pool onto the visible part), categorized into four groups for ease of exposition.

Proposition 13 Equilibria involving volume orders in which buyers pool onto the visiblepart of the order and submit di¤erent (if any) hidden parts can be categorized into the

following four groups:

i) Equilibria that include a combination of iceberg orders and either demand schedules (class

G, H) or supply schedules (classes I, J, K, and Kb).

ii) Equilibria that replicate the discretionary equilibria of class A, B, C, or D (in that they

have almost identical orders, identical expected trades, and thus lead to identical allocations).

iii) Equilibria that include screening both seller types with respect to the second unit (class

L), and that additionally have supply features (class M); equilibria that include screening

with lotteries across units (class N and O).

iv) Equilibria that display non-monotonicity.

The appendix contains the detailed de�nitions and constructions of all equilibrium

classes. The next paragraphs present the intuition behind the di¤erent equilibria.

107

Equilibria in group i) are a combination of iceberg order equilibria and supply schedules

or demand schedules. The pictures below illustrates the demand variants G and H, the

supply variants I and J, and the supply variant Kb (the letter b suggests a bundling condition

that is imposed on the hidden units).

class G

Q


),( LH ppvol , thus hiding a price Lp .Small seller submits a limit order at Hp .Large sellers submits a two unit limit order at Lp ,which executes at Hp , Lp .

Hp

Lp

P

Figure 18: Volume Order Equilibrium Class G.

class H

Q


),( LH ppvol , thus hiding a price Lp .Both seller types submit a two unit limit order at Lp .which executes at Hp , Lp .

Hp

Lp

P

Figure 19: Volume Order Equilibrium Class H.

Equilibria in this group have a feature that resembles iceberg order equilibria: in any

iceberg order equilibrium, the reason the large buyer hides a second unit is that it allows

him to pool with the small buyer on the �rst unit, and thus attain a lower price on that �rst

unit. (The large buyer does not achieve a lower price on the second, hidden unit. In fact, in

the context of interdependent values, the large buyer has to pay a price that is high enough

to induce the small seller to sell the unit, conditional on the buyer type being large.) The

equilibria of classes G through Kb display the same characteristic: the large buyer hides

the second unit in order to be able to pool onto the �rst unit with the small buyer, thus

receiving a lower price on that �rst unit. Moreover, when the second unit is hidden at a

price above that of the �rst, the equilibrium is referred to as one with supply features; when

the price is below that of the �rst unit, the equilibrium is referred to as one with demand

features.

Note that the pictures for the equilibrium pairs G, H and I, J are the same. The

108

class I

Q


),( HL ppvol , thus hiding a price Hp .Small seller submits a limit order at Lp .Large seller submits a volume order ),( HL ppvol ,which executes at Lp , Hp .

Hp

Lp

P

Figure 20: Volume Order Equilibrium Class I.

class J

Q


),( HL ppvol , thus hiding a price Hp .Both seller types submit a volume order ),( HL ppvol ,which executes at Lp , Hp .

Hp

Lp

P

Figure 21: Volume Order Equilibrium Class J.

equilibria di¤er in seller responses. For I and G, the small seller submits a one-unit order at

the hidden price pH , and the large seller submits a volume order priced at the visible and the

hidden price, vol(pM ; pH). For J and H, both sellers submit volume orders at the visible and

hidden price (vol(pM;pH) and vol(pM;pL) respectively), and again the large buyer receives

a relatively lower price on the �rst, displayed, unit by pooling with the small buyer. For

equilibria of classes K and Kb, the large seller submits a volume order at vol(pL; (pH ; pH)),

selling one unit whenever the buyer is small, and two whenever the buyer is large. The

small seller always sells one unit, which the large buyer is able to obtain at a lower price by

pooling with the small buyer.

Much as in the case of private values, supply features (and bundling of hidden units) in

practice essentially allow buyers to engage in price discrimination, while the fact that the

units are hidden mitigates information leakage (for more detail see Section 4.5.1).

The equilibria in group ii) are very similar to discretionary order equilibria, which is

why they are labeled A through D, referencing the corresponding discretionary order equi-

librium. The following pictures illustrate the initial buy orders posted in the book for the

volume order equilibria A and D (classes B and C were already illustrated in the corre-

sponding Section, 4.5, when values are private). While the pictures are identical to those

for the discretionary order equilibria, the volume order equilibria do di¤er from the discre-

109

class Kb

Q


]),[,( HHL pppvol , thus hiding two prices at Hp .Small seller submits a limit order at Lp .Large seller submits a volume order ]),[,( HHL pppvol ,which executes at Lp or Hp , Hp .

Hp

Lp

P

Figure 22: Volume Order Equilibrium Class Kb.

tionary order equilibria, because the underlying orders and execution rules are di¤erent. An

explanation of this di¤erence, using the case of equilibrium class B as an example, follows.

Equilibrium orders:Small buyer submits a limit order at Lp .Large buyer submits a volume order ),( HL ppvol ,thus hiding one unit at Hp .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a order at Lp ,which always executes at Lp .

class A

LpHp

Q

P

Figure 23: Volume Order Equilibrium Class A.

In the discretionary order equilibrium of class B, both buyers pool onto a visible two-

unit order priced at (pL; pL), and the large buyer hides one unit at a higher price pH .

The small seller then submits a one-unit sell order at the high price pH , and the large

seller sells two units at the visible prices. In contrast, for the volume order equilibrium,

the buyers pool onto a visible two-unit order (pL; pL), and the large buyer also hides a

unit at a higher price pH . As a result, the hidden unit may execute both as a single unit

(if a one-unit limit order at the hidden price is submitted) or as the second unit, at pH

(namely, if the sellers submit a volume order vol(pL; pH) with a visible price of pL and a

hidden price of pH). For volume orders, the seller responses in the equilibria of classes

A through D are such that, in terms of on-the-equilibrium-path executions, the buyers

may as well have submitted discretionary orders (at the equilibrium prices). Note, though,

that the equilibrium would not necessarily be sustained if discretionary orders were used

instead of volume orders, as incentive constraints and the set of possible o¤-equilibrium-

path deviations (and consequently, payo¤s from choosing the deviations) would be di¤erent.

110

Equilibrium orders:Small buyer submits a two unit order at Lp .Large buyer submits a volume order )),,(( HLL pppvol ,that is, two units visible at Lp and one unit hidden at Hp .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a two unit limit order at Lp ,which executes at Lp , Lp .

class B

LpHp

Q

P

Figure 24: Volume Order Equilibrium Class B.

Equilibrium orders:Small buyer submits a two unit order at Lp .Large buyer submits a volume order ]),[),,(( HHLL ppppvol ,that is, two units visible at Lp and a hidden bundle of two units at Hp .Small seller submits a two unit limit order at Hp .Large seller submits a two unit limit order at Lp ,which executes at Lp , Lp .

class CHp

Lp

P

Q

Figure 25: Volume Order Equilibrium Class C.

The set of parameter combinations �, for which equilibria of classes A through D exist for

either discretionary or volume orders, will generally di¤er.

The equilibrium of class Cb (not pictured above) is like class C, except that the large

buyer hiding the two units imposes a volume condition, indicating that these hidden units

may only execute as a bundle. Note that the bundling condition weakens the incentive

constraints of the sellers, who now have fewer possible deviations.

Finally, note that the equilibrium of class D can also be interpreted as an iceberg order

equilibrium of class F, as the picture would look identical.

Next, equilibria in group iii) are considered. The pictures below again illustrate the

buyer orders for the equilibria L and M. As can be seen, the picture for L is the same as

that for a discretionary equilibrium of class B; however, seller responses are di¤erent. In

equilibrium L, the small seller submits a volume order priced at vol(pM;pH), and the large

seller submits a two-unit limit order at the visible prices (or a two-unit market order). In

the case of a discretionary order equilibrium of class B, the small seller would have sold one

unit at the hidden price pH (and the large seller would still have sold two units at visible

prices).

Intuitively, what equilibrium L allows the large buyer type to do is separate the seller

types on the second unit: both seller types sell one unit with certainty at the �rst visible

111

Equilibrium orders:Small buyer submits a twounit order at ).,( LH ppLarge buyer submits a volume order )),,(( MLH pppvol ,thus hiding one unit at Mp .Small seller submits a twounit limit order at Mp ,receiving Hp on the first unit and selling a second unit at Mpif the buyer is large.Large seller submits a twounit limit order at Lp ,receiving Hp on the first unit, and selling the second at either

Mp if the buyer is large or Lp if the buyer is small.

class D

Hp

Lp

P

Q

Mp

Figure 26: Volume Order Equilibrium Class D.

class L

Q

Equilibrium orders:Small buyer submits a visible order at LM pp , .Large buyer submits a volume order

)),,(( HLM pppvol , thus hiding a price Hp .Small seller submits a volume order ),( HM ppvol ,receiving Mp on the first unit,and selling a second at Hp if the buyer is large.Large seller submits a two unit limit order at Lp ,which executes at Mp , Lp .

Hp

Lp

P

Mp

Figure 27: Volume Order Equilibrium Class L.

price, while on the second unit, the small seller prefers to gamble for the higher (hidden)

price pH and the large seller type prefers to sell the unit with certainty at the lower (visible)

price pL. Note that this equilibrium does not exist in the case of private values, where the

optimality of the small buyer type�s order implies that the large seller type receives no rent

on the second unit. As a consequence, in any candidate equilibrium, the supermodularity in

payo¤ functions implies that the large seller type would prefer to deviate to the small seller

type�s strategy. When values are interdependent, visible prices on the second unit are such

that the large seller type does receive a rent, which makes the separation of sellers on the

second unit possible. The increase in visible prices is to the detriment of the small buyer.

The large buyer type, instead, bene�ts from pooling with the small buyer type, because the

price on both the �rst and the second unit is reduced, compared to what the large buyer

type would have to o¤er in any separating equilibrium that revealed his type. Finally,

it is important to note that for the equilibrium trades, the execution price on a possible

second unit is sometimes higher than that of the �rst unit traded (namely, whenever the

112

Equilibrium orders:Small buyer submits a volume order

),( ML ppvol , thus hiding a price Mp .Large buyer submits a volume order

),( HL ppvol , thus hiding a price Hp .Small sellers submit a volume order ),( HL ppvol ,selling a first unit at Lp , and a second unit at Hp if thebuyer is large.Large seller submits a volume order ),( ML ppvol ,selling a first unit at Lp , and a second unit either at Hp ifthe buyer is large or at Mp if the buyer is small.

Q

class M

Hp

Lp

P

Mp

Figure 28: Volume Order Equilibirum Class M.

large buyer trades with a small seller). This result can never be achieved with iceberg or

discretionary orders, which is why there is no analogue to this kind of equilibrium in the

previous Sections, 5.3 and 5.5.

In equilibrium class M, both buyer types submit volume orders, with one visible price

at pL and one hidden price: the large buyer type submits vol(pL; pH) (hiding a second unit

at pH) and the small buyer type submits vol(pL; pM ) (hiding a second unit at pM ). The

large seller type submits a volume order vol(pL; pM ) and the small seller type submits a

volume order vol(pL; pH). Thus, both seller types sell one unit at pL, and a second unit at

pH if the buyer is large, while if the buyer is small, only the large seller sells a second unit

(and at a price of pM ). The equilibrium has supply features in the sense that the price pLpaid on the �rst unit is always lower than the prices pM or pH paid on the second unit.

The following picture illustrates the remaining two equilibrium classes, N and O, in

group iii).

Equilibrium class N implements exactly the same expected trades as the discretionary

equilibria of class B calculated in Section 4.4. In equilibria of class B, buyers pool onto

a visible two-unit order (with prices p1 > p2) and the large buyer also hides one unit at

a hidden price, pD1 , which is above the visible prices. The small seller then sells one unit

at the hidden price and the large seller sells two units at the (lower) visible prices. While

this is not what the equilibrium orders in equilibria of class N actually look like, what the

equilibria of class N would resemble is a �ctitious discretionary order equilibrium of class

B in which buyers submit a visible supply schedule (rather than a demand schedule), so

that p2 > p1 and the large buyer hides one unit at pD1 > p2 > p1. Thus, the equilibrium

includes a combination of screening and supply features. To illustrate this, the �gure below

shows the discretionary order equilibrium of class B (left), the volume order equilibrium

113

class N

Q

Equilibrium orders:Small buyer submits a volume order ),( ML ppvol .Large buyer submits a volume order ),( HL ppvol .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a volume order ),( ML ppvol ,selling one unit at Lp , and a second unit either at Hpif the buyer is large, or at Mp if the buyer is small.

Hp

Lp

P

Mp

Figure 29: Volume Order Equilibrium Class N.

class O

Q

Equilibrium orders:Small buyer submits a limit order at Lp .Large buyer submits a volume order ),( HL ppvol .Small seller submits a limit order at Hp ,which executes if the buyer is large.Large seller submits a volume order ),( HL ppvol ,selling one unit at Lp , and a second unit at Hp if thebuyer is large.

Hp

Lp

P

Figure 30: Volume Order Equilibrium Class O.

class N (middle), and the ��ctitious� discretionary order equilibrium (right). Intuitively,

the volume order equilibrium of class N allows buyers to implement a less costly version of

equilibrium class B, by allowing buyers to lower the price on the �rst visible unit.

Note that in class N, the hidden price pH that the large buyer submits is sometimes

paid for the �rst and sometimes for the second unit. In particular, if the large buyer trades

with the small seller, he buys one unit at pH . If the large buyer trades with the large seller,

he buys two units, and pH is the price of the second unit (the �rst unit is bought at pL).

Having the hidden price in equilibrium be applied sometimes to the �rst and sometimes to

the second unit (rather than always to either the �rst unit or the second unit), is something

that can only be achieved with volume orders. That is, volume orders o¤er more �exibility

than both iceberg and discretionary orders.

Moreover, the sellers are screened by being presented with a menu of lotteries across

units. (Rather than being presented with lotteries on a given unit as in discretionary order

equilibria, in which sellers trade o¤ a certain payo¤ with a higher, uncertain payo¤ on a

given unit.) Speci�cally, they face two options: a lottery on the �rst unit (which may or may

114

class N

Q

Hp

Lp

P

Mp

class B

LpHp

Q

P fictitious B

Q

Hp

Lp

P

Mp

Figure 31: Discretionary Equilibrium B, Volume Equilibrium N and Fictitious Equilibrium.

not execute at pH), or a �rst unit sold with certainty and a lottery on a second unit (which

executes sometimes at pM and sometimes at pH). Since the choices involve lotteries across

units, the seller types�relative magnitude of the valuations on the �rst and second unit,

or the �slope� of the supply line de�ned by the seller types�valuation, enters the choice.

Intuitively, the slope of their supply line, can also be thought of as the market�s depth on

the supply side. Thus buyer types using volume orders in equilibrium N are essentially

screening the opposing side of the market (the supply side), with respect to depth.

Equilibrium class O, is relatively more simple than class N, while maintaining the two

main features. Speci�cally, when the buyer type is large, the small seller sells one unit at

pH , while the large seller sells two units at pL and pH . Thus, the price pH is sometimes paid

on the �rst unit, and sometimes paid on the second unit. (As an aside, from the perspective

of the large buyer, the equilibrium class is also similar to class C with added supply features.

On the other hand, though, the small buyer always only buys one unit.) The second main

feature in common with class N is that the sellers are again faced with lotteries across units:

trading o¤ a lottery on the �rst unit (which sometimes executes at pH), with a unit sold

with certainty and a lottery on the second unit (which sometimes executes at pH).

Finally, equilibrium group iv) is similar to equilibrium group iii) in the private values

case. The group contains equilibria that are non-monotonic in the sense that large types

do not always trade weakly more units than small types in each buyer-seller pairing they

are part of. That is, in the corresponding equilibria, the large buyer may trade two units

with the small seller and one with the large seller, while the small buyer trades two units

with the large seller and one unit with the small buyer. The appendix contains numerical

examples. This class is less interesting, and is mentioned here mainly for completeness.


This sections analyzes the e¤ects that the introduction of volume orders has on trading

volume and transparency. The qualitative results are exactly the same as in the case of

private values (Section 4.5.2); the di¤erence is simply that there are now more equilibrium

classes.

115


tionary orders leads to an increase or a decrease in expected trade volume.

The appendix contains a numerical example. Intuitively, much as in the case of discre-

tionary orders, the introduction of volume orders provides buyers with a potentially less

costly alternative to attain a given execution probability. Volume may be reduced following

the introduction of volume orders if, for example, the volume order equilibrium of class B

(group ii) above) is played when previously the optimal strategy for the large buyer was to

submit a limit order with an expected trade volume of 2.

Regarding transparency, a corollary similar to that for discretionary orders holds:

Corollary 12 Volume and transparencyi) It is possible that post- (and pre-) trade transparency is lower for the volume order equilib-

rium than for the corresponding limit order equilibrium, and yet trading volume increases.

ii) It is possible to have the same amount of volume in the volume order equilibrium as in

the corresponding limit order equilibrium, with any remaining uncertainty being about the

buyer in the volume order equilibrium and about the seller in the limit order equilibrium.

The appendix contains numerical examples. The corollary holds because the equilibria

of group ii) involving volume orders are analogous to the discretionary order equilibria B

and C in Section 4.4. The intuition and conclusion about the potential bene�ts of volume

orders in practice is also the same as for discretionary orders (presented in Section 4.4.2).

5.6 Buyer-Optimal Mechanisms

This section considers a principal-agent game in which the buyer is the principal proposing

a contract and the seller is the agent. In the case of private values, Section 4.6 shows that

there is no value to pooling for buyers. That is, buyers could not achieve a higher payo¤ by

pooling onto the same contract rather than submitting separating contracts (as they would

in the full information case). For the case of interdependent values this is no longer true, as

a proposition below illustrates. The �buyer-optimal�contract is now de�ned as a contract

that maximizes a weighted average of the payo¤s for both buyer types (where the weights

on the types may be degenerate�that is, 0 for one type and 1 for the other).

This section is composed of three parts. First, Section 5.6.1 brie�y introduces the

amended notation for the principal agent game when values are interdependent rather than

private. The second part, 5.6.2, characterizes some of the optimal mechanisms and the

third part, 5.6.3, investigates whether the allocations of buyer optimal mechanisms can

be replicated as equilibrium allocations in games with orders as described in Sections 5.2

through 5.5.

