auctions and market design - peter...

Market DesignPeter Cramton

Professor of EconomicsUniversity of Maryland

European University InstituteUniversity of Cologne

www.cramton.umd.edu

3 October 2016

With contributions fromAxel OckenfelsUniversity of Cologne

1

http://www.cramton.umd.edu/courses/market-design/http://www.cramton.umd.edu/papers/spectrum/

Market design

Establishes rules of market interaction Economic engineering

Economics Computer science Engineering, operations research

2

Market design accomplishments

Improve allocations Improve price information Mitigate market failures

Reduce risk Enhance competition

3

Applications

Matching (marriage, jobs, schools, kidneys, homes) Auctions

Electricity markets Communication markets Financial securities Transportation markets Natural resource auctions

(timber, oil, diamonds, pollution emissions) Procurement

4

Innovation in a few words

Electricity Open access Pay for performance

Communications Auction spectrum Open access

Transportation Price congestion

Climate policy Price carbon

Financial securities Make time discrete

5

Resistance to market reforms is the norm: Peter to regulator, "I can save you $10 billion."

Wall Street to regulator, "Hey, thats my $10 billion."Distribution trumps efficiency

Milton Friedman

There is enormous inertiaa tyranny of the status quoin private and especially governmental arrangements. Only a crisisactual or perceivedproduces real change. When that crisis occurs, the actions that are taken depend on the ideas that are lying around. That, I believe, is our basic function [as economists]: to develop alternatives to existing policies, to keep them alive and available until the politically impossible becomes politically inevitable.

6

Lectures

9-10:30am, Monday-Friday, 3-7 October 10-1pm, Thursday, 13 October

7

Assessment

Select a market design application of your choice Draft an essay of less than ten pages in pdf that

develops some aspect of the market design application

Focus on a narrow question or provide a broader summary of issues

8

What is economic engineering?

9

Economic engineering

science of designing institutions and mechanisms that align individual incentives

and behavior with the underlying goals

10

The Economist as Engineer(Roth 2002)

Understand how markets work in enough detail so we can fix them when theyre broken

Build an engineering-oriented economics literature Otherwise, the practical problems of design will be

relegated to the arena of "just consulting" and we will fail to benefit from the accumulation of knowledge which is so evident in other kinds of engineering

See web.stanford.edu/~alroth marketdesigner.blogspot.com

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http://web.stanford.edu/%7Ealroth/http://marketdesigner.blogspot.com/

Useful to distinguish two kinds of complexities

Institutional detail

Behavior

14

Economic engineering

Identifies behavioral and institutional complexities that are relevant for a broader range of real-world contexts

Develops mechanisms, models and methods that can be proven to be robust against such complexities

20

Roth (2010) on why experiments are useful in market design Experiments are a natural part of market design, not

least because to run an experiment, an experimenter must specify how transactions are carried out, and so experimenters are of necessity engaged in market design in the laboratory

Experiments can serve multiple purposes in the design of markets outside of the lab

Scientific use of experiments to test hypotheses Testbeds to get a first look at market designs that may not yet

exist outside of the laboratory (Plott 1987) Demonstrations and proofs of concept

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Game theory

Market designBehavioral and Experimental

Economics

2009 ELINOR OSTROM for her analysis of economic governance, especially the commons

2012 ALVIN E. ROTH and LLOYD SHAPLEY for the theory of stable allocations and the practice of market design

2007 LEONID HURWICZ, ERIC S. MASKIN, and ROGER B. MYERSON for having laid the foundations of mechanism design theory

2005ROBERT J.AUMANN andTHOMAS C. SCHELLING for having enhanced our understanding of conflict and cooperation through game-theory analysis

2002VERNON L. SMITH for having established laboratory experiments as a tool in empirical economic analysis, especially in the study of alternative market mechanisms, and

DANIEL KAHNEMAN for having integrated insights from psychological research into economic science, especially concerning human judgment and decision-making under uncertainty

2001GEORGE A. AKERLOF, A. MICHAEL SPENCE, and JOSEPH E. STIGLITZ, for their analyses of markets with asymmetric information

1996JAMES A. MIRRLEESand WILLIAM VICKREY for their fundamental contributions to the economic theory of incentives under asymmetric information

1994JOHN C. HARSANYI ,JOHN F. NASH andREINHARD SELTEN for their pioneering analysis of equilibria in the theory of non-cooperative games

1978HERBERT A. SIMON for his pioneering research into the decision-making process within economic organizations

Nobel awards

2014 JEAN TIROLE for his analysis of market power and regulation

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ObjectivesEfficiencySimplicityTransparencyFairness

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It is good to keep these goals in mind whenever designing a market

Matching

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One-sided matching

Each of n agents owns house and has strict preferences over all houses

Can the agents benefit from swapping?

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Top trading cycles

Each owner i points to her most preferred house (possibly is own

Each house points back to its owner Creates directed graph; identify cycles

Finite: cycles exist Strict preference: each owner is in at most one cycle

Give each owner in a cycle the house she points to and remove her from the market

If unmatched owners/houses remain, iterate

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Properties of top trading cycles

Top-trading-cycle outcome is a core allocation No subset of owners can make its members better off by

an alternative assignment within the subset

Top-trading-cycle outcome is unique Top-trading-cycle algorithm is strategy proof

No agent can benefit from lying about her preferences

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Example

Owners ranking from best to worst1: (h3, h2, h4, h1)2: (h4, h1, h2, h2)3: (h1, h4, h3, h3)4: (h3, h2, h1, h4)

1

2

3

4

h1

h2

h3

h4

3

1

2

4

h2

h4

4

2

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Two-sided matching

Consider a set of n women and n men Each person has an ordered list of some members of

the opposite sex as his or her preference list Let be a matching between women and men A pair (m, w) is a blocking pair if both m and w prefer

being together to their assignments under . Also, (x, x)is a blocking pair, if x prefers being single to his/her assignment under

A matching is stable if it does not have any blocking pair

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Example

Lucy Peppermint Marcie Sally

Charlie Linus Schroeder Franklin

Schroeder Charlie

Charlie Franklin Linus

Schroeder Linus

Franklin

Lucy Peppermint

Marcie

Peppermint Sally

Marcie

Marcie Sally

Marcie Lucy

Stable!

Charlie Linus

Franklin

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Deferred Acceptance Algorithms

In each iteration, an unmarried man proposes to the first woman on his list that he hasnt proposed to yet

A woman who receives a proposal that she prefers to her current assignment accepts it and rejects her current assignment

This is called the men-proposing algorithm

(Gale and Shapley, 1962)

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Example



Schroeder Charlie


Schroeder Linus

Franklin

Lucy Peppermint

Marcie

Peppermint Sally

Marcie

Marcie Sally

Marcie Lucy

Stable!

Charlie Linus

Franklin Schroeder Charlie


Schroeder

Charlie Linus

FranklinSchroederCharlie

CharlieLinus

Franklin Linus

Franklin

LucyPeppermint

Marcie

MarcieSally

MarcieLucy

PeppermintSally

Marcie

MarcieLucy

LucyPeppermint

Marcie

PeppermintSally

Marcie 35

Theorem 1. The order of proposals does not affect the stable matching produced by the men-proposing algorithm

Theorem 2. The matching produced by the men-proposing algorithm is the best stable matching for men and the worststable matching for women This matching is called the men-optimal matching

Theorem 3. In all stable matchings, the set of people who remain single is the same

Classical Results

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Applications of stable matching

Stable marriage algorithm has applications in the design of centralized two-sided markets

National Residency Matching Program (NRMP) since 1950s Dental residencies and medical specialties in the US, Canada, and parts

of the UK National university entrance exam in Iran Placement of Canadian lawyers in Ontario and Alberta Sorority rush Matching of new reform rabbis to their first congregation Assignment of students to high-schools in NYC

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Question: Do participants have an incentive to announce their real preference lists?

Answer: No! In the men-proposing algorithm, sometimes women have an incentive to be dishonest about their preferences

Incentive Compatibility

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Example



Schroeder Charlie


Schroeder Linus

Franklin

Lucy Peppermint

Marcie

Peppermint Sally

Marcie

Marcie Sally

Marcie Lucy

Stable!