116

5.6.1 Principal-Agent-Game Setup

The setup of the principal-agent game setup in Section 4.6 carries over almost entirely;

the only necessary changes needed to be made to capture interdependent values are with

respect to the de�nition of the payo¤s.

As before, let U ij(�nm) and V

ij (�

nm) be, respectively, the buyer�s and seller�s payo¤

when the buyer type is i and the seller types is j, and the contract �nm = (xnm; tnm) is

implemented because the buyer and seller reported n and m in the third stage of the game.

With interdependent values, the payo¤s are now as below (for � = 0, the case of private

values is recovered).

U ij(�nm) =

2Pk=1

(vBik + � � vSjk)xnmk � tnm

V ij (�nm) = t

nm �2Pk=1

(vSjk + � � vBik)xnmk

5.6.2 Characterization of Optimal Mechanisms

This section analyzes the nature of the optimal mechanisms when values are interdependent.

Note that all numerical examples provided are examples of buyer-optimal mechanisms in

which the weight on the large buyer type is 1 (that is, they are the optimal mechanisms for

the large buyer type).

Proposition 14 The optimal mechanism for buyers may involve pooling onto a contract

at the contract proposal stage.

See the appendix for a numerical example. The intuition for the proposition is that with

interdependent values the large buyer has an incentive to hide his type by pooling with the

small buyer. Optimal pooled contracts may include �volume reduction�for the large buyer:

that is, the large buyer may want to pool onto a contract that has a lower expected trade

volume than that which he would have chosen if his type were known. Pooling is largely to

the detriment of the small buyer, who cannot costlessly signal his type. Note that for all

mechanisms (or, interchangeably, all allocations calculated in this section, the mechanism

design analogue of the intuitive criterion holds. Thus, the set of equilibrium allocations) for

the contract proposal game all weakly dominate the RSW allocation. That is, the payo¤

each buyer receives in any pooling mechanism exceeds the payo¤ the buyer could have

achieved by proposing an optimal separating contract.

The following proposition is analogous to Proposition 7 in the private values case. In

particular, there are instances in which the optimal pooling contract for the buyers is to

submit a contract of the form �i (as if it were a separating contract). The general pooling

contract, though, is � = (�S ; �L). The special case in question arises when the optimal

contracts in � satisfy �S = �L, which in turn must hold if the expected trade probabilities

and quantities for the �i are the same. Now if the pooling contract looks like a separating

117

contract, the proposition states that (as was the case for the private values setting) the

transfers prescribed may resemble a supply schedule in that the transfer (price) speci�ed

on a second unit to be traded may be higher than that on the �rst unit.

Proposition 15 If the optimal pooling contract has the form of a separating contract (that

is, � = �i), payo¤s associated with those contracts cannot always be implemented with

limit orders. Moreover, this will always be true for the case in which the transfers in the

optimal (separating) mechanism resemble a supply schedule (such that the implied price on

the second unit is higher than that on the �rst).

The appendix contains numerical examples. The intuition for the proposition is the

same as in the private values setting. First, the optimal mechanism may have a payment

schedule that can be implemented only with a supply schedule because of the nature of

prices on the �rst versus the second unit. Second, it may be that an optimal separating

mechanism involves bundling in a way that cannot be replicated with limit orders. For

example, the buyer submitting the contract may optimally not buy anything from the small

seller (tiS = 0), while buying two units from the large seller. In order to implement this

mechanism with limit orders, buyers would have to submit a two-unit limit order, priced at

p1, p2 (with p1+p2 = tLL). In the mechanism design framework, the sellers�options are to

reject the mechanism or to accept it and report one of two types. In the buyer-seller game,

though, the sellers can also decide to deviate and sell only one unit, at p1. In order to make

this deviation least attractive, the buyers will choose to set p1 = p2 = tLL=2. As is shown

in the appendix, it may be that even with this choice of prices, sellers will prefer to deviate

to selling at p1, so that the optimal mechanism cannot be implemented with limit orders.

5.6.3 Optimal Mechanisms Allocations as Equilibrium Outcomes

This section analyzes whether allocations of optimal contracts can be replicated as equilibria

with games involving orders, as considered in Sections 5.2 through 5.5.

It is important to make one remark regarding contracts before attempting to replicate

the allocation of an optimal contract. Any contract de�nes quantities xijk to be traded for

each pair of buyer and seller types, and it also speci�es the transfers tij to be exchanged

between the pair. For contracts in which buyers pool, though, constraints for all buyer and

seller types are averaged over the unknown type of the other player. As a result, players�

payo¤s depend on expected transfers, which will be denoted TBi for buyer type i andTSj for seller type j. In particular:

TBi = � � tiL+ (1��) � tiS for i 2 fL; Sg, and TSj = � � tLj + (1� �) � tSj .for j 2 fL; Sg.The fact that players�decisions depend on the expected transfers implies that optimal

contracts may have transfers tij that could never be the equilibrium prices of a game in-

volving trade by crossing orders. For example, it may be that the transfer tij is positive

118

although buyer i and seller j trade no units. Or the transfer tij may be zero although the

buyer-seller pair does in fact trade one or two units.

At the same time, there is a relationship between the transfers tij and the expected

transfers. Speci�cally, let

M :=

0BBBB@� 1� � 0 0

0 0 � 1� �� 0 1� � 0

0 � 0 1� �

1CCCCA,

transfers t =

0BBBB@tLL

tLS

tSL

tSS

1CCCCA, and expected transfers T =0BBBB@TBL

TBS

TSL

TSS

1CCCCA.Then M � t = T .The following remark holds, and allows re�nements of optimal contracts to be consid-

ered.

Observation 9 The matrix M has rank 3.

The fact that the matrix does not have full rank implies that for any optimal contract

� with transfers t, it is possible to �nd other transfers ~t such that M � t = T =M � ~t. Thisis done simply by adding a vector in the core of M to the original transfers t.

With this remark in mind, optimal mechanisms � = (x; t) can be categorized inequivalence classes: for any mechanism � = (x̂; t̂), its class is de�ned as those mechanisms� = (x̂; ~t) that satisfyM �t̂ =M �~t. In the following, verifying whether an optimal mechanism� = (x̂; t̂) can be implemented as the equilibrium of a game with orders, implies verifying

whether there is at least one mechanism in the equivalence class that is replicable. In many

cases, there will be one mechanism in the equivalence class that is the obvious candidate

for verifying whether the class is replicable. For example, if a buyer and seller pair i; j does

not trade any units, it would be optimal to consider the (unique) mechanism in the class

for which tij = 0.

The appendix contains a numerical example of a speci�c contract that is not imple-

mentable but is equivalent to an implementable contract, so that the equivalence class of

the speci�c contract is implementable.

The reason for considering equivalence classes is that in answering replicability, it is not

desirable to exclude mechanisms that are essentially implementable, in the sense that only

a small modi�cation (to a mechanism in the equivalence class) would be necessary in order

to implement the original mechanism.

Next, note that it may be possible that a particular optimal mechanism is not ex-post

implementable, but a mechanism in the equivalence class is (this is also illustrated in the

119

example in the appendix). It is important to note that many optimal mechanisms do not sat-

isfy ex-post implementability. Any optimal contract must satisfy Bayesian implementability,

meaning that buyer and seller types receive at least as much by accepting the mechanism

as choosing their next best outside option. Buyer and seller types calculate expected pay-

o¤s from the mechanism by averaging over the other player�s type (with the exception of

when the buyer�s proposed mechanism reveals his type). Bayesian implementability thus

requires that participation constraints for the optimal mechanism must hold on average,

and depends on the expected transfers.

Ex-post implementability, on the other hand, imposes stricter constraints. In par-ticular, a mechanism is ex-post implementable if both the buyer and seller would like to

implement the allocation rather than quitting in stage three of the game once they were

told the other player�s type. That is, it must hold that for all buyer types i, V ij (�ij) � 0

8j, and for all seller types j, U ij(�ij) � 0 8i.As an aside, while ex-post implementability is not a necessary requirement for replica-

bility of a given optimal contract with orders, an intermediate concept �between Bayesian

implementability and ex-post implementability�would be. In particular, optimal mecha-

nisms would have to satisfy the no-regret condition (as introduced by Green and La¤ont(1987) and Chakravorti (1992)).The condition, as applied to this game states that when

a player trades a given quantity, it must be that his expected payo¤ from trading that

quantity is positive. That is, if a seller sells one unit only to the small buyer type (and

two units to the large buyer type), then the seller�s payo¤ must be positive conditional on

trading one unit, and conditional on trading two units. If instead the seller type only ever

trades one unit, that is, he trades one unit whether the buyer type is small or large, then it

must be that the seller�s payo¤ is positive trading one unit. That is, if the seller knew the

buyer type to be large, his payo¤ might be negative when trading one unit (and positive

when he knew the buyer type to be small), implying that this mechanism would not satisfy

ex-post implementability.

The no-regret condition seems a natural condition one would expect to be satis�ed in

a game with orders. If it was violated, and, say, a seller knew that he always loses in

equilibrium when trading two units, then the buyer may not submit a two unit order (that

is, a price on the second unit). Note that this would be a possible deviation in any game

with orders, but it would not be possible to deviate in an analogous way as a seller in

the principal agent game, as sellers only have the option to accept or reject the proposed

contract as a whole.

The following proposition states that not all optimal mechanisms can be replicated as

equilibria involving orders.

Proposition 16 Consider an admissible set of orders to consist of both limit and marketorders, as well as either iceberg orders, discretionary orders or volume orders. Then there

are optimal contracts (more precisely, equivalence classes of optimal contracts) whose cor-

120

responding allocation and payo¤s cannot be implemented with an admissible set of orders.

In particular these contracts may satisfy ex-post implementability.

The appendix contains a numerical example of a mechanism that cannot be imple-

mentable with orders, although the mechanism is ex-post implementable. The proof involves

showing that there is no set of transfers in the equivalence class that has the structure that

the prices in any candidate order equilibrium would have to have. For the speci�c optimal

contract presented, buyer-seller pairs always trade two units, except when both the buyer

and the seller type is small, in which case they trade one unit. The same allocation in terms

of quantities can be achieved in a separating equilibrium with limit orders, an iceberg equi-

librium of class F, a discretionary order equilibrium of class D, and volume order equilibria

of classes D, L, and M. It is shown that none of these equilibria can exist, though, because

no set of transfers in the equivalence class satis�es any of the requirements imposed on

prices in those equilibria.

For example, if an iceberg equilibrium of class F were to exist, then the transfers would

have to satisfy tLL = tSL, because the large seller always sells at the visible prices (pH ; pL),

no matter what the buyer type. Moreover, it must be that tLS=2 = tSS , since the large

buyer pays 2 � pH to the small seller (pH on both a visible and a hidden unit), whereas

the small buyer only buys one unit at the visible price of pH from the small seller. For the

example provided, no set of transfers satis�es these equations jointly.

Similarly, a volume order equilibrium of class L would require that tLL = tLS = pL+pH ,

since the large buyer always pays pL on the visible unit and pH for the hidden unit, no matter

what the seller type. Moreover, tSL = pL+pM and tSS = pL would be the mapping between

prices and the other transfers and in an equilibrium of class L, it must be that tSS < tSL=2:

As it turns out, though, for the example optimal contract provided, then tLL = tLS , then

tSS > tSL=2. Thus, again, such an equilibrium cannot exist.

The fact that some of the optimal contracts cannot be implemented with orders even

though ex-post implementability is given has to do with the kind of deviations that are

allowed in a game with orders, compared to the deviations that are possible in the principal-

agent game. Sellers in the principal-agent game have only the option to accept or reject the

mechanism, and may deviate to the other seller type�s strategy in the third stage. Sellers

in a game with orders may have additional deviations available to them, namely to respond

to only part of an equilibrium order.

121

6 Appendix for Private Values

6.1 For Section 4.2 (Private Values with Only Limit Orders)

Proof of Proposition. 1Part i): Shown in the text preceding the proposition.

Part ii): Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:5; 0:6; 7; 5; 8; 7:7; 5:5; 6; 1:4; 2g

For this set of parameters, the buyer types separate: the small buyer type submits a

one-unit buy order with an execution probability of � = 0:6, and the large buyer type

submits a two-unit buy order with an execution probability 1 + � = 1:6.

In contrast, when the probability � of the seller being large is equal to � = 0:7 (while

all other parameters remain the same), the large buyer type�s optimal order is a two-unit

buy order with an execution probability of 2� = 2 � 0:7 = 1:4 which is lower. Moreover,

with � = 0:7, the small buyer type�s order is identical with that of the large buyer type (so

that buyers pool).

6.2 For Section 4.4 (Private Values with Discretionary Orders)

Proof of Proposition. 3The candidate equilibrium classes B and C have already been identi�ed in the text, it

remains to show that these classes do, and that for a given parameter combination, both

an equilibrium of class B and class C may exist.

As a reminder, the �gure below illustrates the buyer orders graphically, and the table

below illustrates how equilibrium orders look like in classes B and C. As for notation, p1; p2refer the prices on the �rst and second unit respectively and and pi(pD) means there is also

a hidden price on unit i.

class Cclass B

Q Q

Lp LpHp HpP P

Figure 32: Discretionary Order Equilibrium Classes B and C.

122

B order vol

Large buyer (pL(pH); pL) 1 + �

Small buyer (pL; pL) 2�

Large seller (pL; pL) 2

Small seller (pH) �

C order vol

Large buyer (pL(pH); pL(pH)) 2



Small seller (pH ; pH) 2�

Table 11: Equilibrium orders in equilibirum classes B and C.

Equilibrium BLet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:35; 0:7; 5:5; 5; 7; 6; 5:6; 6:5; 1:5; 3g

Then there exists and equilibrium of class B in which the visible order is (pL; pL) = (3; 3)

and the large buyer also hides a price on the �rst unit, at pH = 5:6:(Note that pL = vSL2,

which is the large seller type�s valuation on the second unit.)

For completeness note that the corresponding limit order equilibrium for this parameter

combination � would have the same strategies and payo¤s for the small buyer and the large

seller type. For the speci�c value of �, the large buyer would submit the same order as the

small buyer, with an execution probability 2�. Thus, volume in equilibrium B is higher

than in the corresponding limit order equilibrium (namely equal to �(1 + �) + (1� �)(2�)compared to �(2�) + (1� �)(2�) = 2�).

Equilibrium CFor the following parameter combinations , and equilibrium of class C exists with a

visible order of (pL; pL) = (3; 3) and two hidden units at (pH ; pH) = (6; 6). (Again, from

the optimality of the small buyer type�s order, it follows that pL = vSL2.)

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 5:5; 5; 9; 8; 4; 5:5; 1:5; 3g

In this equilibrium, p1 = p2 = 3 and pH1 = pH2 = 6.

For completeness note that in the corresponding limit order equilibrium, strategies and

payo¤s for the small buyer and the large seller type would always be the same as in C (by

construction). For the speci�c value of � the large buyer�s optimal strategy in the limit

order equilibrium would be to submit a two unit order with an execution probability of

1 + �. (The optimal strategy for the small buyer is a two unit order with an execution

probability of 2�). Thus, volume in equilibrium C is increased compared to the limit order

equilibrium.

123

Equilibrium B and C jointlyFor the parameter combination � below, both equilibria B and C exist.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 5; 4:5; 6:4; 6; 5:2; 5:5; 1:5; 3g

Speci�cally, the visible orders are (pL; pL) = (3; 3). The hidden price is equal to pH = 5:2

for equilibrium class B and the hidden price for both units in class C is equal to pH = 5:5.

For the given value of � the large buyer�s optimal strategy in the corresponding limit order

equilibrium is to submit the same order as the small buyer, namely an order with an expected

volume to be traded of 2�. Thus, volume equilibria B and C is increased compared to the

limit order equilibrium.

Proof of Corollary. 3An equilibrium of class B exists for:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:25; 0:5; 4:4; 4:2; 9:1; 8:9; 4:5; 5; 2:5; 3g

In the limit order equilibrium, the optimal order for the small buyer, priced at (pL; pL) =

(3; 3), has an expected volume of 2� (as always, given the existence of equilibrium B). The

optimal strategy for the large buyer type in the limit order equilibrium is to buy two units

for sure (with an order priced at (5; 5), thus achieving a trade volume of 2).

If equilibrium B is played, the large buyer�s optimal order is to pool onto the visible

order (pL; pL) = (3; 3), and to submit a hidden price of pH = 4:5 on the �rst unit, which

yields him an expected trade volume of 1 + � on the the entire order. Total trade volume

is thus reduced to �(1 + �) + (1 � �)(2�) in B, compared to the limit order equilibrium(where total trade volume is �(2) + (1� �)(2�)).

Proof of Corollary. 4For i):For the parameter combination � below, equilibrium of class C is exists, and the optimal

strategy of the large buyer in the corresponding limit order equilibrium is to submit an order

(pBL1 ; pBL2 ) with pBL2 > pBL1 yielding and an expected trade volume of 1 + �:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 6; 5; 9; 8; 4; 5:5; 1:5; 3g

124

In this equilibrium, the visible order is (pL; pL) = (3; 3), and the large buyer type hides

a price of pH = 6 on both units.

Total trade volume in equilibrium C is �(2) + (1� �)(2�); and thus larger than in thelimit order equilibrium (where it is �(1+�)+ (1��)(2�)). As for transparency: pre-trade,no (buyer) types are known in C. In contrast, there is separation of buyer types in the limit

order equilibrium. Moreover, as illustrated in the following table, post-trade transparency

is also lower in equilibrium.


when vol(xBL) = 1 + �; vol(xBS) = 2�

seller S seller L

buyer SS; S

(0)

S;L

(2)

buyer LL; S

(1)

L;L

(2)


seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 12: Post-trade transparency in limit order equilbrium and equilibirum of class C.

For ii):For the parameter combination � below, an equilibrium of class C is exists and the

optimal strategy of the large buyer in the corresponding limit order equilibrium is to submit

an order with same expected trade volume as in class C (namely a trade volume of 2). Thus,

total trade volume in C and with limit orders is the same (equal to �(2) + (1� �)(2�)).

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:5; 5:2; 4:8; 9:8; 9:3; 5:3; 5:5; 1:5; 3g

The visible order in the discretionary equilibrium is (pL; pL) = (3; 3), and the hidden

part of the large buyer type�s discretionary order is (pH ; pH) = (5:5; 5:5). In the limit order

equilibrium, the two unit order that the large buyer submits, is priced at p1 = p2 = 5:5.