Charlie Linus

Franklin Schroeder Charlie

Lucy Peppermint

Marcie

MarcieSally

MarcieLucy

PeppermintSally

Marcie

Charlie Linus

Franklin

MarcieSally

LinusFranklin

MarcieLucy

SchroederCharlie

Lucy Peppermint

Marcie

CharlieLinus

Franklin

PeppermintSally

Marcie

PeppermintSally

Marcie

Charlie FranklinLinus

Schroeder

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Next Question: Is there any truthful mechanism for the stable matching problem?

Answer: No!Roth (1982) proved that there is no mechanism for the stable marriage problem in which truth-telling is the dominant strategy for both men and women

Incentive Compatibility

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Auctions

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Why auctions?

Auctions solve complex problems in infrastructure industries

(communications, energy, transport) in B2B markets in internet markets in financial markets

in an economic way high revenues high efficiency revelation of hidden information

A basic framework with one buyer

c vp

Payoff of sellerp - c

Payoff of buyerv - p

Trade is efficientGains from trade = v c > 0

Money

Price determines the distribution of the surplus

and with many buyers

c v2

Money

v1v3v5 v4

Why auctions?

What selling mechanism results in the highest expected price (of a single item)?

(Seemingly) Trivial answer in case of complete information: take-it-or-leave-it offer

With incomplete information, an ascending auction typically does better than take-it-or-leave-it offer

Why auctions?

Instead of guessing how much buyers are willing to pay, an auction lets buyers name their own price, revealing how much the item is worth

E.g., price = value of strongest competitor

If there are many buyers, this is almost as good as with complete information ideal

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Single-item auctions(assuming that the pre-auction decisions, such as product design, have been made)

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Auction formats English (clock) auction: the price rises

continuously until there is only one bidder left willing to pay

Dutch (clock) auction: the price drops continuously until the first bidder accepts

First price auction: all bidders submit an offer unaware of what the other bidders are offering. The highest bid wins. The price equals the highest bid

Second price auction: As first price auction, but the price equals the second highest bid

Bidding behavior: English auction Which auction format is the best one?

To find out the optimal auction, one has to analyze bidding behavior under the different formats

Winner i gets vi p, and all others get 0

In the English auction, it is optimal to stay in the auction as long as the price is below ones own value

The bidder with the highest value will win at a price equal to the second highest value

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The second price auction looks weird. Why should a seller be satisfied with the second highest bid?

But, in fact, the second-price auction is similar to (the sealed-bid equivalent of) the English auction

Each bidder will bid her value

As a consequence, the bidder with the highest value will win at a price equal to the second highest value

Second-price auction

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0 1vibi max b-i

(Zero-) payoff unchanged

0 1vibimax b-i

Payoff unchanged

0 1vibi max b-i

loss

1) bi < vi < max b-i

2) max b-i < bi < vi

3) bi < max b-i < vi

Should I bid less than my value?


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0 1vi bi max b-i

(Zero-) Payoff unchanged1) vi < bi < max b-i

2) max b-i < vi < bi

3) vi < max b-i < bi

0 1vi bimax b-i

Payoff unchanged

0 1vi bimax b-i

loss

Should I bid more than my value?


52


Bidding ones value never hurts and is sometimes rewarded

So it is a dominant strategy to bid value

The underlying idea is that bidders are price takersa winner cannot influence ones price, but only the probability of winning

By bidding her value, she makes sure that shell win if and only if the price is below her value

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First-price auction (pay-as-bid) Because the winners bid determines what she has to pay, the

only way to make positive surplus is to bid less than ones value

Thus, bidders will shade their bids. But by how much?

The winner must think about how much he needs to bid to win, which is difficult because of the risk-return trade-off she faces

She wants to just outbid the bidder with the second-highest value

So, under risk neutrality (an ex ante symmetry), she will bid the expected value of the second-highest value, conditional on winning (only then this is relevant)

54

First-price auction (pay-as-bid)

Because the bidder with the highest value wins (in any equilibrium), the revenue equals the expected second-highest value

Thus, expected revenue is the same in the first-price, second-price and English auction

55

Dutch auction

The Dutch auction is strategically equivalent to the first price auction

You must decide how much to bid at a time when you do not know how much the others bid

So, all these auctions lead to the same expected revenue

The winner pays the (expected) value of the strongest competitor

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The revenue equivalence theorem

All auction protocols with the properties that the bidder with the highest value wins, and bidders with the lowest-possible value make zerolead to the same expected revenue to the seller!