Thus, the large buyer prefers equilibrium C, as he can buy two units for the lower price of

3 whenever the seller type is large.

Pre-trade, no (buyer) types are known in class C, and there is separation of buyers in

the limit order equilibrium. The table below illustrates that any remaining uncertainty with

limit orders is about the sellers. In contrast, any remaining uncertainty in class C is about

the buyer types (as shown in the previous table).

125



seller S seller L

buyer SS; S

(0)

S;L

(2)

buyer LL; ?

(2)

L; ?

(2)

Table 13: Post-trade transparency with limit orders.

6.3 For Section 4.5 (Private Values with Volume Orders)

Proof of Proposition. 4Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:5; 0:7; 2; 1:5; 8; 7; 2; 3; 1:8; 2:2g

In this case, buyer types separate on the visible part of their orders. The small buyer

type submits an order priced at p1 = 1:8 with execution probability �. The large buyer

submits a volume order with a visible price of p1 = 2; and two hidden prices pV1 = pV2 = 2:1

with a volume condition (so that both hidden units have to execute at once). Note that the

large buyer type�s volume order is thus vol(p1; [p2; p2]).

If in the above case, vSL2 = 1:85 instead, then the large buyer is indi¤erent between

submitting either a volume order which looks like a demand schedule (that is p1 = 2 and

pV1 = pV2 = 1:925. or vol(2; [1:925; 1:925])), or the optimal limit order with the same

execution probability of 1 + � (with prices (~p1; ~p2) = (2; 1:85). Note that 2 � pV1 � p1 =2�1:925�2 = 1:85, illustrating how volume orders can be used in place of demand schedulesas well as in place of supply schedules.

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:5; 0:7; 7:8; 6:9; 8; 7; 2; 3; 1:8; 2:2g

In this case, it is optimal for buyers to pool onto a volume order with visible price p1 = 2

and and two hidden prices pV1 = pV2 = 2:1 with a volume condition (so that both hidden

units have to execute at once).Note that the volume order is vol(2; [2:1; 2:1])

126

Proof of Proposition. 5:Consider two cases:

A) Buyer types pool onto a visible part, but only one buyer type submits a volumeorder by hiding at least one more unit.

B) Buyer types pool onto a visible order and both buyer types submit volume orders,where the hidden parts of the volume orders di¤er.

Note that with the exception of candidate equilibria in case B.2 to be analyzed below,

supermodularity�that is, the structure of the buyer and seller valuations�pins down which

buyer type submits which order (because in any equilibrium, the large buyer type�s order

has to have a higher execution probability than that of the small buyer type). In particular,

whenever only one buyer hides units, it is the large buyer. Whenever both buyer types hide

one unit, the large buyer�s hidden unit has a higher price so that the large buyer achieves

a higher expected trade volume in equilibrium. (Section B.2 addresses the case when one

buyer type hides one unit and the other type hides two units, at varying prices).

In the following, candidate equilibria are determined, existence is ruled out for some and

proven with numerical examples for the others. The candidate equilibrium classes are listed

below, organized by the groups i, ii and iii identi�ed in the proposition. The descriptions

found in brackets are (intuitive) labels for the di¤erent classes.

Group i) :class I (supply with separation)

class J (supply with pooling)

class G (demand with separation)

class Kb (supply with bundling and separation)

Group ii)class B (like discretionary B)

class Cb (like discretionary C with bundling)

Group iii)class 22 (non-monotonic)

Case A:

This case includes all candidate equilibrium classes in which both buyer types pool onto

the same visible order, but the large buyer also hides one or two units (that is, the large

buyer type submits a volume order, whose visible part is identical to the simple limit order

submitted by the small buyer). There are four subcases, labeled 1-4.

A.1. One unit hidden, with pvol1 > p1 (hidden unit priced above the �rstvisible unit):

There are two subcases, 1.1 and 1.2 which di¤er by the orders the buyer types submit.

The pictures below illustrate these subcases; letters next each graph denote the candidate

127

classes that are part of the corresponding subcase, brackets indicate that the particular

class will be shown not to exist with private values.

Case A.1.2:classes B, (L)

Case A.1.1:classes I, J, (A), (O)

Q

P

LpHp

Q

LpHpP

Figure 33: Volume Order Equilibrium Candidate Classes, Cases A.1.1 and A.1.2.

For subcase A.1.1:Buyer orders: The small buyer type submits a one-unit limit order at pL , the large

buyer type submits a volume order vol(pL; pH).

Possible seller responses to the buyer orders are:

�One unit limit order priced at pH�One unit limit order priced at pL�Volume order vol(pL; pH) or a two unit limit order priced at pL

If the small seller submits pH , the large seller cannot submit the same order, as thenthe visible order submitted by the small buyer would never execute, thus there are only two

options. Thus, the large buyer has to submit either and order priced at pL (candidate class

A) or an volume order vol(pL; pH) (candidate class G). Both classes are discussed next. As

for notation, used from now on, (NO) means that the equilibrium is shown not to exist;

(Yes) means it does exist, in which case the group and the intuitive description of the

equilibrium (as in the list above) is added.

Class A (No) The small seller type submits a limit order at pH and the large seller

type submits a limit order at pL.

This equilibrium cannot exist with private values, as the price pL would have to be such

that the large seller receives no rent (because it would be the small buyer�s optimal buy

order with execution probability �). Thus, the large seller type would have an incentive

to deviate to the small seller type�s strategy (which would give him positive pro�ts). Note

that this is exactly the reason that discretionary order equilibria with one visible unit do

not exist in the private values setting.

Class O (No).The small seller submits a limit order at pH and the large seller submitsa volume order vol(pL; pH).

Again, optimality of the small buyer type�s strategy implies that the large seller would

receive no rent on the �rst unit. Thus, since vSL2 > vSL1, the large seller type would have an

128

incentive to deviate to the small seller type�s strategy (receiving pH � vSL1 with probability� rather than pH � vSL2 with the same probability).

If the small seller submits pL, the large seller has to submit the volume order bysupermodularity (as he has to trade more volume in equilibrium than the small seller). This

is the case in the next candidate equilibrium class.

Class I (supply with separation)The small seller submits a limit order at pL, andthe large seller submits a volume order vol(pL; pH).

Consider the following parameter combination:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:7; 0:6; 4; 1:5; 9:5; 7; 3:07; 6:5; 3; 5:7g

In this case, the optimal limit order equilibrium has both buyers buying one unit at

p1 = 3:07. With volume orders, the small buyer submits the same limit order, while the

large buyer also hides one unit at pv1 = 5:7. This equilibrium has supply function features

given pv1 > p1 and it also increases trade volume as the large buyer now buys two units from

the large seller.

Note that if the buyers had the ability to submit supply functions with limit orders

(rather than only demand functions), then the large buyer may as well have displayed the

hidden unit.

If the small seller submits a volume order vol(pL; pH), so must the large seller,by supermodularity.

Class J (supply with pooling) Both sellers submit a volume order vol(pL; pH).Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:6; 0:3; 4:5; 2:5; 6:5; 5:5; 3:5; 5; 3:5; 4:7g

In this case, the optimal limit order equilibrium has both buyer types buying one unit at

p1 = 3:5. With volume orders, both buyers submit the same visible order and hide another

unit at pv1 = 5. This equilibrium has supply function features given that pv1 > p1, and it

also increases trade volume as the large buyer now buys two units from the seller types.

In this case, again, if the buyer types had the ability to submit supply functions, then

the large buyer type may as well display the hidden unit.

For subcase A.1.2:Buyer orders: The small buyer type submits a two-unit limit order at (pM ; pL), the large

buyer type submits a volume order vol((pM ; pL); pH).

Possible seller responses to the buyer orders are as below. Note that In this subcase,

pM � pL.

129

One unit limit order priced at pHOne unit limit order priced at pMTwo unit limit order priced at pLVolume order vol(pL; pH)

If the small seller type submit a one unit limit order at pH , the only responsefor the large seller that guarantees positive execution probability for both visible units, is

to submit a tow-unit limit order at pL.

Class B ( like discretionary B).The small seller submits a limit order at pH and thelarge seller submits a two unit limit order at pL.

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:15; 0:7; 2:5; 2; 9; 8:5; 2:5; 3; 1:2; 1:5g

In this case, equilibrium B exists, and the large buyer type�s optimal order is to pool

onto visible prices p1 = p2 = 1:5 and to submit a hidden price at pH1 = 3, achieving an

expected trade volume of 1 + � on the the entire order.

In the limit order equilibrium, the optimal order for the small buyer (priced at (p1; p2) =

(1:5; 1:5)) has an expected volume of 2�, while the optimal strategy for the large buyer type

is to buy two units for sure, at pBL1 = pBL2 = 3.

Total trade volume is thus reduced to �(1 + �) + (1 � �)(2�) in class B, compared to�(2) + (1� �)(2�) in the limit order equilibrium.

The intuition for this class is that the volume order allows the large buyer to separate

sellers, exactly like in the case of discretionary orders. In fact, in this case, the equilibrium

with volume orders is identical with the equilibrium with discretionary orders, because the

associated payo¤s and execution probabilities are the same for all relevant deviations.

If the small seller type submit a one unit limit order at pM , no equilibriumexists, because there would be no order the large seller could submit, so that all visible and

hidden units would have positive execution probability. If the small seller type submita two-unit limit order at pL, no equilibrium can exist either, because the large seller

then would have to submit the same response, and no units would execute at hidden prices.

If the small seller type submit a volume order at vol(pM ; pH), the large seller hehas to either submit a volume order or the two-unit limit order. If the large seller chooses

the volume order, the second visible unit would never execute, thus he would have to submit

a two unit limit order.

Class L (No)The small seller submits a volume order vol(pM ; pH) and the large sellersubmits a two unit limit order priced at pL.

130

Since the large seller trades weakly more volume in equilibrium, This equilibrium cannot

exist in the private values case, because by the optimality of the buy order for the small

buyer, pL will leave the large seller no rent when trading the second unit at pL, so the large

seller will have an incentive to deviate to the small seller�s strategy (pH has to be high

enough for the small seller to want to trade, which implies that the large seller would make

positive rent playing the small seller�s strategy).

A.2. One unit hidden, with pvol1 < p1 (hidden unit priced below the �rstvisible unit):

There are two subcases, as illustrated below.

Case A.2.2:class (D)

Case A.2.1:classes G, (H)

Q

P

LpHp

Q

LpHpP

Mp


For case A.2.1:Buyer orders: The small buyer type submits a limit order priced at pH , and the large

buyer type submits a volume order vol(pH;pL).

Possible seller responses are:

�One unit limit order priced at pH�Volume order vol(pH ; pL) or a two unit limit order priced at pL

If the small seller type submit a limit order at pH , the large seller type has tosubmit a two-unit limit order at pL so that he trades weakly more in equilibrium an all

units submitted by the large buyer type have positive execution probability.

Class G (demand and separation) The small seller submits a limit order at pH ,and the large seller submits a two-unit limit order at pL.

Consider the following parameter combinations:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:7; 0:5; 6; 1:5; 9:5; 7; 5:5; 6:5; 5; 5:2g

131

A volume order equilibrium exists in which the visible price is equal to pS1 = pL2 = 5:5

and the hidden price submitted by the large buyer type is pv2 = 5:2.

The optimal limit order equilibrium has separation., with the small buyer type buying

one unit at pS1 = 5:5; and the large buyer type submitting a two-unit order priced at

(pL1 ; pL2 ) = (5:5; 5:2).

Thus, trading volume in the volume order and limit order equilibrium are the same.

The volume order equilibrium has demand features (because pv1 < p1), and the large buyer�s

payo¤ is the same in both equilibria (that is, he does not bene�t from pooling) and he also

may as well have displayed the hidden unit.

If the small seller type submit a two-unit limit order at pL, the large seller hasto do the same by supermodularity.

Class H (No) Both seller types submit a two-unit limit order priced at pL (or a volumeorder vol(pH ; pL)).

In this case, the large buyer would be buying two units with certainty. In order to buy

the second unit from the small seller, the price pL would have to be above vSS2. Given that

pH > pL must hold, this implies that the large buyer would be better o¤ displaying the

units.

For case A.2.2:Buyer orders: The small buyer type submits a two-unit limit order priced at (pH ; pL),

and the large buyer type submits a volume order vol((pH ; pL); pM ).

Note that In this case, pH � pM > pL.


�One-unit limit order priced at pH�Two-unit limit order priced at pM or volume order vol(pH ; pM )

�Two-unit limit order priced at pL

If the small seller type submit a two-unit limit order at pM , the large seller hasto submit either a two-unit limit order at pM or a two-unit limit order at pL. In the former

case, the second unit would never execute at the visible price pL. Thus, only candidate

equilibrium D below is left in this case.

Class D (No)The small seller submits a two unit limit order at pM (or equivalently, a

volume order vol(pH ; pM )) and the large seller submits a two unit limit order at pL.

This equilibrium cannot exist in the private values case: the optimality of the small

buyer type�s limit order implies that the large seller type would make no rent trading the

second unit at pL, and if the small seller type sold a second unit in equilibrium at pM , then

the large seller type would �nd it pro�table to deviate to the small seller type�s strategy.

132

If the small seller type submit a limit order at pH , no equilibrium exists, becausethere would be no order the large seller type could submit, such that all parts of the orders

submitted by the buyer types would have positive execution probability.

A.3. Two units hidden, with pvol1 = pvol2 > p1 (hidden unit priced above the�rst visible unit):

There are two cases, as illustrated below. Note that there are no cases in which two

units are hidden at a price below the visible price, because the second hidden unit would

never execute.

Case A.3.2:class (C)

Case A.3.1:class (K)

Q

P

LpHp

Q

Lp

HpP

Mp


For case A.3.1:Buyer orders: The small buyer type submits a limit order priced at pL, and the large

buyer type submits a volume order at vol(pL; (pH;pH)).


�One unit limit order priced at pH�Two unit limit order priced at pH�One unit limit order priced at pL�Volume order vol(pL; pH) (or vol(pL; (pH ; pH)), which executes the same way)

�Volume order vol(pL; [pH ; pH ])

There is no equilibrium in which a seller submits a volume order at vol(pL; pH): It would

be preferable for the seller to submit vol(pL; [pH ; pH ])., because that order would give him

a higher price on the �rst unit whenever two units execute.

There is no equilibrium in which a seller submits a limit order at pH : The only order

for the other seller that would guarantee that the second hidden unit executes would be

vol(pL; [pH ; pH ]), but if that was the seller order, then the large buyer type�s unit priced at

pL would never execute.

There is no equilibrium in which the small seller submits a two unit limit order at pH .

If so, the large seller would have to be submitting a volume order that gives him higher

133

execution probability than the small seller type�s order. The optimal volume order would be

vol(pL; [pH ; pH ]) (as this would leave the large seller type strictly better o¤ than submitting

vol(pL; pH) or vol(pL; (pH ; pH)). Given the seller�s orders, though, the large buyer�s unit

unit priced at pL would never execute.

There is no equilibrium in which both sellers submit a volume order vol(pL; [pH ; pH ]),

or else the visible unit of the large buyer�s order would never execute.

If the small seller type submits a limit order at pL, then the large seller mustsubmit a volume order that gives him a higher expected trade volume than submitting pL.

The optimal volume order is vol(pL; [pH ; pH ]), and the corresponding equilibrium candidate

is considered next.

Class K (No).The small seller submits a one unit limit order at pL, and the largeseller submits a volume order vol(pL; [pH ; pH ])

The small buyer type must be submitting his optimal limit order, with execution proba-

bility 1. In the absence of volume orders, the optimal limit order with execution probability

1 is pL = vSS1. This price can never be part of a volume order equilibrium considered here,

because the small seller would be left with no rent selling at this price, so he would always

gamble and submit a one unit limit order at pH . Thus, pL would have to satisfy pL > vSS1in any candidate equilibrium. Choose any candidate equilibrium pL, then the small buyer

would always have a pro�table deviation submitting epL = pL � ", where " < pL � vSS1.For case A.3.2:Buyer orders: The small buyer type submits a two-unit limit order priced at (pM ; pL),

and the large buyer type submits a volume order at vol((pM ; pL); (pH;pH)).


�One unit limit order priced at pH�Two unit limit order priced at pH�One unit limit order priced at pM�Two unit limit order priced at pL (or volume order vol(pL; [pH ; pH ]) which executes

the same way)

�Volume order vol(pM ; pH) (or vol(pL; (pH ; pH)), which executes the same way)

In this case, the small buyer type cannot be submitting a one unit limit order at pH or

pM or a volume order at vol(pM ; pH) (as there would be no possible response for the large

seller type such that all submitted units would have positive execution probability). The

small seller type cannot submit a two-unit limit order at pL either, as the large seller then

would have to do the same, implying that there would be no reason for the large buyer to

hide units.

134

If the small seller type submits a two-unit limit order at pH , then the largeseller has to submit a two-unit limit order priced at pL, or else not all units submitted buy

the buyer types have positive execution probability.

Class C (No)The small seller type submits a two unit limit order at pH , and the largeseller type submits a two unit limit order at pL:

First note that a volume order vol((pL; pL); [pH ; pH ]) would execute the same way as

the two-unit limit order at pL. In equilibrium, the small buyer type must be submitting

his optimal limit order. If that optimal order(pM ; pL) has a price pL that does not leave

zero rent for the large seller on the second unit, then the small buyer type has a pro�table

deviation to an order (pM ; epL) with epL = pL � " > vSL2. Thus, the small buyer type�s

optimal order has p2 = vSL2, and the large seller type receives no rent on that unit. Then,

though, the large seller will always have an incentive to deviate to submitting vol(pM ; pH)

rather than submitting (pL; pL) (which executes at (pM ; pL)).

A.4. Two units hidden, with pvol1 > p1 (hidden unit priced above the �rst visibleunit) and volume condition:

Note that below, arrows on the lines representing the hidden prices indicate that the

buyer has included a (minimum) volume condition of �2�: the hidden units may only execute

if both execute at the same time (as a bundle).

Case A.4.2:class Cb

Case A.4.1:class Kb

Q

P

LpHp

Q

Lp

HpP

Mp


Volume Order Equilibrium Candidate Classes, Cases A.4.1 and A.4.2.

For case A.4.1:Buyer orders: The small buyer type submits a limit order priced at pL, and the large

buyer type submits a volume order vol(pL; [pH ; pH ]).