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But strong assumptions are required

Single item: one item is auctioned Risk neutrality: bidders are risk-neutral Independence: bidders values are private

information and statistically independent Symmetry: values are drawn from the same

probability distribution

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These assumptions never hold in practice, yet the revenue equivalence theorem is incredibly useful!

William Vickrey

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... received the Nobel prize 1996 for the revenue equivalence theorem!

Illustration with complete information

An item is worth 15 to bidder A and 10 to bidder B

In the English auction, B drops out at a price 10

In the second price auction, both bidders bid their value, so that the price is 10

In the first price auction, A bids just enough to win (10 + epsilon), and B bids 10

The Dutch auction is strategically equivalent to the first price auction

60

Remarks Standard single unit auction protocols fall under the

revenue equivalence theorem Bidders adjust their behavior in a way that does not allow the seller

to get more revenue by some sophisticated auction procedures

However, when the underlying assumptions do not hold, (e.g. risk aversion, asymmetries, etc.) the choice of mechanism matters!

Still, the RET is useful

Similarities to matching: if you ask me for my preferences, the critical question is what are you going to do with this information

61

Remarks:The optimal auction and reserve prices

The optimal auction has a reserve price that is equal to the optimal take-it-or-leave-it price in case of only one bidder

All four auction formats with this reserve price are optimal

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Remarks: Interdependent values

Here is a jar with cents Please write down

Your name: Peter Your estimate of number of cents: E = nnn Your bid in 1st price auction (in cents): 1st = nnn Your bid in 2nd price auction (in cents): 2nd = nnn

The highest bid wins in each auction

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0

5

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15

20

25

30

35

40

45

50

1 2 3 4 5 6 7 8 9 10 11

Remarks: The winners curse

Real value: 3,00

Distribution of bids

Average bid: 2,70

Highest bid: 11,50

Diagramm1

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34

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13

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11

4

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Nagel

SpieltheoretikerZeitungsleserTemplate

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Nagel (2)

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Zahlen der Zeitungsumfrage(Hufigkeiten in %)

Template

Zahlen von Spieltheoretikern(Hufigkeiten in %)

Template

Zahlen der Zeitungsumfrage(Hufigkeiten in %)

Hufigkeit in %

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Nagel (2)

Hufigkeit in %

Aktienindizes

Historical

NEMAX 50 PRC AUC

Type:Stock IndexFamily:NemaxISIN:DE0009665141Epic or local ID:Symbol:Country: Germany

YearFirst trading day.HighLowLast trading day.Average

2003355.74596,24306,31560,240

20021150.131295,15303,93355,99

20012834.262958,76638,481145,03

20005112.949665,812666,492859,28

19994341.065230,583272,435074,93

Source:onvista

Historical

NEMAX 50 PRF AUC

Type:Stock IndexFamily:NemaxISIN:DE0009665133Epic or local ID:Symbol:NMDXCountry: Germany

YearFirst trading day.HighLowLast trading day.

2003358.54604.37308.73567.88

20021,155.231,300.89306.32358.79

20012,843.902,968.82641.311,150.10

20005,127.899,694.072,675.562,869.01

19994,352.155,245.873,281.125,089.76

istorical

DAX (PERFORMANCEINDEX)

Type:Stock IndexFamily:DaxISIN:DE0008469008Epic or local ID:5738566Symbol:Country: Germany

YearFirst trading day.HighLowLast trading day.