�One unit limit order priced at pL�Two unit limit order priced at pH

135

�Volume order vol(pL; (pH ; pH)) (or volume order vol(pL; [pH ; pH ]) which executes the

same way)

Using the same arguments as for case A.3.1, one can see that there is no equilibrium in

which a seller type submits a volume order at vol(pL; pH), a limit order pH or a two-unit

limit order pH .

No equilibrium exists in which both seller types submit a volume order vol(pL; [pH ; pH ]),

or else the visible unit of the large buyer type�s order would never execute.

If the small seller type submits a limit order at pL, then the large seller has tosubmit a volume order vol(pL; [pH ; pH ]) or else not all units submitted by the buyer types

execute.

Kb (supply and bundling).The small seller submits a limit order at pL and the largeseller submits a volume order vol(pL; [pH ; pH ]):

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:4; 0:4; 9; 4; 11; 10; 5; 12; 2:5; 7g

A volume order equilibrium exists with visible price p1 = 5 and hidden prices (pv1; pv2) =

(6; 6). Note that the hidden prices are set so that the large seller type is indi¤erent between

submitting vol(pL; [pH ; pH ]) and pL, as they both yield the same payo¤ of 0:4(12 � 9:5) +0:6(5� 2:5) = 2:5:

For the same parameters, the optimal limit order equilibrium has separation:the small

buyer type submits an order with execution probability 1 (priced at p1 = 5), and the large

buyer type submits an order with execution probability 1 + � (priced at (p1; p2) = (7; 7)).

For case A.4.2:Note that in this case, pvol1 = pvol2 > p1 � p2.

Buyer orders: The small buyer type submits a two-unit limit order at (pM ; pL) and the

large buyer type submits a volume order at vol((pM ; pL); [pH ; pH ]).


�Two unit limit order priced at pH�One unit limit order priced at pM�Two unit limit order priced at pL (or vol(pL; (pH ; pH)), which executes the same way)

Much like in the analysis for case 4.1 (without the volume condition), there is no equi-

librium in which the small seller submits a one unit limit order at pM , because there is no

response by the large seller that would guarantee that all submitted orders have positive

136

execution probability. There is also no equilibrium in which the small seller submits a two-

unit limit order at pL, as then the large seller would have to do the same, and thus the large

buyer would not have any reason to hide a unit.

If the small seller type submits a two-unit limit order at pH , then the largeseller has to submit a two-unit limit order at pL, or else not all units submitted buy the

buyer types execute.

Class Cb (like discretionary C with bundling)The small seller type submits atwo-unit limit order at pH and the large seller type submits a two-unit limit order at pL (or

vol(pL; (pH ; pH)), which executes the same way).

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 4; 3:5; 5; 4:5; 3; 4; 1:5; 2:5g

A volume order equilibrium exists with visible prices (p1; p2) = (2:5; 2:5) and hidden

prices are (pv1; pv2) = (4; 4).

The corresponding limit order equilibrium has separation; the small buyer type submits

an order with execution probability 2� (priced at (p1; p2) = (2:5; 2:5)), and the large buyer

type submits an order with execution probability 1+�. Volume thus is higher in the volume

order equilibrium of class C, compared to the limit order equilibrium.

Case B:B.1. Both buyers hide one unit:There are three cases, as illustrated below. Note that there is no case in which two units

are visible. All parts of an order submitted must have positive execution probability, which

implies that the seller responses must involve volume orders (or limit orders) at the two

hidden prices. There is no order that allows for two visible units to execute in addition to

a hidden unit (that would be a three unit trade, which is excluded).

Case B.1.2:Candidates 1, 2

Case B.1.1:classes (M), (N)

Q

P

LpHp

Q

LpHpP

Mp

Case B.1.3:Candidate 3

Q

LpHpP

MpMp

Figure 37: Volume Order Equilibrium Candidate Classes, Cases B.1.1, B.1.2 and B.1.3.

137

For case B.1.1:Buyer orders: the small buyer type submits a volume order vol(pL; pM ), and the large



�One unit limit order priced at pH�One unit limit order priced at pM�One unit limit order priced at pL�Volume order vol(pL; pH)

�Volume order vol(pL; pH)

There are no equilibria in which the small buyer submits a one-unit limit order at pM or

at pL (because there would be no response for the large seller that would guarantee positive

execution probability on all units submitted by buyers). There is also no equilibrium in

which the small seller submits vol(pL; pH), as then the large seller would have to do the same

by supermodularity, and thus the buyer submitting vol (pL;pM ) would have an inventive to

deviate and submit vol (pL;pM ) instead.

If the small seller type submits a limit order at pH , then the large seller has tosubmit a volume order vol(pL; pM ) to guarantee positive execution probability on all units.

Class L (No) The small seller submits a one-unit limit order at pH , and the largeseller submits a volume order vol(pL; pM ).

For the small seller type to sell the unit, it must be that pH � vSS1. Next, if must bethat the large seller does not want to deviate to submitting a volume order of vol(pM;pH)

(if he did, he would sell one one with certainty: at a price of pM if the buyer type was small

and at a price of pH if the buyer type was large). In equilibrium, the large seller type�s

incentive constraint corresponding to the deviation to vol(pM;pH) is: pL + (1 � �)pM +

�pH � vSL1 � vSL2 > (1� �)pM + �pH � vSL1.Thus it must hold that pL > vSL2. But if this holds, then the large buyer has a pro�table

deviation, namely submitting a limit order with the same execution probability of 1 + � as

the equilibrium volume order (the order would be priced at (p1; p2) = (vSS1; vSL2) which is

less costly than the equilibrium volume order because vSL2 < pL and vSS1 < pH).

If the small seller type submits a volume order vol(pL; pH), then the large sellerhas to submit a volume order vol(pL; pM ) to guarantee positive execution probability on all

units.

Class M (No)The small seller submits a volume order vol(pL; pH); and the large sellersubmits a volume order vol(pL; pM ).

For the small seller type to sell the unit, it must be that pH � vSS2. Next, if must

be that the small seller type does not want to deviate to submitting a volume order of

138

vol(pM;pH).The corresponding incentive constraint is: pL � vSS1 + �(pH � vSS2) > (1 ��)(pM + pH)� vSS1.

Thus it must hold that pL > (1 � �)pM + �vSS2. Since pL < pM by assumption, this

implies pL > vSS2. But if this inequality holds, then the large buyer type has a pro�table

deviation, namely to submit a limit order with the same execution probability of 2 as the

equilibrium volume order (the order would be priced at (p1; p2) = (vSS2; vSS2) which is less

costly than the equilibrium volume order because vSS2 < pH and vSSS < pL)

For case B.1.2:Buyer orders: The small buyer type submits a volume order vol(pM ; pL) and the large

buyer type submits a volume order vol(pM ; pH).


�One unit limit order priced at pH�One unit limit order priced at pM�Volume order vol(pM ; pH)

�Volume order vol(pM ; pH) or two unit limit order at pL

There are no equilibria in which the small seller type submits a one unit limit order

at pM or a volume order vol (pM;pL). In the �rst case, this is because there would be no

response for the large seller that would guarantee positive execution probability on all units

submitted by buyers. In the second case it is because the large seller type would have to also

submit vol (pM;pL) and, again, some units submitted by the buyers would never execute.

The following two equilibria candidates 1 and 2 are in fact identical to equilibria B

and L respectively. This holds because the orders submitted in candidates 1 and 2 would

essentially have the same execution behavior given incoming seller responses as the orders

submitted in equilibria B and L, that is, they would have the same outcomes (that is, prices

and quantities). The candidate classes 1 and 2, and classes B and L are pictured next to

each other below, more precise arguments will be given next.

Case B.1.2:Candidates 1, 2

Q

LpHpP

Mp

Case A.1.2:classes B, L

Q

LpHpP

Figure 38: Volume Order Equilibrium Candidate Classes, Cases B.1.2 and A.1.2.

139

If the small seller type submits a limit order at pH , then the large seller typehas to submit a volume order vol(pM ; pL) to guarantee positive execution probability on all

units submitted by the buyer types.

candidate class 1 (No, same as class B).The small seller submits a one unit limitorder at pH , and the large seller submits a volume order vol(pM ; pL).

This equilibrium exists would be identical to the equilibrium in which seller responses

are as in the candidate equilibrium and both buyers submit a visible order at (pM ; pL) and

the large buyer type hides a unit at pH (which is equilibrium class B).

If the small seller type submits a volume order vol(pM ; pH), then the large sellertype has to submit a volume order vol(pM ; pL) to guarantee positive execution probability

on all units submitted by the buyer types.

candidate class 2 (No, same as class L).The small seller submits a volume ordervol(pM ; pH);.and the large seller submits a volume order vol(pM ; pL).


are as in the candidate equilibrium and both buyers submit a visible order of (pM ; pL) and

the large buyer type hides a unit at pH (which is equilibrium class L).

For case B.1.3:Buyer orders: The small buyer type submits a volume order vol(pH ; pL) and the large

buyer type submits a volume order vol(pH ; pM ).


�One unit limit order priced at pH�Volume order vol(pH ; pM )

�Volume order vol(pH ; pL)

There are no equilibria in which the small seller type submits a one unit limit order

at pH or a volume order vol (pH;pL). In the �rst case, this is because there would be no

response for the large seller type that would guarantee positive execution probability on all

units submitted by buyers. In the second case it is because the large seller type would have

to also submit vol (pH;pL), and again some units submitted by the buyer types would never

execute.

If the small seller type submits a volume order vol(pH ; pM ), then the large sellertype has to submit a volume order vol(pH ; pL) to guarantee positive execution probability

on all units submitted by the buyer types.

candidate class 3 (No, same as D)The small seller type submits a volume ordervol(pH ; pM ), and the large seller type submits a volume order vol(pH ; pL).


are as in the candidate equilibrium and both buyers submit a visible unit at (pH ; pL) and

the large buyer type hides a unit at pM (which is equilibrium class L). The picture below

illustrates candidate class 3 and equilibrium class D next to each other.

140

Case B.1.3:Candidate 3

Q

LpHpP

Mp

Case A.2.2:class D

Q

LpHpP

Mp

Figure 39: Volume Order Equilibrium Candidate Classes, Cases B.1.3 and A.2.2.

B.2. One buyer hides one unit, the other hides two units:There are three general cases, as illustrated below. First, no two hidden units can be

priced below the visible price. If this were the case, then the visible unit would always

execute �rst. As a consequence, if there was no volume condition on the hidden order, only

one unit would execute (so that the second hidden unit would not have positive execution

probability in equilibrium). If there was a volume condition on the hidden units (in that they

have to execute both at once), then none of the units would execute,because higher-priced

units would have to be executed �rst�so this cannot be the case in equilibrium.

The three candidate classes below are illustrated without a volume condition on the two

hidden units, though equilibria in with a volume condition on the two hidden units are also

analyzed below.

Note that for these equilibria it is not immediately obvious which of the buyer types

and which seller types would be submitting which of the orders.

Case B.2.2:Nonmonotonic 3

Case B.2.1:Nonmonotonic,(1) and (2)

Q

P

LpHp

Q

LpHpP

Mp

Case B.2.3:Nonmonotonic,(4) and (5)

Q

LpHpP

MpMp

Figure 40: Non-Monotonic Volume Order Equilibria.

For case B.2.1:First, consider the case in which a volume condition is imposed on the two

hidden units.Buyer orders: One buyer type submits a volume order vol (pL;pM ) and one buyer type

141

submits a volume order vol (pL;[pH ; pH ]).


1. Volume order vol(pL; pM )

2. Two unit limit order priced at pH3. Volume order vol (pL;[pH ; pH ]) or vol (pL;(pH ; pH))

4. One unit limit order priced at pL5. One unit limit order priced at pM6. Volume order vol (pM;[pH ; pH ]) or limit order pM7. Volume order vol (pL;(pM ; pM )) or two unit limit order at pL

There can be no equilibrium in which the seller types submit the same responses. If the

seller types both responded with 1-5, then some of the units submitted by the buyer types

would never execute. If the seller types responded with 6 or 7, then there are pro�table de-

viations for the buyer types. For example, the buyer type submitting the volume order with

two hidden units would prefer to deviate to submitting vol(pL; [pM ; pM ]) and vol(pL;(pL))

(or an iceberg order at pL), respectively, if the sellers pool onto 6 or 7.

There can be no equilibria in which one of the seller types responds with 4 or 5, as then

there is no response by the other seller type so that all buyer units execute. There can be

no equilibrium in which a seller type submits option 3, as it would always be better for the

seller to deviate to response 6 (same execution probability, but higher sell prices).

This leaves 1, 2, 6 and 7 as possible seller responses, and six possible combinations of

these responses.

There can be no equilibrium in which the seller responses are (1 and 7), as then the buyer

type submitting vol (pL;[pH ; pH ]) would have an incentive to deviate to vol (pL;[pM ; pM ]).

There can be no equilibrium in which the seller responses are (2 and 6), or (2 and 7). If

there was, the transacted prices and units would be as illustrated in the following tables.

buyer/seller (S): 2 (L): 6

(S): vol(pL; pM ) 0 pM

(L): vol(pL; [pH ; pH ]) pH ; pH pH ; pH


(S): vol(pL; pM ) 0 pL; pM


Table 14: Case B.2.1 with Volume Condition, Seller Responses (2 and 6) and (2 and 7).

In the table, (L) and (S) denote the type of the buyer and seller, both of which are

pinned down in these candidate equilibria, due to supermodularity (that is, the structure of

buyer and seller type valuations, which implies that large types have to trade more units in

equilibrium). Since the small seller type in both candidate equilibria knows that he will sell

two units only if the buyer type is large, it must be that pH must be above the reservation

value of the small seller conditional on the buyer type being large. Since the large buyer

type pays pH ; pH though, whenever he trades (that is, also when trading against the large

seller type), he would be (weakly) better o¤ displaying the units.

142

Similarly, there can be no equilibrium in which the seller responses are (6 and 7). The

associate matrix would be as below. Buyer and seller types would be pinned down, and the

large buyer type would be (weakly) better o¤ displaying the units.


(S): vol(pL; pM ) pM pL; pM


Table 15: Case B.2.1 with Volume Condition, Seller Responses (6 and 7).

The two remaining candidate equilibria, involving the seller responses (1 and 2) and (1

and 6), can be excluded by showing that they lead to a contradiction as to which types of

sellers are submitting the orders. In particular, the corresponding matrices would be:


vol(pL; pM ) pL; pM 0

vol(pL; [pH ; pH ]) pL pH ; pH

buyer/seller 1 6

vol(pL; pM ) pL; pM pM

vol(pL; [pH ; pH ]) pL pH ; pH


For the case in which the seller responses are (1 and 2), note the following. It must be

that the seller type submitting 1 is small, otherwise he would deviate to vol(pL; (pM ; pM )),

which would execute at pL; pM (when the buyer submitted vol(pL; pM )),and pH ; pH (when

the buyer submitted vol(pL; [pH ; pH ])). That is, if the seller type submitting 1 would prefer

to trade only one unit at pL, rather than two units at a per unit price pH > pL, while the

other seller does sell two units at pH , then it must be that the seller type submitting 1 is

small. Otherwise, he would have a pro�table deviation in submitting 7 (two units at pL).

Now if the seller type submitting 1 in is small, then the seller type submitting 2 must be

large. But if so, the large seller type would prefer to deviate to submitting 7 instead of 2,

since that would allow him to sell two units at pL; pM like the small seller type (rather than

0) when the buyer submitted vol(pL; pM ). (while conserving the same price when the buyer

submitted vol(pL; [pH ; pH ])).

The arguments for the case when the seller responses are (1 and 6) are the same as

when they are (1 and 2). That is, the seller type submitting 1 must be small, or else he

would have a pro�table deviation to 7. But in this case, 7 would pose a pro�table deviation

for the large seller type. (One can also argue stating that the seller type submitting 6 or 2

must be small, or else 7 is a pro�table deviation for him, but then 7 constitutes a pro�table

deviation for the large seller type that submitted 1).

Next, consider the case in which there is no volume condition imposed onthe two hidden units.

143

Buyer orders: One buyer type submits a volume order vol (pL;pM ) and one buyer type

submits a volume order vol (pL;(pH ; pH)).


1.Volume order vol(pL; pM ) or vol (pL;(pM ; pM )) or two unit limit order at pL2.Two unit limit order priced at pH3.Volume order vol (pL;[pH ; pH ])

4.One unit limit order priced at pL5.One unit limit order priced at pM6. Volume order vol(pL; pH) or vol (pL;(pH ; pH))

7. Volume order vol (pM;[pH ; pH ])

8. One unit limit order priced at pH9. Volume order vol (pM;pH) or two unit order at pM

There can be no equilibria in which one of the seller types responds with 4, 5 or 8, as

then there is no response by the other seller so that all buyer units execute. There can

be no equilibria in which a seller type submits 3, 6 or 9, as it would always be better to

submit 7. For the remaining responses 1, 2 and 7, there can be no equilibrium in which the

seller types both submit the same order, as then some buyer units would never execute. It

remains to check the seller response combinations (1 and 2), (1 and 7), and (2 and 7).

There can be no equilibrium in which the seller responses are (2 and 7). If there was,

the transacted prices and units would be as illustrated in the following table.


(S): vol(pL; pM ) 0 pM

(L): vol(pL; (pH ; pH)) pH ; pH pH ; pH

Table 17: Case B.2.1, Seller Responses (2 and 7).

Since large types have to trade more units in equilibrium, the buyer and seller types are

pinned down as in the table. Now, since the small seller in both candidate equilibria knows

that he will sell two units only if the buyer type is large, pH must be above the reservation

value of the small seller conditional on the buyer type being large. Since the large buyer

pays (pH ; pH) though, whenever he trades (that is, also when trading against the large seller

type), the large buyer would be (weakly) better o¤ displaying the units.

Next consider the response combinations (1 and 2), (1 and 7).

NM1 (No) The large buyer type submits a volume order vol(pL; (pH ; pH)), the smallbuyer type submits vol(pL; pM ). The large seller type submits vol(pL; pM ) (1) and the small

seller type submits a two-unit limit order at pH (2).

The table below illustrates execution prices and quantities.

144

buyer/seller (L): 1 (S): 2

(S): vol(pL; pM ) pL; pM 0

(L): vol(pL; (pH ; pH)) pL; pH pH ; pH



pinned down as in the table.

In order for the small seller type to want to sell the two units, it must be that pH > vSS2.