20032,898.683,996.282,188.753,965.16

20025,155.265,467.312,519.302,892.63

20016,431.146,795.143,539.185,160.10

20006,961.728,136.166,110.266,433.61

19994,978.386,992.924,601.076,958.14

19984,270.696,224.523,822.595,006.57

19972,836.824,477.702,815.504,224.30

19962,284.862,916.162,284.862,880.07

19952,079.452,317.011,910.962,253.88

19942,267.982,271.111,960.592,106.58

19931,531.332,266.681,516.502,266.68

19921,601.881,811.571,420.301,545.05

19911,366.101,715.801,322.681,577.98

19901,814.381,968.551,334.891,398.23

19891,335.011,790.371,271.701,790.37

1988943.881,340.41931.181,327.87

19871,000.001,000.001,000.001,000.00

Hufigkeit in %

Template

Template

dax 95-04

DateOpenHighLowCloseclose .00sVolumeAdj. Close*DateClose

6/1/043924.303929.883856.0438641803864.1819951/2/952021

5/3/043972.884029.323710.0239214103921.412/1/952102

4/1/043858.344156.893857.0439852103985.213/1/951922

3/1/044026.194163.193692.4038567003856.704/3/952015

2/2/044062.794150.563960.4140181604018.165/2/952092

1/2/043969.044175.483969.0440586004058.606/1/952083

12/1/033752.723996.283752.7239651603965.167/3/952218

11/3/033657.613814.213576.5237459503745.958/1/952238

10/1/033255.763675.783217.4036559903655.999/1/952187

9/1/033493.343676.883202.8732567803256.7810/2/952167

8/1/033481.813588.533299.7734845803484.5811/1/952242

7/1/033217.213487.863119.3534878603487.8612/1/952253

6/2/032990.933324.442990.9332205803220.5819961/2/962470

5/2/032938.723068.082769.4529826802982.682/1/962473

4/1/032426.243004.792395.722942402942.043/1/962485

3/3/032553.742731.572188.7524238702423.874/1/962505

2/3/032750.402802.932433.152547502547.055/2/962542

1/2/032898.683157.252563.9227478302747.836/3/962561

12/2/023332.733476.832836.0128926302892.637/1/962473

11/1/023152.193443.492987.8533203203320.328/1/962543

10/1/022772.963299.012519.3031528503152.859/2/962651

9/2/023698.693698.692719.492769302769.0310/1/962659

8/1/023699.313930.963235.3837129403712.9411/1/962845

7/1/024377.104483.033265.9637001403700.14199712/2/962888

6/3/024818.074850.383946.7043825604382.561/2/973035

5/2/025043.015126.334741.9548183004818.302/3/973259

4/2/025379.645379.644929.3550412005041.203/3/973429

3/1/025025.965467.315003.0653972905397.294/1/973438

2/1/025109.485165.974706.015039805039.085/2/973562

1/2/025155.265352.164974.5851076105107.616/2/973766

12/1/014986.585341.864872.2551601005160.107/1/974405

11/1/014556.715217.474481.5549899104989.918/1/973919

10/1/014315.064874.314157.6045591304559.139/1/974154

9/3/015195.675220.103539.1843081504308.1510/1/973753

8/1/015859.685930.455124.6951881705188.1711/3/973972

7/2/016053.816131.975551.1458611905861.19199812/1/974224

6/1/016122.966278.045758.6960583806058.381/2/984442

5/2/016272.036337.475960.4661232606123.262/2/984693

4/2/015843.376271.205383.9962645106264.513/2/985097

3/1/016212.666342.545351.4858299505829.954/1/985107

2/1/016788.576788.576071.3562082406208.245/4/985569

1/2/016431.146795.146172.4467951406795.146/2/985897

12/1/006379.266813.306110.266433616842106433.617/1/985873

11/1/007093.847185.666361.4363723310454546372.338/3/984833

10/2/006800.077087.926297.497077443095237077.449/1/984474

9/1/007221.417456.716468.4667981210476196798.1210/1/984671

8/1/007194.037395.816954.6072447910434787244.7911/2/985022

7/3/006912.977503.326842.617190373333337190.3712/1/985002

6/1/007117.577486.406851.8468824406882.4419991/4/995159

5/2/007407.537571.726794.0871096707109.672/1/994911

4/3/007599.787641.536890.9674146807414.683/1/994884

3/1/007645.308136.167411.7675993907599.394/1/995393

2/1/006841.127813.206841.1276445507644.555/3/995069

1/3/006961.727306.486388.9168356006835.606/1/995378

12/1/995891.276992.925836.7369581406958.147/1/995101