Next, in order for the large seller type not to want to deviate to vol(pM ; pH), it must be

that hid payo¤ from that deviation is below the equilibrium payo¤. That is, it must be

true that pL � vSL1 + (1 � �)pM + �pH � vSL2:(1 � �)pM + �pH � vSL1 + �(pH � vSL2).Which means pL > vSL2 + �(pH � vSL2) > vSL2. (The last inequality holds because

pH > vSS2vSL2). This implies that the small buyer type would have a pro�table deviation

to (p1; p2) = (vSL2; vSL2), which would yield him at least the execution probability of 2�

that the equilibrium order has.

NM2 (No) The large buyer type submits a volume order vol(pL; (pH ; pH)), the smallbuyer type submits vol(pL; pM ). The large seller type submits vol(pL; pM ) (1) and the small

seller type submits vol(pM ; [pH ; pH ]) (7).



(S): vol(pL; pM ) pL; pM pM




pinned down as in the table. It must again be that pH > vSS2 so that the small seller type

sells the second unit. As a consequence, the large seller type will again have an incentive to

deviate unless pL > vSL2 + �(pH � vSL2) > vSL2. Next, it must be that pM > vSS1, or else

the small seller type will deviate to submitting a two unit order at pH . Also, it must be that

pM > vSL2 or else the large seller type would deviate to vol(pL; pH): It follows that the small

buyer type will have a pro�table deviation in submitting (p1; p2) = (max(vSS1; vSL2); vSL2),

because it holds that pM > max(vSS1; vSL2) and pL > vSL2:


hidden units.

145

Buyer orders: One buyer type submits a volume order vol (pL;pH) and one buyer type

submits a volume order vol (pL;[pM ; pM ]).


1.Volume order vol(pL; pM ) or vol (pL;pH)

2.Two unit limit order priced at pM or volume order vol ((pM;pM ); pH)

3.Volume order vol (pL;[pM ; pM ])

4.One unit limit order priced at pL5.One unit limit order priced at pM6. Volume vol (pL;(pM ; pM )) or two unit limit order at pL7. One unit limit order priced at pH

There can be no equilibria in which one of the sellers responds with 4, 5 or 7, as then

there is no response by the other seller so that all buyer units execute. There can be

no equilibria in which a seller submits 3, as it would always be better to submit 2. For

the remaining responses 1, 2 and 6, there can be no equilibrium in which the sellers both

submit 1 or 2, as then some buyer units would never execute. There can be no equilibria

in which both sellers submit 6 because then the buyer submitting vol(pL; pM ) would have

an incentive to deviate to vol(pL; pM ).

The remaining combinations to check are (1 and 2), (1 and 6), and (2 and 6).

There can be no equilibria in when seller responses are (1 and 6), or (2 and 6). If so,

the corresponding matrices would be as follows, with buyer and seller types pinned down

since large types have to trade more units in equilibrium.


(L): vol(pL; pH) pL; pH pL; pH

(S): vol(pL; [pM ; pM ]) pl pM ; pM


(S): vol(pL; pH) pH pL; pH

(L): vol(pL; [pM ; pM ]) pM ; pM pM ; pM


If seller responses are (1 and 6), then the small buyer type has an incentive to deviate

to vol(pL; (pM ; pM )), so that he would pay only pL; pM when the seller type is large. When

seller responses are (2 and 6), the small buyer type has an incentive to deviate to vol(pL; pM ),

so that again he would pay pL; pM rather than pL; pH when the seller type is large.

There can be no equilibria in which the seller responses are (1 and 2). If so, the corre-

sponding matrix would be as follows.


(S): vol(pL; pM ) pL; pH pH

(L): vol(pL; (pH ; pH)) pL pM ; pM

Table 21: Case B.2.2 with Volume Condition, Seller Responses (1 and 2).

146

The seller type submitting 2 must be small. Otherwise, 6 (that is, vol (pL;(pM ; pM )))

would be a pro�table deviation, as the seller type would be selling two units at pL; pH if

the buyer submitted vol(pL; pM ) (while selling the same as before in the case where the

buyer submitted vol(pL; (pH ; pH))). Given this, though, the large seller type would �nd

it pro�table to deviate to 6, and thus sell two units at pM ; pM when the buyer submitted

vol(pL; (pH ; pH)).


Buyer orders: One buyer type submits a volume order vol (pL;pH) and the other buyer

submits a volume order vol (pL;(pM ; pM )).


1.Volume order vol (pL;pH)

2.Two unit limit order priced at pM or volume order vol ((pM;pM ); pH)

3.Volume order vol (pL;[pM ; pM ])

4.One unit limit order priced at pL5.One unit limit order priced at pM6. Volume order vol(pL; pM ), vol (pL;(pM ; pM )) or two unit limit order at pL7. One unit limit order priced at pH8. Volume order vol (pM;pH)

There can be no equilibria in which one of the seller types responds with 4, 5 or 7, as

then there is no response by the other seller type so that all buyer units execute. There

can be no equilibria in which a seller type submits 3, as it would always be better for him

to submit 2. For the remaining responses 1, 2, 6 and 8, there can be no equilibrium in

which the seller types both submit the same response, as then some buyer units would

never execute.

There are six remaining combinations to check, namely: (1 and 2), (1 and 6), (1 and 8),

(2 and 6), (2 and 8), (6 and 8). There can be no equilibria with response pairs (1 and 6)

or (2 and 8) as some buyer units would then never execute. There can be no equilibrium

involving (2 and6), (1 and 8) or (6 and 8), as then the buyer type submitting vol (pL;pH)

would have an incentive to deviate to vol (pL;pM ), which would yield the same execution

probability at lower prices.

NM3 (non-monotonic) One buyer type submits a volume order vol (pL;pH) and theother buyer submits a volume order vol (pL;(pM ; pM )). One seller type submits vol(pL; pH)

(1) and the other seller type submits a two unit limit order at pM (2).


147

buyer/seller 1 2

vol(pL; pH) pL; pH pH

vol(pL; (pM ; pM )) pL pM ; pM


Let:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:8; 0:3; 9; 8; 11; 9; 6; 7; 2; 6g

The following is an equilibrium: the large buyer type submits vol(pL; pH) = vol(4; 7:5),

the small buyer type submits vol(pL; (pM ; pM )) = vol(4; (6:5; 6:5)). The large seller type

submits two units at pM and the small seller type submits vol(pL; pH). Note that in the

private values setting, the structure of the buyer and seller valuations implies that in equi-

librium, large types always trade more units in expectation than small types. In the speci�c

example provided, � < 0:5, so the large buyer type trades two units with the small, achiev-

ing a trading volume of 2 � �, which is larger than the expected trading volume of 1 + �for the small buyer type. Similarly, as � > 0:5, the large seller type trades two units with

the large buyer type , achieving an expected trading volume of 1 + �; which is larger than

the expected trading volume or 2� � for the small seller type.

For case B.2.3:

First, consider the case in which a volume condition is imposed on the twohidden units.

Buyer orders: One buyer type submits a volume order vol (pM;pL) and one buyer type

submits a volume order vol (pM;[pH ; pH ]).


1.Volume order vol(pM ; pL) or a two unit order at pL2.Two unit limit order priced at pH3.Volume order vol (pM;[pH ; pH ]) or vol (pM;(pH ; pH))

4.One unit limit order priced at pM

There can be no equilibria in which one of the seller types responds with 4, as then there

is no response by the other seller type so that all buyer units execute. For the remaining

responses 1, 2, and 3, there can be no equilibrium in which the seller types both submit the

same response, as then some buyer units would never execute. There are three remaining

combinations to check, namely: (1 and 2), (1 and 3), and (2 and 3). There can be no

equilibrium involving (2 and 3), as some buyer units would never execute.

148

There can be no equilibrium in which the seller responses are (1 and 2) or (1 and 3). If

there was, the transacted prices and units would be as illustrated in the following tables.


(S): vol(pM ; pL) pM ; pL 0



(S): vol(pM ; pL) pM ; pL pM



Both the type of the buyer and seller are pinned down in the candidate equilibria, as the

large types trade more units in equilibrium. Since the small seller type in both candidate

equilibria knows that he will sell two units only if the buyer type is large, pH must be above

the reservation value of the small seller type conditional on the buyer type being large.

Since the large buyer type pays (pH ; pH), though, whenever he trades (that is, also when

trading against the large seller type), the large buyer type would be (weakly) better o¤

displaying the units.


Buyer orders: One buyer type submits a volume order vol (pM;pL) and one buyer type

submits a volume order vol (pM;(pH ; pH)).


1.Volume order vol(pM ; pL) or a two unit order at pL2.Two unit limit order priced at pH3.Volume order vol (pM;[pH ; pH ])

4.One unit limit order priced at pM5.Volume order vol(pM ; pH)

There can be no equilibria in which one of the seller types responds with 4, as then

there is no response by the other seller so that all buyer units execute. For the remaining

responses 1, 2, 3, and 5, there can be no equilibrium in which the seller types both submit

the same response, as then some buyer units would never execute. There are six remaining

combinations to check, namely: (1 and 2), (1 and 3), (2 and 3), (1 and 5), (2 and 3), (2

and 5), (1 and 5) and (3 and 5). There can be no equilibrium involving (2 and 3), (3 and

5) or (2 and 5) as some buyer units would never execute. It remains to analyze (1 and 2),

(1 and 3) and (1 and 5).

NM4 (No) The large buyer type submits a volume order vol(pM ; (pH ; pH)) and thesmall buyer type submits vol(pM ; pL). The large seller type submits vol(pM ; pL) (1) and the

small seller type submits a two unit limit order at pH (2).

149




(L): vol(pM ; (pH ; pH)) pM ; pH pH ; pH




The small buyer type trades 2� units in equilibrium. Next, pL > vSL2, or else the large

seller type would deviate to submitting vol(pM ; pH). But then, since pM > pL, the small

buyer type would always have a pro�table deviation to submitting (pL; pL), which would

have at least an execution probability of 2�.

NM5 (No) The large buyer type submits a volume order vol(pM ; (pH ; pH)), the smallbuyer submits vol(pM ; pL). The large seller submits vol(pM ; pL) (1) and the small seller

submits vol(pM ; [pH ; pH ]) (3).



(S): vol(pM ; pL) pM ; pL pM





The small buyer type trades 1 + � units in equilibrium. Next, pL > vSL2, or else the

large seller type would deviate to submitting vol(pM ; pH). Moreover, pM > vSS1, or else

the small seller type deviates to submitting a two-unit limit order at pH . Finally pM > pL

by assumption in this case. Together, this implies that the small seller type would have a

pro�table deviation to (p1; p2) = (max(vSS1; vSL2); vSL2).

Proof of Corollary. 5The following parameter combination �, a volume order equilibrium exist that has lower

trade volume than the corresponding limit order equilibrium. Speci�cally, a volume order

150

equilibrium of class B exists.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:15; 0:7; 2:5; 2; 9; 8:5; 2:5; 3; 1:2; 1:5g

In the limit order equilibrium, the optimal order for the small buyer (priced at (p1; p2) =

(1:5; 1:5)) has an expected volume of 2� (this has to be the case, given the existence of

equilibrium B). The optimal strategy for the large buyer in the limit order equilibrium is to

buy two units for sure, by submitting an order priced at (pBL1 ; pBL2 ) = (3; 3). In equilibrium

B, the large buyer�s optimal order is to pool onto visible order (pL; pL) = (1:5; 1:5) and to

submit a hidden price at pH = 3, achieving an expected trade volume of 1 + � on the the

entire order volume order. Total trade volume is thus reduced to �(1 + �) + (1� �)(2�) inB, compared to �(2) + (1� �)(2�) in the limit order equilibrium.

In contrast, for the following �, volume would be increased compared to the limit order

equilibrium if a volume order equilibrium of class B was played.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:15; 0:8; 3; 2:5; 5; 4; 2:5; 3; 1:2; 1:5g

In particular, consider the equilibrium of class B in which the visible order is (pL; pL) =

(1:5; 1:5) and the large buyer hides a price of pH = 3 on the �rst unit.

The corresponding limit order equilibrium has the same strategies and payo¤s as equi-

librium class B for the small buyer and the large seller type. On the other hand, though, the

large buyer�s optimal in the limit order equilibrium is the same as that of the small buyer

in that equilibrium, namely to submit an order with an execution probability 2� (priced

at (p1; p2) = (1:5; 1:5)). As volume in equilibrium B is equal to �(1 + �) + (1 � �)(2�), itfollows that this volume is higher than in the corresponding limit order equilibrium (where

it is �(2�) + (1� �)(2�) = 2�).

Proof of Corollary. 6For i):The following is an example in which, compared to the corresponding limit order equilib-

rium, volume increases wit a volume order equilibrium of class C, yet pre-trade transparency

and post trade transparency is reduced.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 4; 3:5; 5; 4:5; 3; 4; 1:5; 2:5g

151

In this case, the optimal limit order equilibrium has separation. The small buyer submits

an order with execution probability 2� (priced at (p1; p2) = (2:5; 2:5)), and the large buyer

submits an order with execution probability 1 + � (priced at (p1; p2) = (3; 2:5)). In the

volume order equilibrium, the visible prices are (pL; pL) = (2:5; 2:5) and the hidden prices

are (pH ; pH) = (4; 4).

For ii):For the parameter combination � below, volume order equilibrium C is exists, and the

optimal strategy of the large buyer in the corresponding limit order equilibrium is to submit

an order with same expected trade volume as in class C (of 2).

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; 0:2; 0:7; 4; 3:5; 9; 8; 3; 3:5; 1:5; 2:5g

Speci�cally, the optimal buy order for the large buyer type is to submit (p1; p2) =

(3:5; 3:5), while for the small buyer type it is to submit (p1; p2) = (2:5; 2:5) (which has an

execution probability of 2�). Thus total trade volume in C and with limit orders is the

same (equal to �(2) + (1� �)(2�)).The visible orders in the volume order equilibrium is (pL; pL) = (2:5; 2:5), and the large

buyer type hides a price of pH = 4 on both units.

Pre-trade, no (buyer) types are known in class C. There is separation of buyers in

the limit order equilibrium. The table below illustrates that, post-trade, any remaining

uncertainty in class C is about the buyer types, while any remaining uncertainty with limit

orders is about the sellers.



seller S seller L

buyer SS; S

(0)

S;L

(2)

buyer LL; ?

(2)

L; ?

(2)


seller S seller L

buyer SS; S

(0)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 26: Post-trade Transpareny with Limit Ordes and in Equilibrium Class C.

6.4 For Section 4.6 (Private Values and Optimal Mechanisms)

Proof of Proposition. 6Overview: I will show that buyers can never achieve payo¤s in pooling equilibria that

are higher than payo¤s in separating equilibria. Pooling occurs when both buyers submit

152

a contract � = (�L; �S). For any optimal pooling contract, I will construct separating

contracts such that at least one of the buyers will (weakly) prefer deviating to the separating

contract rather than adhering to the pooling contract.

The proof has a four steps. Step 1 shows that the small seller�s IR and the large seller�s

IC are binding for any optimal pooling contract � = (�L; �S). Step 2 constructs separating

contracts e�L; e�S based on the optimal pooling contracts. If e�S ; e�L were feasible (satisfyingall necessary constraints), at least one of the buyers would prefer proposing e�i to proposingthe pooling contract �. The separating contracts e�L; e�S already satisfy a number of theincentive and individual rationality constraints for sellers. Step 4 shows how to, if necessary,

modify e�L; e�S such that the modi�ed contracts, ee�L; ee�S , satisfy all the constraints imposedby sellers and give buyers a (weakly) higher payo¤ than e�L; e�S . As a consequence, (atleast) one of ee�L; ee�S will be (weakly) preferred to the pooling contract by (at least) one ofthe buyers.

Step 1: Show that for an optimal pooled contract � =��S ; �L

�with �i = (�iS ;�

iL) =

(xSi; tSi;xLi; tLi) it must be that:

a) the small seller�s participation constraint binds and

b) the large sellers Incentive constraint binds.

Suppose the optimal pooling contract � does not satisfy a). Then construct another

contract e� equal to � except that all transfers are reduced by "; etij = tij � ", where " issmall enough so that the small seller�s IR is not violated. Note that since all payments have

been reduced by the same amount ", e� satis�es all seller incentive constraints in the sameway that � did. Moreover, e� also satis�es the large seller�s IR, as both (expected) sellerpayments, tSL and tLL, fell by the same amount and the large seller�s IR must have been

slack to begin with (due to supermodularity and the fact that the optimal contract satis�ed

the large seller�s IC).

Finally, the new contract e� increases buyer payo¤s by " compared to �, so the individualrationality constraints hold. Incentive constraints for buyers also hold; more precisely they

are una¤ected, since all (expected) buyer payments, tiL and tiS (for i 2 fL; Sg) fell by thesame amount.

Suppose the optimal pooling contract � does not satisfy b). Then construct another

contract e� equal to � except for etLL = tLL�" and etSL = tSL�", where " is small enough sothat the large seller�s IC is not violated. Note that the small seller�s IR remains una¤ected.

Thus, the large seller�s IR also still holds (due to supermodularity and because the large

seller�s IC holds). The large seller�s contract only became more unattractive, so the small

seller�s IC still holds. The new contract e� will satisfy buyers�IR�s (it is actually better forthe buyers, as it has lower transfers. Buyer�s incentive constraints also hold: the ranking ofe�S versus e�L is the same as for �S versus �L because expected payo¤s for both buyer typesfell by the same amount, ".

Step 2. Given a optimal pooling contract, construct two separating contracts for the

153

buyers, e�S and e�L. Let U i(�nm) and Vj(�nm) be, respectively, the buyer�s and sellers payo¤when the buyer type is i and the seller types is j, and the contract �nm is implemented

because the buyer and seller reported n and m in the third stage of the game.

First, de�ne rL; rS as below, using the small sellers IR constraint, which is binding in

equilibrium for the optimal contract � (as shown in step1).

� � US(�LS)| {z }rL

+ (1� �) � US(�SS)| {z }rS

= 0

Next, de�ne cL; cS as below, using the large sellers binding incentive constraint, which

is binding as well (step 1):

� ��UL(�

LL)� UL(�LS)

�| {z }cL

+ (1� �) ��UL(�

SL)� UL(�SS)

�| {z }cS

= 0

Construct two separate contracts e�L and e�S to have the same allocations as � for givenpairs of buyers and sellers (so exjik = xjik 8i; j; k), but with new transfers. Speci�cally, letetLS = tLS�rL, etSS = tSS�rS ,etLL = tLL�rL�cL, and etSL = tSL�rS�cS . By construction,e�L and e�S satisfy the small seller�s IR73 constraints and the large seller�s IC constraints74.Note that seller constraints are now conditional on the buyer type (that is, type-by-type

constraints rather than one constraint that has to hold in expectation). The large seller�s IR

constraints are also trivially satis�ed (again, due to supermodularity of the payo¤ functions

in vSJk and .xjik ).