11/1/995518.746032.675474.125896405896.048/2/995270

10/1/995166.575557.915078.5955254005525.409/1/995149

9/1/995286.585530.765069.8751498305149.8310/1/995525

8/2/995088.655455.844948.0852707705270.7711/1/995896

7/1/995428.345686.555035.2051018705101.8712/1/996958

6/1/995083.625500.934980.6353785205378.5220001/3/006835

5/3/995372.685449.225003.9750698305069.832/1/007644

4/1/994852.395397.644779.0453931105393.113/1/007599

3/1/994923.315151.014605.2748842004884.204/3/007414

2/1/995193.035287.964749.4549118104911.815/2/007109

1/4/994991.955509.344768.4151599605159.966/1/006882

12/1/984986.815085.474451.2050023905002.397/3/007190

11/2/984722.845176.274617.4150227005022.708/1/007244

10/1/984389.214722.093833.7146711204671.129/1/006798

9/1/984727.355161.404357.4244745104474.5110/2/007077

8/3/985811.425818.154752.4048338904833.8911/1/006372

7/1/985863.256199.585810.9558739205873.9212/1/006433

6/2/985569.895921.405512.5858974405897.4420011/2/016795

5/4/985251.415664.845172.955569805569.082/1/016208

4/1/985084.105436.804992.1051074405107.443/1/015829

3/2/984696.505136.804612.1050973005097.304/2/016264

2/2/984481.004736.104474.3046939004693.905/2/016123

1/2/984270.704458.304053.9044425004442.506/1/016058

12/1/974000.004320.004000.0042243004224.307/2/015861

11/3/973802.703986.403645.8039721003972.108/1/015188

10/1/974154.604363.703366.0037537003753.709/3/014308

9/1/973942.504196.803783.1041549004154.9010/1/014559

8/1/974422.204460.203849.9039198003919.8011/1/014989

7/1/973782.704477.703782.6044055004405.5012/1/015160

6/2/973551.003836.903544.5037669003766.9020021/2/025107

5/2/973448.703695.303448.7035627003562.702/1/025039

4/1/973336.103452.603192.3034381003438.103/1/025397

3/3/973264.503475.003260.2034291003429.104/2/025041

2/3/973034.303287.803032.3032596003259.605/2/024818

1/2/972855.203041.602833.8030352003035.206/3/024382

12/2/962854.802914.602760.8028887002888.707/1/023700

11/1/962675.402845.502665.9028455002845.508/1/023712

10/1/962647.502739.302645.9026593002659.309/2/022769

9/2/962540.802672.602510.3026519002651.9010/1/023152

8/1/962492.202571.702491.0025438002543.8011/1/023320

7/1/962563.902583.702446.2024734002473.4012/2/022892

6/3/962526.802578.702524.4025614002561.4020031/2/032747

5/2/962490.702571.802455.2025428002542.802/3/032547

4/1/962495.402554.802487.1025053002505.303/3/032423

3/1/962494.702526.302400.5024859002485.904/1/032942

2/1/962472.002484.502375.1024736002473.605/2/032982

1/2/962271.402471.702271.4024701002470.106/2/033220

12/1/952259.402289.902232.5022539002253.907/1/033487

11/1/952157.902256.002154.3022428002242.808/1/033484

10/2/952205.402219.302096.1021679002167.909/1/033256

9/1/952233.402320.202163.202187002187.0010/1/033655

8/1/952222.302277.402208.2022383002238.3011/3/033745

7/3/952083.402243.602081.8022187002218.7012/1/033965

6/1/952118.302155.802081.7020839002083.9020041/2/044058

5/2/952029.602110.702015.9020922002092.202/2/044018

4/3/951915.102030.701914.1020159002015.903/1/043856

3/1/952114.502130.401893.6019226001922.604/1/043985

2/1/952037.702145.702037.7021022002102.205/3/043921

1/2/952084.102092.202017.8020213002021.306/1/043864

dax 95-04

Close

Dax-Verlauf 1995 - 2004

winners curse

Hufigkeit

0,00-0,497

0,50-0,9920

1,00-1,4934

1,50-1,9927

2,00-2,4945

2,50-2,9924

3,00-3,4913

3,50-3,9915

4,00-4,4911

4,50-4,994

5,00-5,497

5,50-5,995

6,00-6,492

6,50-6,99

7,00-7,491

7,50-7,99

8,00-8,491

8,50-8,991

9,00-9,491

9,50-9,99

10,00-10,492

10,50-10,99

11,00-11,492

11,50-11,99

12,00-12,49

12,50-12,99

13,00-13,49-

13,50-13,99

14,00-14,49

14,50-14,99

15,00-15,492

15,50-15,99

>15,993

Close


Close


winners curse

Hufigkeit

Hufigkeiten der Gebote

winners curse (2)