As for the (remaining) small seller incentive constraint, two cases will be considered: a)e�L and e�S satisfy them and b) e�L and/or e�S do not.Step 3: Next, I will show that if the separating contracts e�L and e�S were implemented

as is, both buyers would do exactly as well as in the pooling contract, or one of them would

do strictly better. Speci�cally, e�L and e�S would yield buyer payo¤s as below, where Vj(�nm)are the payo¤s from the optimal pooling contract �:

� � Vj(e�jL) + (1� �) � Vj(e�jS) = � � Vj(�jL) + (1� �) � Vj(�jS) + rj + � � cj75This implies that the di¤erence in payo¤s between the separating (e�i) and the pooling

contract � for the large buyer (1.) and the small buyer (2.) are:

1: rL + � � cL

2: rS + � � cS

Note that by construction:

73The small seller IR constraint with respect to e�LS and e�SS are, respectively:US(e�LS) � 0 () US(�

LS)� tLS + etLS � 0 () rL � rL � 0

US(e�SS) � 0 () US(�SS)� tSS + etSS � 0 () rS � rS � 0

74The large seller�s Incentive constraints with respect to e�L and e�S are, respectively:UL(e�LL)� UL(e�LS) � 0 () cL + etLL � tLL + tLS � etLS � 0 () cL � rL � cL + rL = 0UL(e�SL)� UL(e�SS) � 0 () cS + etSL � tSL + tSS � etSS � 0 () cS � rS � cS = rS = 0

75

� � Vj(e�jL) + (1� �) � Vj(e�jS)= � � Vj(�jL) + (1� �) � Vj(�

jS) + � � (tjL � etjL) + (1� a) � (tjS � etjS)

= � � Vj(�jL) + (1� �) � Vj(�jS) + � � (rj + cJ) + (1� �) � rj

= � � Vj(�jL) + (1� �) � Vj(�jS) + r

j + � � cj

154

3: � � rL + (1� �) � rS = 04: � � cL + (1� �) � cS = 0Now it cannot be that both 1: and 2: are negative. In fact, 1: and 2:are either both

equal to zero, or, if one is positive then the other is negative.

Thus, e�L and e�S would be separating contracts such that they either give buyers payo¤sequal to the optimal mechanism (if 1: and 2: are both equal to zero) or such that one of the

buyers has a higher payo¤ than with pooling (one of 1: or 2: is positive, the other negative).

In the latter case, the buyer who would do better with e�i would prefer separating ratherthan pooling onto � (so the pooling equilibrium could not have existed).

Step 4: In step 2 it was shown that, by construction, e�L, e�S always satisfy seller�sindividual rationality constraint and the large seller�s incentive constraint. The remaining

seller constraint was the small sellers incentive constraint.

a) Assume that e�L, e�S also satisfy the small seller�s incentive constraint. Then thereis no value to pooling because either buyers do as well in pooling as in separation, or one

of the buyers will strictly prefer to deviate and propose e�i, which he may do as e�i wouldsatisfy all seller constraints.

b) Assume that a given separating contract e�i for buyer i does not satisfy the smallseller�s incentive constraint. Then the buyer submitting e�i could instead submit ee�i, withee�i = (ee�ij ; ee�ij), where ee�ij is either equal to e�iS or e�iL, depending on which of the twoyields a higher payo¤ to buyer i. Note that the sellers incentive constraints will be trivially

satis�ed. (because the contracts ee�iL and ee�iS are identical). The small seller�s IR will be

satis�ed in both cases. For ee�ij = e�iS this holds by construction, for ee�ij = e�iL it holdsbecause the small seller�s individual rationality constraint was satis�ed for e�iS and sincethe small seller�s incentive constraint was violated by assumption, the small seller preferrede�iL to his own contract e�iS . The large seller�s IR is also satis�ed, given that the small

seller�s IR is satis�ed.

Note that by construction, ee�L, ee�S yield (weakly) higher payo¤s to buyers than contractse�L, e�S . Buyers were either achieving payo¤s equal to pooling payo¤s or one buyer wasachieving a strictly higher payo¤ than with pooling. Thus, there is no value to pooling

because with separating contracts ee�L, ee�S either separation yields (weakly) as much aspooling to both buyers, or one buyer will strictly prefer to separate (as before, whenever the

preference for the separating contract is strict for at least one buyer the pooling equilibrium

could not have existed).

Comment: In the case of interdependent values, the proof would not work. Speci�cally,

buyers could not just pick a contract e�i and expect the seller constraints to be satis�ed, assellers would update their beliefs about the buyer types and those beliefs enter the valuation

function.

155


� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; �; 0:5; vBS1; vBS2; 9; 8; 5; 7; 4; 6g

.

(Here, �; vBS1; vBS2 can take any value that is consistent with the general model as-

sumptions.)

Then for the large buyer, the optimal mechanism is � involves o¤ering to buy one unit

at tLS = 5 and two units at tLL = 11. That is, tLL � tLS > tLS , so that these transfers

would correspond to a supply schedule.

For the following parameter

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0; �; 0:5; vBS1; vBS2; 2; 1:7; 1:6; 3; 1; 1:5g

the optimal mechanism for the large buyer involves bundling. Speci�cally, the optimal

mechanism prescribes not to buy any units from the small seller (tLS = 0) and to buy to

units from the large buyer at tLL = 2:5. Notice though, that it is not possible to implement

the result of the optimal mechanism with limit orders. In order to do so, the large buyer

would have to submit a limit order priced at p1 = p2 = tLL=2 = 1:25. Then, though, the

large seller would �nd it pro�table to only sell one unit at p1, rather than selling both units

at p1 = p2, as would prescribed by the optimal mechanism (selling one unit would leave the

large seller type with a payo¤ of 0:25, selling both leaves him no rent). Also note that for

any limit order with p1 > p2, instead of p1 = p2, the large seller type�s incentive to deviate

would only be larger.

Proof of Proposition. 8Consider any optimal mechanism �i with transfers tiL and tiS , and some trade quan-

tities per buyer-seller type (xijk )k2f1;2g;j2fL;Sg (since there is no value to pooling, these are

separating contracts for buyers). Due to supermodularity, for each buyer type i, there are

six possible trade quantity combinations with the large and small seller type, labeled (a-f)

and illustrated in the following table. Speci�cally, the trading volume for buyer i per seller

type j is denoted xij1 + xij2 .

156

a b c d e f

xiL1 + xiL2 0 1 1 2 2 2

xiS1 + xiS2 0 1 0 2 0 1

Table 27: Possibe combination of trade volume per seller type, for a given buyer.

.

For case a), it is easy to see that it would be optimal for buyer i not to submit an order

at all. For case b), he optimally submits an order priced at p1 = tiL = tiS , which would

have an execution probability of 1 (because tiL = tiS must hold for the optimal mechanism,

or else one of the seller incentive constraints (IC�s) would be violated.

For case d), the optimal order would again be priced at p1 = tiL, but in this case, the

optimal mechanism has tiS = 0 so that the execution probability of the order would only

be �. Note that this order satis�es all relevant individual rationality (IR) constraints and

IC�s because the optimal mechanism does.

For case d) the optimal order would be a supply schedule with a price p1 low enough

that it would never execute, and p2 > p1 such that p1+p2 = tiL = tiS (again tiL = tiS must

hold, or else one of the seller�s IC would be violated). These prices would guarantee that

both sellers are willing to sell two units, but that no seller would like to deviate and only

sell one unit at p1. Note that the seller IC�s are trivial in this case, and that the IR�s are

satis�ed because the mechanism is optimal (and was thus accepted by the sellers).

For case e), the optimal buy order would again be a supply schedule with p1 low enough

so that it would never execute, and a p2 > p1 such that p1 + p2 = tiL (note that in this

case tiS = 0). Much like in case d), the large seller would now like to sell two units at

p1 + p2 = tiL, but would not want to deviate to selling only one unit(all other constraints

are satis�ed because the mechanism was optimal).

For case f), the optimal buy order is p1 = tiS and p2 = t1L� tiS . Note that in this case,the small seller IR binds. The large seller IC binds because the corresponding IC in the

optimal mechanism did bind as well. The large seller IR is satis�ed due to supermodularity

and the fact that the IC binds. Note that as stated in Proposition 7, it may be that p1 < p2.

157

7 Appendix for Interdependent Values

7.1 For Section 5.2 (Interdependent Values with Only Limit Orders)

Numerical example for Remark 4Let � be as below:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:5; 6:5; 5; 7; 5:5; 4:5; 5:5; 3; 4g

For this �, there exists a pooling equilibrium in which both buyers submit an order with

execution probability 1 + �, priced at (p1; p2) = (6:465; 5:515). It is also an equilibrium

for buyers to pool onto an order with same execution probability but slightly higher prices

(p1; p2) = (6:565; 5:515).

Let � be as below:

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:7; 11; 4; 12; 8; 3:5; 3:7; 1:1; 1:3g

.

Then the small and the large buyer pool onto and order of (p1; p2) = (4:5; 1:98), and

the large buyer hides a unit at pvol2 = 4:61. In the absence of volume orders, there exists a

pooling equilibrium in which both buyers pool onto (p1; p2) = (4:61; 1:74). Thus with the

presence of volume orders, volume is increased in equilibrium D.

Note that this is not quite like equilibrium D in which the hidden price had to be equal

to p1. In order for the equilibrium to exist, though, it is still necessary that p11[X;A] >

p21[X; 1], which means that buyer valuations decrease a lot but the seller valuations do not

increase much from the �rst to the second unit.

Proof of Corollary. 9For i): The fact that the large buyer type always trades more volume in equilibrium

than small buyer type follows directly from supermodularity of the payo¤s (analogously for

the large seller type versus small seller type).

For ii): Trade volume may be decreasing in the probability � that the seller is large

because buyers have an incentive to game and submit low-priced limit orders whenever

they think that the order�s execution probability is high because the seller is likely to be of

the large type. Given the following parameter combination, trade volume decreases when

going from � = 0:5 to � = 0:7.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; �; 6:5; 5; 7; 5:5; 4:5; 5:5; 3; 4g

158

For � = 0:5 there exists a pooling equilibrium in which both buyers submit an order

with execution probability 1+� (priced at (p1; p2) = (6:465; 515)). For � = 0:7 there exists

a pooling equilibrium in which both buyers pool onto an order with execution probability

2� (the order is priced at (p1; p2) = (5:15; 5:15)).

7.2 For Section 5.3 (Interdependent Values with Iceberg Orders)

Proof of Proposition. 10Note that in this model, iceberg orders have at most one hidden unit (a second hidden

unit would never execute). Next, buyers iceberg orders do not provide buyers with the

ability to pool onto a visible unit and then separate on the hidden part of the order (unlike

in the case of volume orders), because any hidden unit must be hidden at the highest visible

price. Moreover, there are no equilibria in which buyers separate on the visible part of the

orders, as in this case it would be weakly better for them to disclose the hidden unit. Thus,

it must be that buyers pool on one visible order and one of the buyers also hides a unit.

Thus, there are two possible combinations of buyer orders that can be part of an iceberg

order equilibrium, as illustrated below.

class E

Q

Lp

P class F

Q

Hp

Lp

P

Figure 41: Iceberg Equilibria Classed E and F.

Supermodularity implies that the order submitted by each buyer and seller type and

the associated volume that each type trades is as follows:

class E order vol

Large buyer (pL(pL)) 1 + �

Small buyer pL 1

Large seller (pL; pL) 1 + �

Small seller pL 1

class F order vol

Large buyer (pH(pH); pL) 2

Small buyer (pH ; pL) 1 + �


Small seller (pH ; pH) 1 + �

Table 28: Equilibirum Orders in Iceberg Equilibrium Classes E and F.

It remains to show that such equilibria exist. Let

159

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:3; 8; 3; 11; 9; 5; 9; 3; 4g

in this case, an iceberg equilibrium of class E exists, with a visible price of p1 = 7:49.

Next, let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:4; 0:3; 9; 6; 11; 7:5; 5; 5:5; 2:5; 3:5g

Then an equilibrium of class F exists. The visible order in the iceberg equilibrium is

(p1; p2) = (7:94; 5:48).

A pooling equilibrium with limit orders exists in which both buyer types submit an

order with execution probability 1 + � and priced exactly like the visible prices in the

iceberg equilibrium (that is, priced at (p1; p2) = (7:94; 5:48)). Compared to that limit order

equilibrium, equilibrium of class E increases trading volume.

Proof of Corollary. 7For i):

Let � be as below:


In this case, there exists an iceberg equilibrium of class E in which buyers pool onto a

visible price of p1 = 8:71 and the large buyer hides another unit at that price.

A separating limit order equilibrium also exists, in which the large buyer buys two units

and the small buyer buys one unit. The payo¤s for the small buyer type, the small seller

type and the large seller type are reduced from 4:66 to 4:54, 0:17 to 0, and 2:47 to 2:022,

respectively.

Volume traded is lower in the iceberg equilibrium (namely ��(1+�)+(1��)�1 comparedto � � 2 + (1� a) � 1).

For ii): Let


In this case, there exists an iceberg equilibrium of class E (with a visible price of p1 =

7:49).

A separating limit order equilibrium with the same expected volume as the iceberg

equilibrium also exists. In the limit order equilibrium, the large buyer type submits a limit

order (p1; p2) = (8:3; 6:7) with execution probability 1+�, and the small buyer type submits

a limit order (p1) = 7:4 with execution probability 1.

160

Note that both equilibria have the same traded volume, post-transparency, though, is

lower in the iceberg equilibrium, as illustrated below.


when vol(xBL) = 1 + �; vol(xBS) = 1

seller S seller L

buyer SS; S

(1)

S;L

(1)

buyer LL; S

(1)

L;L

(2)

post-trade transparency equilibrium E

seller S seller L

buyer S?; ?

(1)

?; ?

(1)

buyer L?; ?

(1)

l; L

(2)

Table 29: Post-Trade Transparency in Limir Order Equilibrium and Equilibrium E.

7.3 For Section 5.4 (Interdependent Values with Discretionary orders)

Proof of Proposition. 11Step 1: Structure or all equilibriaThis step is exactly like in the private values case. Speci�cally, it was shown in the

corresponding Proposition 3 for the case of private values, that there are four possible

classes of equilibria, named A-D and with buyer orders, as illustrated below.

class Dclass Cclass Bclass A

Q Q Q Q

P

Lp Lp Lp LpHp Hp Hp Hp

P P P

Figure 42: Buyer Orders in Discretionary Equilibrium Classes A, B, C, and D.

The full set or orders as well as volume traded for each buyer and seller in each equilib-

rium is spelled out in the tables below:

161

A order vol

Large buyer (pL(pH)) 1

Small buyer pL �

Large seller pL 1

Small seller pH �

B order vol

Large buyer (pL(pH); pL) 1 + �



Small seller pH �

C order vol

Large buyer (pL(pH); pL(pH)) 2



Small seller (pH;pH) 2�

D order vol

Large buyer (pH ; pL(pH)) 2

Small buyer (pH ; pL) 1 + �


Small seller (pH;pH) 1 + �

Table 30: Equilibrium Orders in Discretionary Order Equilibria A Through D.

Step 2: Calculating equilibriaIn this interdependent values setup, equilibria of types A, B, C and D exist. Below are

numerical examples.

Equilibrium ALet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:3; 7; 2; 11; 6; 5:7; 8; 2:5; 3:5g

In this case, pL = 5:04 and pH = 9.

The intuition for why this equilibrium exists when values are interdependent is that the

large seller receives a rent when selling at pL (this is not true in the private values setting).

For the given parameter combination, a separating limit order equilibrium exists in which

the large buyer submits an order with execution probability 1+� and the small submits an

order with execution probability �. Thus the introduction of discretionary orders can have

led to a decrease in trading volume.

Equilibrium BLet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:8; 4:1; 3:5; 8:6; 7:2; 3:3; 4:6; 1:2; 1:7g

.

In this case, p1 = p2 = 2:087 and pD1 = 5:32.

Note that in this case, a pooling equilibrium with limit orders would exist, in which both

buyers submit a limit order with execution probability 2�, priced at (p1; p2) = (2:087; 2:087):

162

Equilibrium CLet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f 0:2; 0:1; 0:5; 6; 3:5; 10; 8; 4; 4:2; 1:1; 1:6g

.

In this case, the visible order is (pL; pL) = (2:75; 2:75) and the large buyer hides tow

units at pH = 6:8.

For the same parameters, a separating limit order equilibrium exists in which the large

buyer submits an order with execution probability 2 (priced at (p1; p2) = (6; 5:8)) and the

small buyers submits an order with execution probability 2� (priced at (p1; p2) = (2:3; 2:3)).

Volume is thus the same in the discretionary order and the limit order equilibria, but

transparency has switched in the discretionary order equilibrium, where �switched�is used

in the sense �rst described in Section 4.4.2.

Equilibrium DLet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:15; 0:4; 9; 5:4; 10; 7; 5:7; 5:8; 2:5; 2:7g

.

In this case, the visible order is (pH ; pL) = (7:53; 4:3425) and the hidden discretionary

price on the second unit is equal to pH .

Proof of Corollary. 8For cases B and C, this proof is exactly like the proof for the analogous Proposition 2

in the private values section. The following table illustrates that the statement also holds

for equilibria of classes A and D.

post-trade transparency equilibrium A

seller S seller L

buyer SS; S

(0)

?; L

(1)

buyer LL; S

(1)

?; L

(1)

post-trade transparency equilibrium D

seller S seller L

buyer SS; S

(1)

?; L

(2)

buyer LL; S

(2)

?; L

(2)

Table 31: Post-Trade Transparency in Equilibria A and D.

Proof of Corollary. 9

163

The following example has exactly the same structure as that given for the proof of the

analogue Corollary 3 in the private values case.

In particular, an equilibrium of class B exists for the following parameter combination

�.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:2; 0:7; 3; 2:7; 9:5; 9:3; 2:5; 2:9; 1:2; 1:4g

For the given �, a separating limit order equilibrium exist, in which the optimal order

for the small buyer type has an expected volume of 2� (priced at (p1; p2) = (1:67; 1:67)),

and the optimal strategy for the large buyer type involves buying two units for sure, at

pBL1 = pBL2 = 3:83.

If equilibrium B is played, the large buyer type�s optimal order is to pool onto visible

prices p1 = p2 = 1:884 and to submit a hidden price at pH1 = 3:83; with a total expected

trade volume on the order of 1 + �. Total trade volume is thus reduced to �(1 + �) + (1��)(2�) in class B compared to the limit order equilibrium (where trading volume is equal

to �(2) + (1� �)(2�)).

In contrast, volume may also be increased when discretionary order are introduced. Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:8; 4:1; 3:5; 8:6; 7:2; 3:3; 4:6; 1:2; 1:7g

.

In this case, and equilibrium of class B exists, with a visible order of (pL; pL) =

(2:087; 2:087) and a hidden price at pH = 5:32.

Fro the same �, a pooling equilibrium with limit orders would exist, in which both buyers

submit a limit order with execution probability 2�, priced at (p1; p2) = (2:087; 2:087). This

pooling equilibrium would thus have lower expected trade volume than the discretionary

order equilibrium.

Proof of Corollary. 10The following two examples for statement i) and ii) have exactly the same structure as

those given for the proof analogue Corollary 4 in the private values case. The proof there.

The parameter values have to be adjusted only slightly to be as stated below.

For i):

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:7; 5:4; 3; 13; 9; 3:5; 7; 1:4; 1:6g

164

In this case, equilibrium C exists, with visible prices (p1; p2) = (2:518; 2:518) and hidden

prices (pD1 ; pD2 ) = (7:9; 7:9).

Fro the given �, a separating limit order equilibrium exists in which the large buyer type

submits a limit order with execution probability 1 + � (priced at (p1; p2) = (4:8; 2:5)) and

the small buyer type submits an order with execution probability 2� (priced at (p1; p2) =

(1:94; 1:9)). Thus, pre- and post-trade transparency would be reduced with discretionary

orders, but trade volume would increase.

For ii) Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f 0:2; 0:1; 0:5; 6; 3:5; 10; 8; 4; 4:2; 1:1; 1:6g

.

An equilibrium of class C exists with pL = 2:75and pH = 6:8. A separating limit order

equilibrium also exists, in which the large buyer type submits an order with execution

probability 2 (priced at (p1; p2) = (6; 5:8)) and the small buyer type submits an order with

execution probability 2� (priced at (p1; p2) = (2:3; 2:3)). Volume is thus the same in both

equilibria, but transparency has �switched�in the sense described in Section 4.4.2.

7.4 For Section 5.5 (Interdependent Values with Volume Orders)

Proof of Proposition. 12i) In the following case, buyers pool onto a volume order

Let


The optimal volume order is for both buyers to submit a visible price of p1 = 7:99 and

hide two units at (pvol1 ; pvol21 ) = (8:54; 8:54): The large seller type then sells two units at the

hidden price and the small seller type sells one unit at the visible price.

ii) In the following case, buyer types separate on the visible part of the order, and one

of the buyers submits a volume order. In equilibrium the seller will know the hidden part

to any visible order. As a consequence, the buyer types may as well disclose the hidden

part of their order. In the case where the volume order maps into a demand schedule (with

the price on the second unit below that of the �rst unit), buyers could submit limit orders

instead of the volume order.

An example of a volume order implementing a supply schedule instead is below.

165

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:3; 0:5; 16; 7; 18; 15; 4; 13:5; 1; 9g

Here the small buyer type submits a unit priced at p1 = 8:8. The large buyer type

submits a volume order with visible price p1 = 9:4 and two hidden units at (pvol1 ; pvol21 ) =

(11:45; 11:45) with a volume condition. The seller responses are as follows: the small seller

type always submits a one unit market order (that is, he sells one unit), the large seller type

submits a one unit market order when the buyer type is small and a two-unit limit order

at the hidden price when the buyer type is large.

Proof of Proposition. 13The structure of the proof is the same as that of the proof for the corresponding Propo-

sition 5 in the private values setting. In addition, none of the arguments used in the proof

of Proposition 5 to identify the set of candidate equilibrium classes relied on values be-

ing private. Thus, for the case of interdependent values, one can restrict attention to the

candidate classes identi�ed in Proposition 5 for private values.

In the following, the equilibrium candidate classes identi�ed in Proposition 5 will be

considered one by one, in the same order as in the private values setting. First, though,

a list of all equilibria that are veri�ed to exist is added below. Equilibria are organized

by the groups i through iv identi�ed in Proposition 13, descriptions found in brackets are

(intuitive) labels for the class.

This is how the cases below relate to the groups in the proposition:

Group ii)class A (like discretionary A)

class B (like discretionary B)

class C (like discretionary C)

class Cb (like discretionary C and bundling)

class D (like discretionary D)

class21a (like discretionary C and supply)

class 23a (like discretionary C and demand)

Group i)class G (iceberg and demand and separation)

class H (iceberg and demand and pooling)

class I (iceberg and supply and separation)

class J (iceberg and supply and pooling)

class K (iceberg and supply )

166

class Kb (iceberg and supply and bundling)

Group iii)class L (screening on 2nd)

class M (pooling, supply and screening on 2nd)

class N (pooling, supply and screening with lotteries across units)

class O (pooling, supply and screening with lotteries across units)

Group iv)class 22 (non-monotonic)

There are two cases:

A) Buyer types pool onto a visible part, but only one buyer type submits a volumeorder by hiding at least one more unit.

B) Buyer types pool onto a visible order and both buyer types submit volume orders,where the hidden parts of the volume orders di¤er.

Case A:First note that the large buyer has to be the one submitting the volume order, because

supermodularity implies that he has to trade more in equilibrium.

A.1.One unit hidden, with pvol1 > p1 (hidden unit priced above the �rst visibleunit):

There are two subcases i) and ii) as pictured below (left and right picture). The numbers

next to the pictures denote the classes of equilibria that can arise with these buyer orders.

Case A.1.2:classes B, L

Case A.1.1:classes A, I, J, O

Q

P

LpHp

Q

Lp

HpP

Mp


For case A.1.1:Buyer orders: The small buyer type submits a one-unit limit order at pL , the large


Class A (like discretionary A). The small seller submits a limit order at pH and

the large seller submits a limit order at pL.

Let � be:

167

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:2; 0:2; 5:4; 2; 11; 3; 5; 13; 2:5; 10g

In this case, the buyers pool onto the visible price pL = 4:304. The small buyer only sub-

mits the visible unit, while the large buyer submits a volume order (pL; pH) = (4:304; 7:2).

Class O (pooling, supply and screening with lotteries across units )The smallseller submits a limit order at pH and the large seller submits a volume order vol(pL; pH).

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:5; 5; 1:8; 12; 8:8; 6; 13:3; 1; 3:5g

In this case, the buyers pool onto the visible price pL = 2:864. The small buyer only sub-

mits the visible unit, while the large buyer submits a volume order (pL; pH) = (2:864; 9:6).

Note that the prices for an order with execution probability 1+� for the large in a separating

equilibrium would be (p1; p2) = (9:6; 6:14):

Class I (iceberg and supply and separation) The small seller submits a limit orderat pL, and the large seller submits a volume order vol(pL; pH).

Let


For these sets of parameters, buyer types pool onto a visible price of pL = 5:36 and the

large buyer type hides another unit at pH = 7:7.

The intuition is that the volume order allows the large buyer to receive a lower price on

the �rst unit by pooling with the small buyer and to submit something resembling a supply

schedule (which may be optimal).

This equilibrium is dominated though in that the large buyer could submit a volume

order with a visible price of pL and two hidden units at p�H = (pL + pH)=2 and impose a

volume condition. Seller responses would be: small seller sells at pL and large seller submits

a volume order with two hidden prices at p�H and one visible at pL.

Class J (iceberg and supply and pooling) Both sellers submit a volume ordervol(pL; pH).

For


The volume order equilibrium has a visible price of pL = 6:21 and a hidden price of

pH = 10:8. In this case, the volume order allows the large buyer to both receive a lower

168

price on the �rst unit and to submit a supply schedule.

( For the given �, a separating equilibrium exists in which the large buyer type buys

units, priced at (p1; p2) = (10:8; 10:8) and the small buyer type buys one unit priced at

pL = 5:4. This equilibrium has the same expected trade volume, but the large buyer type

has a higher payo¤ in the volume order equilibrium, because the prices he pays there are

lower.)

The volume order equilibrium is dominated, though, in that the large buyer could submit

a volume order with a visible price of pL and two hidden units at p�H = (pL + pH)=2 and

impose a volume condition. Seller responses would be: both sellers submits a volume order

with two hidden prices at p�H and one visible at pL.

For case A.1.2:Buyer orders: The small buyer type submits a two-unit limit order at (pM ; pL), the large

buyer type submits a volume order vol((pM ; pL); pH).

Class B (like discretionary B).The small seller submits a limit order at pH and the

large seller submits a two unit limit order at pL.

Let


:

In the volume order equilibrium, both buyer types pool onto the visible order (pL; pL) =

(2:635; 2:635) and the large buyer type also hides a price pH = 5:2 (that is, the large buyer

type submits a volume order vol((pL; pL); pH) ). The intuition here is that the volume order

allows the large buyer type to separate sellers, much like in the case of discretionary order

equilibrium B.

Note that in this case, a pooling equilibrium with limit orders exists, in which both buyer

types submit a limit order with execution probability 2�, priced at (p1; p2) = (2:385; 2:385).

Class L.(iceberg and supply and screening on 2nd) The small seller type submitsa volume order vol(pM ; pH) and the large seller type submits a two unit limit order priced

at pL.

Let


.

In this case, the buyer types pool onto (p1; p2) = (8:34; 6:072), while the large seller type

hides a unit at pvol1 = 8:7.

169

A separating equilibrium exists in which the large buyer type submits an order with

execution probability 2 and the small buyer type submits an order with execution probability

1+ � (In the volume order equilibrium, the large buyer type is better o¤, as he would have

to pay (p1; p2) = (8:7; 8:7) on both units in the separating equilibrium. The small buyer is

worse o¤, as he would have paid (p1; p2) = (8:3; 5:78) on a limit with the same execution

probability, of 1 + �, as his order in the volume order equilibrium).

A.2. One unit hidden, with pvol1 < p1 (hidden unit priced below the �rstvisible unit):

There are two subcases as pictured below:

Case A.2.2:class D

Case A.2.1:classes G, H

Q

P

LpHp

Q

LpHpP

Mp


For case A.2.1:Buyer orders: The small buyer type submits a limit order priced at pH , and the large

buyer type submits a volume order vol(pH;pL).

Class G (iceberg and demand).The small seller type submits a limit order at pHand the large seller type submits a two-unit limit order at pL.

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:4; 9:5; 2; 15; 11; 5; 9; 1; 3g

In this case, the buyer types pool onto the visible order priced at p1 = 8:015 and the

large buyer type also hides a unit at pvol2 = 6:3.

The large buyers best order with separation would be one of execution probability 1+�,

but the price on the �rst unit he would have to pay then would be much higher, the two

unit order would be (p1; p2) = (9:5; 6:3).

The intuition for the volume order equilibrium is that the volume order allows the large

buyer type to receive a better price on the �rst unit by pooling with the small buyer.

170

Class H.(iceberg and demand and pooling) Both seller types submit a two-unitlimit order priced at pL (or a volume order vol(pH ; pL)).

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:25; 0:4; 0:3; 13:4; 4; 15:6; 12:8; 5:9; 6; 2; 5:4g

.

In this case, the buyer types pool onto p1 = 9:47 and the large buyer hides a unit at

pvol2 = 9:2. In this case, the volume order allows the large buyer to receive a lower price on

the �rst unit.

(A separating limit order equilibrium exists in which the large buyer type submits an

order with execution probability 2 priced at (p1; p2) = (9:8; 9:2), and the small buyer type

submits an order with execution probability 1 priced at p1 = 9:25. The prices the large buyer

type pays to buy two units with certainty are thus lower in the volume order equilibrium.)

For case A.2.2:Buyer orders: The small buyer type submits a two-unit limit order priced at (pH ; pL),

and the large buyer type submits a volume order vol((pH ; pL); pM ).

Note that In this case, pH � pM > pL.

Class D (like discretionary D)The small seller type submits a two-unit limit orderat pM and the large seller type submits a two unit limit order at pL.

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:7; 11; 4; 12; 8; 35; 3:7; 1:1; 1:3g

.

Then the small and the large buyer pool onto and order of (p1; p2) = (4:5; 1:98), and the

large buyer hides a unit at pvol2 = 4:61.

A pooling equilibrium with limit orders exist in which both buyer types pool onto

(p1; p2) = (4:61; 1:74). Thus volume is large in the volume order equilibrium.

(Note that in order for the volume order equilibrium to exist, the vBik of the buyer

types on the second unit are signi�cantly lower than on the �rst unit, while the vSjk of

the seller types do not increase much from the �rst to the second unit, so that the sellers

reservation price on the �rst unit, conditional on the buyer type being unknown, is higher

than that on the second unit, conditional on the buyer type being large. essentially, in this

interdependent values case, seller valuations (vSjk + � � (� � vBLk + (1 � �) � vBSk)) ) arehigher on the �rst than on the second unit.)

A.3. Two units hidden, with pvol1 = pvol2 > p1 (hidden unit priced above the

171

�rst visible unit):There are two cases, as illustrated below. Note that there are no cases in which two

units are hidden at a price below the visible price, because the second hidden unit would

never execute.

Case A.3.2:class C

Case A.3.1:class K

Q

P

LpHp

Q

Lp

HpP

Mp


For case A.3.1:Buyer orders: The small buyer submits a limit order priced at pL, and the large buyer

submits a volume order at vol(pL; (pH;pH)).

Class K (iceberg and supply).The small seller type submits a one-unit limit orderat pL, and the large seller type submits a volume order vol(pL; [pH ; pH ])

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:4; 0:4; 9; 4; 11; 10; 5; 12; 2:5; 7g

In the volume order equilibrium, the visible price is p1 = 7:525 and the hidden prices

are (pv1; pv2) = (8:0125; 8:0125).

For case A.3.2:Equilibria involving two hidden units at a price pvol1 = pvol2 > p1.Buyer orders: The small buyer type submits a two-unit limit order priced at (pM ; pL),

and the large buyer type submits a volume order at vol((pM ; pL); (pH;pH)).

Class C (YES, like discretionary C).The small seller type submits a two-unit limitorder at pH , and the large seller type submits a two unit limit order at pL:

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:1; 0:7; 7; 6; 9; 8; 4; 4:2; 1:5; 2:2g

172

In the volume order equilibrium, the visible prices are (pM ; pL) = (3:792; 3:792) and the

hidden prices are (pv1; pv2) = (7:32; 7:32).

A.4. Two units hidden, with pvol1 > p1 (hidden unit priced above the �rstvisible unit) and volume condition:

Note that below, arrows on the lines representing the hidden prices indicate that the

buyer has included a (minimum) volume condition of �2�: the hidden units may only execute

if both execute at the same time (as a bundle).

Case A.4.2:class Cb

Case A.4.1:class Kb

Q

P

LpHp

Q

Lp

HpP

Mp


For case A.4.1:Buyer orders: The small buyer submits a limit order priced at pL, and the large buyer

submits a volume order vol(pL; [pH ; pH ]).

Class Kb (iceberg and supply and bundling).The small seller type submits a limitorder at pL and the large seller type submits a volume order vol(pL; [pH ; pH ])

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:4; 0:4; 9; 4; 11; 10; 5; 12; 2:5; 7g

In the volume order equilibrium, the visible price is p1 = 7:44 and the hidden prices are

(pv1; pv2) = (7:97; 7:97).

For case A.4.2:Note that in this case, pvol1 = pvol2 > p1 � p2.Buyer orders: The small buyer submits a two-unit limit order at (pM ; pL) and the large

buyer submits a volume order at vol((pM ; pL); [pH ; pH ]).

Class Cb (like discretionary C with bundling).The small seller type submits atwo-unit limit order at pH and the large seller type submits a two-unit limit order at pL.

173

In this case, the large buyer has an incentive to pool with the small in order to be able

to separate sellers, much like in the case of discretionary orders.

Let


.

A volume order equilibrium exists in which both buyer types pool onto the visible order

of (pL; pL) = (2:978; 2:978) and the large buyer type hides a price of pH = 6:82 on both

units.

A separating limit order equilibrium also exists, in which the large buyer type submits

an order with execution probability 2 (priced at (p1; p2) = (6:3; 6:3)) and the small buyer

type submits an order with execution probability 2� (priced at (p1; p2) = (2:3; 2:3)). Volume

is thus the same in both equilibria, but transparency has switched in the sense de�ned in

Section 4.4.2.

Case B:B.1. Both buyers hide one unit:In the private values analysis, it was shown that only one subcase remains, which is

illustrated.

Case B.1.1:classes M, N

Q

P

LpHpMp

Figure 47: Volume Order Equilibrium Candidate Classes, Cases B.1.1.

For case B.1.1:Buyer orders: the small buyer type submits a volume order vol(pL; pM ), and the large


Class N (pooling, supply and screening with lotteries across units)The smallseller type submits a one-unit limit order at pH and the large seller type submits a volume

order vol(pL; pM ).

Let

174


:

In this case, both buyer types pool onto the visible price p1 = 8:8, the large buyer type

also hides pvol;H2 = 12:8; and the small buyer type hides pvol;M2 = 11:15. In this equilibrium

the large buyer type bene�ts from pooling with the small type as he receive a lower price

on the �rst unit than he would have if he revealed his type.

Class M (pooling, supply and screening on 2nd)The small seller type submits avolume order vol(pL; pH).and the large seller type submits a volume order vol(pL; pM ).

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:12; 0:1; 0:5; 12; 9:5; 16; 14; 3; 5; 1:5; 4g

.In this case, both buyers pool onto the visible price p1 = 5:046. The large buyer also hides

pvol;H2 = 7:62 and the small buyer hides pvol;M2 = 5:14. This again is an equilibrium in which

pooling allows the large buyer type to receive a lower price on the �rst unit he buys. More

intuition is found in the text in Section 5.5.1.

For case B.1.2 and B.1.3:It was shown in the private values setting that the only possible candidate equilibria in

these cases are equivalent to the equilibria B, L and D.

B.2. One buyer hides one unit, the other hides two units:

The three general cases, as illustrated below without a volume condition on the two

hidden units, though equilibria in with a volume condition on the two hidden units are also

analyzed below. (Remember that as noted in the private values setting, it is not immediately

obvious which of the buyer types and which seller types would be submitting which of the

orders.)


hidden units.As has been shown in the private values case, no equilibrium in this case can exist. (note

that none of the steps in the proofs of the private values case necessitated that the values

were private rather than interdependent).


175

Case B.2.2:Nonmonotonic 3

Case B.2.1:Nonmonotonic,1 and (2)

Q

P

LpHp

Q

LpHpP

Mp

Case B.2.3:Nonmonotonic,4 and (5)

Q

LpHpP

MpMp

Figure 48: Non-Monotonic Volume Order Equilibria.

Buyer orders: One buyer type submits a volume order vol (pL;pM ) and one buyer type

submits a volume order vol (pL;(pH ; pH)).

Equilibrium NM2 may exist, a numerical example is given below for equilibrium NM1.

NM1 (non-monotonic) The large buyer type submits vol(pL; (pH ; pH)) and the smallbuyer type submits vol(pL; pM ). The large seller type submits vol(pL; pM ) (1) and the small




(S): vol(pL; pM ) pL; pM 0



Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:1; 0:75; 3:8; 3:5; 9:5; 9; 3:5; 4; 1:4; 2:5g

.

In this case, both buyer types pool onto the visible price p1 = 4:25. The large buyer type

also hides two units at pvol;H2 = 6:7 and the small buyer type hides one unit at pvol;M2 = 4:25.

Note that pvol;H2 = 6:7 is also the reservation value of the small seller type on the second

unit when he knows that the buyer type is large.


hidden units.Using exactly the same arguments as in the private values case, one can exclude all

candidate equilibria in this case.

176


Buyer orders: One buyer type submits a volume order vol (pL;pH) and one buyer type

submits a volume order vol (pL;(pM ; pM )).

NM3 (non-monotonic) One buyer type submits a volume order vol (pL;pH) andthe other buyer type submits a volume order vol (pL;(pM ; pM )).One seller type submits

vol(pL; pH) (1) and the other seller type submits a two-unit limit order at pM (2).


buyer/seller 1 2

vol(pL; pM ) pL; pH pH

vol(pL; (pH ; pH)) pL pM ; pM


Let:


In this equilibrium, the large buyer type submits vol(pL; pH) = vol(6:44; 9:14), the small

buyer type submits vol(pL; (pM ; pM )) = vol(6:44; (8:2; 8:2)). The large seller type submits

two units at pM and the small seller type submits vol(pL; pH).


hidden units.Using exactly the same arguments as in the private values case, one can exclude all

candidate equilibria in this case.


Buyer orders: One buyer type submits a volume order vol (pM;pL) and the other buyer

type submits a volume order vol (pM;(pH ; pH)).

Equilibrium NM5 may exist, a numerical example is given below for equilibrium NM4.

NM4 (non-monotonic) The large buyer type submits vol(pM ; (pH ; pH)) and the smallbuyer type submits vol(pM ; pL). The large seller type submits vol(pM ; pL) (1) and the small



177





Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:12; 0:2; 0:6; 8; 5:8; 10:7; 7:1; 5:9; 6:5; 1:5; 3:5g

.

In this case, both buyer types pool onto the visible price p1 = 4:346. The large buyer

type also hides two units at pvol;H2 = 7:352 and the small buyer type hides one unit at

pvol;M2 = 4:296.

Proof of Proposition. 11i) Example where total trade volume went down:

Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:4; 4:5; 3:5; 10; 8:5; 3:4; 4:5; 1:2; 2g

.

In this case, a volume order equilibrium of class B exists, where buyers pool onto a

visible order (p1; p2) = (2:65; 2:65) and the large buyer hides a price pH1 = 5:35. In the

absence of volume orders, a separating equilibrium would exist, in which the large buyer

buys two units at (p1; p2) = (5:35) and the small buyer submits an order with execution

probability 2�. priced at (p1; p2) = (2:35; 2:35).

ii) The following is an example for the case in which a volume order equilibrium has

higher trading volume than a limit order equilibrium for the same parameters. Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:7; 11; 4; 12; 8; 3:5; 3:7; 1:1; 1:3g

.

Then a volume order equilibrium of class D exists. The small and the large buyer type

pool onto and order of (p1; p2) = (4:5; 1:98), and the large buyer hides a unit at pvol2 = 4:61.

A pooling equilibrium with limit orders exist in which buyers pool onto an order with

execution probability 1 + �, priced at (p1; p2) = (4:61; 1:74). Thus volume is higher in

equilibrium of class D.

178

(Note that in order for the volume order equilibrium to exist, the vBik of the buyer

types on the second unit are signi�cantly lower than on the �rst unit, while the vSjk of

the seller types do not increase much from the �rst to the second unit, so that the sellers

reservation price on the �rst unit, conditional on the buyer type being unknown, is higher

than that on the second unit, conditional on the buyer type being large. essentially, in this

interdependent values case, seller valuations (vSjk + � � (� � vBLk + (1 � �) � vBSk)) ) arehigher on the �rst than on the second unit.)

Proof of Corollary. 12For i):

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:1; 0:6; 5:4; 3; 13; 9; 3:6; 7; 1:4; 1:6g

In this case, equilibrium C exists, with visible prices (pL; pL) = (2:518; 2:518) and hidden

prices (pH ; pH) = (7:9; 7:9).

A separating limit order equilibrium exists, in which the large buyer submits a limit

order with execution probability 1 + � (priced at (p1; p2) = (4:9; 2:5)) and the small buyer

submits an order with execution probability 2� (priced at (p1; p2) = (1:94; 1:9)). Thus, pre-

and post-trade transparency would be reduced with discretionary orders, but trade volume

would increase.

For ii) Let


.

In this case, a volume order equilibrium of class C exists with visible order (pL; pL) =

(2:978; 2:978) and hidden prices (pH ; pH) = (6:82; 6:82).

A separating limit order equilibrium exists in which the large buyer type submits an

order with execution probability 2 (priced at (p1; p2) = (6:3; 6:3)) and the small buyer type

submits an order with execution probability 2� (priced at (p1; p2) = (2:3; 2:3)). Volume

is thus the same in both equilibria, but transparency has switched in the sense de�ned in

Section 4.4.2.

7.5 For Section 5.6 (Interdependent Values and Optimal Mechanisms)


179

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:4`; 0:5`; 5:5`; 4:4`; 7; 6; 1:56`; 3; 1; 1:5`g

.

In this case, the RSW allocation is as follows. The large buyer proposes a mechanism

�L = (xL; tL) with trade quantities

xL = fxLL1; xLL2; x�LS1; xLS2g = f1; 1; 1; 1g (that, is he buys two units from both the

small and the large seller type) and with transfers tL = ftLL; tLSg = f8:46; 8:46g. Thepayo¤ achieved by the large buyer is 5:599, the total transfers paid are 8:46. The small

buyer proposes a mechanism �S = (xS ; tS) with trade quantities

xS = fxSL1; xSL2; xSS1; xSS2g = f1; 1; 1; 0g (buying two units from the large and one

unit from the small seller) and with transfers tS = ftSL; tSSg = f6:42; 3:6g. The payo¤achieved by the small buyer is 3:299, the total transfers paid are � �tSL+(1��) �tSS = 5:01..

The optimal mechanism � = (x; t), when weight one is put on the large buyer type, has

trade quantities of

x = fxLL1; xLL2; xLS1; xLS2; xSL1; xSL2; xSS1; xSS2g = f1; 1; 1; 1; 1; 1; 1; 0g. Now the

payo¤ achieved by the large buyer type is 6:184 and by the small buyer (as before) is

3:259. Since the volume traded as well as the payo¤ for the small buyer is the same as in

the RSW allocation, his total transfers remains the same as well. Transfers paid by the

large buyer are reduced by 0:585 to � � tLL + (1 � �) � tL = 7:875, thus transfers receivedby the sellers are reduced by that amount.

For completeness, the following is an example in which the buyers pool onto a contract

� = (x; t), such that both buyers trade the same volume (namely two units).

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:3; 0:102`; 0:7; 9:3; 9; 9:5; 9:31`; 2:5`; 2:88; 1:16`; 1:35`g

.

Speci�cally here the associated trade quantities are x = f1; 1; 1; 1; 1; 1; 1; 1g.

Proof of Proposition. 15The following is an example in which there is no pooling mechanism and the separating

mechanism for one of the buyers has the characteristics of a supply schedule. Let

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:4; 0:5; 1:5; 1:4; 3; 2; 1:1; 3; 1; 1:5g

In this case, there is no pooling mechanism when the weight on the large buyer type is

1. The separating mechanisms are as follows.

The small buyer proposes �S with trade quantities xS = fxSL1; xSL2; xSS1; xSS2g =f1; 0; 1; 0g,

180

and transfers tS = ftSL; tSSg = f1:25; 1:25g.The large buyer proposes �L with trade quantities xL = fxLL1; xLL2; xLS1; xLS2g =

f1; 1; 1; 0g,and transfers tL = ftLL; tLSg = f3:1; 1:4g.The contract for the large buyer has the characteristics of a supply schedule because

tLL � tLS = 1:7 > tLS = 1:4.

The following is an example in which no pooling mechanism exists (when the weight on

the large buyer is 1), and where the separating mechanism for one of the buyers features

bundling.

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:1; 0:4; 0:5; 1:5; 1:4; 2; 1:75; 1:56; 3; 1; 1:5g

The separating mechanisms are as follows. The small buyer proposes �S = (xS ; tS) with

trade quantities

xS = fxSL1; xSL2; xSS1; xSS2g = f1; 0; 0; 0g and transfers tS = ftSL; tSSg = f1:15; 0g.The large buyer proposes �L = (xL; tL) with

xL = fxLL1; xLL2; xLS1; xLS2g = f1; 1; 0; 0g and tL = ftLL; tLSg = f2:875; 0g. Thecontract for the large buyer has bundling features: if the large buyer tried to submit a limit

order priced at p1 = p2 = tLL=2 = tLS = 1:4375, then the large buyer type would prefer to

deviate and sell only one unit instead of 2. This is because the payo¤ to the large seller in

the mechanism is zero, whereas it would be 0:2375 if he was able to sell one unit at p1 =

1:4375.

Numerical example #1: Mechanism allocation that is replicable as equilib-rium allocation

Let� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:2; 0:6; 4; 2:5; 4:2; 2:6; 1:8; 3:5; 1; 3:2g

Then the buyer types pool onto a contract � = fx; tg, where all possible buyer and sellertype combinations always trade one unit. That is x = fxL; xSg = f1; 0; 1; 0; 1; 0; 1; 0g.Moreover, the associated transfers could be

t = ftL; tSg = ftLL; tLS ; tSL; tSSg = f4:34667; 0; 2:17333; 3:26g.This particular contract is not expost-implementable, and also not implementable with

limit orders, as a zero transfer would have to be associated with the trade of one unit

(between the large buyer and the small seller type).

On the other hand, the following contract in the same equivalence class can be imple-

mented with orders: ~� = fx; tg, with the same x as in �, and transfers~t = f2:608; 2:608; 2:608; 2:608g:

181

These transfers lead to the same expected transfers for each of the player types as with

t, because M � t = M � ~t. Moreover, the following strategies in the basic game (involvingonly limit orders) construct an equilibrium that has the same traded quantities and prices

as the optimal contract: both buyer types submit a one unit order priced at p1 = 2:608.,

and both seller types sell at that price (by submitting either a one unit market order or a

one unit limit order at p1).

To verify that these strategies are in fact part of a pooling equilibrium as stated, one has

to verify that both seller types would be willing to sell at that price, and that both buyer

types would prefer to pool onto that order rather than deviating to another limit order.

Given that both seller types accept the contract, it must be optimal for the sellers to sell

one unit in the equilibrium. As for the buyer types, the payo¤ for the large buyer type is

1:856 in the pooling equilibrium, and for the small type it is 1:656. The best deviation of the

large buyer type would yield him 1:536 (it consists of deviating to an order with execution

probability of �). Note that only equilibria satisfying the intuitive criterion are considered

throughout. (Thus, the seller�s belief when observing a deviation is that the type of the

deviating buyer is large unless the large buyer would prefer his equilibrium payo¤ to the

payo¤ associated with the observed deviation.)

The large buyer bene�ts in the pooling equilibrium, because the price on the one-unit

order with execution probability of 1 if his type was known would be higher than in the

pooling equilibrium (speci�cally the price would have to be at least equal to vSS1+��vBL1 =2:64).

Numerical example #2: Mechanism that is not ex-post implementableLet

� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:2; 0:6; 5; 3; 6; 3:3; 2:2; 5; 2; 2:5g

Then the buyer types pool onto a contract � = fx; tg, wherex = fxL; xSg = f1; 1; 1; 0; 1; 1; 1; 0g:That is, the large seller always sells two units, and the small seller always sells one

unit. In the optimal contract, the payo¤ for the small seller is USS = 0. As a re-

sult, in order to guarantee ex-post implementability, for the small seller, transfers t =

ftLL; tLS ; tSL; tSSg have to satisfy�(vSS1+��vBL1)+tLS = 0 and�(vSS1+��vBS1)+tSS = 0.The unique transfers for which these two conditions hold are t = ftLL; tLS ; tSL; tSSg =f6:24533; 3:4; 6:37867; 3:2g. These transfers, though, do not satisfy ex-post implementabil-ity for the large seller type when he trades with the large buyer type, as �(vSL1 + vSL2 +� � (vBL1 + vBL2)) + tLL = 0:114667 < 0. In other words, there is no set of transfers for

which the two conditions imposed by the small seller hold and, additionally, the large seller�s

constraint holds

182


� = f�; �; �; vBS1; vBS2; vBL1; vBL2; vSS1; vSS2; vSL1; vSL2g= f0:2; 0:2; 0:6; 4; 2:5; 5; 4; 1:8; 3; 1; 2:2g

Then the buyer types pool onto a contract � = fx; tg, wherex = fxL; xSg = f1; 1; 1; 1; 1; 1; 1; 0g:That is, all buyer-seller pairs trade two units, except for the pair consisting of the small

buyer and the small seller type (who trade one unit only).

One possible set of transfers in the equivalence class is t = f10:04; 0; 4:44; 4:25g. Notethat this speci�c contract could never be replicated with orders, as tLS = 0 although the

large buyer type buys two units from the small seller type. The associated expected transfers

are T = fTBL; TBS ; TSL; TSSg = f5:56; 3:4; 6:024; 4:364g.Consider the list of the possible equilibria that would lead to the same traded quantities:

1) Limit orders only: A separating equilibrium in which the large buyer type submits

a two unit order with execution probability 2, and the small buyer type submits a two

unit-order with execution probability 1 + �.

2) Iceberg order equilibrium of class F

3) Discretionary order equilibrium of class D

4) Volume order equilibrium class D, L and M, NM2 or NM5

In order to prove that the mechanism allocation cannot be replicated, it is necessary to

exclude that equilibria of any of the above classes can exist with prices that are equal to a

set of transfers in the equivalence group of the optimal contract.

For 1):

In a separating equilibrium, the large buyer would have to submit a two-unit order with

prices of at least (p1; p2) = (7:6; 7:6), where p1 = vSS2 + � � vBL2. Thus the large buyerwould be paying at least 7:6 in expectation, which is di¤erent from the expected transfer

TBL = 6:024 that he would pay under the optimal contract, implying that a separating

equilibrium cannot replicate the optimal contract.

For 2);

In an iceberg equilibrium of class F, it must hold that tLL = tSL (that is, the large seller

always receives the same payo¤ of (p1; p2), no matter what the buyer type). Moreover, it

must be that tLS=2 = tSS = tSS , since the large buyer pays 2 � p1 to the small seller (p1 onboth a visible and a hidden unit), whereas the small buyer only buys one unit at the visible

price of p1 from the small seller.

If tLL = tSL is imposed, then the contract in the equivalence class which satis�es the

constraint has t = f5:56; 6:72; 5:56; 2:57g, so that tSL=2 = 6:72=2 = 3:36 > 2:57 = tSS ,

implying in turn that an iceberg equilibrium of this class cannot replicate the optimal

contract.

For 3)

183

A discretionary order equilibrium of class D has the same structure of prices as an

iceberg equilibrium of class F, so it cannot implement the optimal contract either.

For 4)

First, consider a volume order equilibrium of class L. In this case, it must be that

tLL = tSL = pM + pL and tSS = pM > tSL=2. Note though, that in 2) it was shown that if

tLL = tSL, then t = f5:56; 6:72; 5:56; 2:57g, so tSS = 2:57 < 2:78 = 5:56=2 = tLL=2. Thus,no equilibrium of this class can exist.

Next, consider an equilibrium of class D. Here tLL = tSL = pH + pL. Moreover, tLS =

pH + pM and tSS = pH . Thus, it must hold that tSS > tLS=2. But when tLL = tLS , then

t = f5:56; 6:72; 5:56; 2:57g, so tSS = 2:57 < 3:36 = 6:72=2 = tLS=2:Consider an equilibrium of class M. Here, it must be that tLL = tLS = pL + pH ,

and also tSL = pL + pM and tSS = pL so that tSS < tSL=2: When tLL = tLS , then

t = f6:024; 6:024; 5:444; 2:744g, so tSS = 2:744 > 2:722 = 5:444=2. Thus, again, such an

equilibrium cannot exist.

Next consider an equilibrium of class NM2. In equilibrium, the following holds: tSL =

pL + pM ; tSS = pM ; tLL = pL + pH ; tLS = 2 � pH . Thus, tSL � tSS = pL = tLL � 0:5 � tLS :For the transfers that satisfy this equation,

(t = f5:89263; 6:22105; 5:47684; 2:69474g), though, tSL � tSS = 2:78211 = pL > tSS =

2:69474 = pM . Thus, no equilibrium of this class could exist that implements the contract.

Finally consider an equilibrium of class NM5. In equilibrium, the following holds: tLL�tSS = pH = 0:5 � tLS . For the transfers that satisfy this equation,

t = f5:82909; 6:31636; 5:49273; 2:67091g, though, tSL � tSS = 2:82182 = pL > tSS =

2:67091 = pM , Thus no equilibrium of this �nal candidate class can implement the contract

either.

184

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