Hufigkeit

7

Remarks: Interdependent valuesYou want to bid on this used car ...

but you dont know the true value

65


You estimate the value somewhere between 0 and 12.000 (every value has the same probability)

The seller knows the true value, but he needs money and is willing to sell at a price of 2/3 of the value (otherwise he rejects your bid)

How much do you bid?

66

0 6.000 12.000

The expected value is 6.000

4.000

Assume you bid 4.000

If you win at this price, is this a bargain?


67

0 6.000 12.000

Assume seller accepts your bid

4.000

The (expected) value is thus only 3.000 That is, independent of your bid, if you win you must always

expect to make a loss!

3.000

Then you know that the value of the car cannot exceed 6.000, because otherwise your bid would be rejected (2/3 6.000 = 4.000)

68


Remarks: "The winners curse"UMTS-auctions

MobilCom gave its license back for which it paid 8,5 billion Euros

Road construction and tender ...Book manuscriptsOil fieldsBaseball players and ManagerseBay....

69

Remarks: "The winners curse"Bidders typically have different pieces of information

about the true value

Winning is bad news, because it means that others (other bidders or the seller) have more negative information about the value

Thus, bids need to be substantially shaded: "what is the value of the item conditional on all opponents estimating this value to be lower?"

This bid shading reduces revenues

70

Remarks: "The winners curse" What does this imply for auction design if bidders are

rational? Reduce uncertainty!

Each bidder would revise his value if he also had the information about other bidders

This information might be inferred by the competitors bids in open (but not in sealed-bid) auctions

Better information and less uncertainty allows bidders to bid more aggressively

Thus, the English auction, theoretically, yields larger revenues

71

Multiple Item Auctions

72

Outline Equilibrium of games with incomplete information Mechanism design Auctioning many similar items Revenue equivalence and optimal auctions Applications

Electricity Spectrum Mobile communications Transportation Financial securities

73

First some mathCalculating equilibrium behavior in games with

incomplete information

Warning: A little knowledge is a dangerous thing!

74

Equilibrium in first-price auction

Each of n bidders private value v drawn from distribution F Bidder's expected profit:

(v,b(v)) = (v - b(v))Pr(Win|b(v))

By the envelope theorem,

But then d/dv = Pr(Win|b(v)) = Pr(highest bid)= Pr(highest value) = F(v)n-1

75

ddv b

bv v v

=

+

=

Equilibrium in first-price auction

By the Fundamental Theorem of Calculus,

Substituting into (v,b(v)) = (v - b(v))Pr(Win|b(v)) yields

76

( ) ( ) ,v F(u) du F(u) dun 1v

n 1v

= +XZY

=XZY

00 0

b v v vPr(Win)

v F(v) F(u) du(n 1) n 1v

( ) ( ) .= = XZY

0

Example

v U on [0,1] Then F(v) = v, so

b(v) = v - v/n = v(n-1)/n The optimal bid converges to the value as n, so in

the limit the seller is able to extract the full surplus In equilibrium, the bidder bids the expected value of

the second highest value given that the bidder has the highest value

77

Bargaining: Simultaneous offers(Chatterjee & Samuelson, Operations Research 1983)

A seller and a buyer are engaged in the trade of a single item worth s to the seller and b to the buyer

Valuations are known privately, as summarized below

78

TradersValue Distributed Payoff

PrivateInfo

CommonKnowledge

Strategy(Offer)

Seller s sF on [s, s] u =P s s F, G p(s)

Buyer b bG on [ , ]b b v =b P b F, G q(b)

Traders

Value

Distributed

Payoff

Private

Info

Common Knowledge

Strategy (Offer)

Seller s

s(F on

[

s

,

s

]

u =P s

s

F, G

p(s)

Buyer b

b(G on

[

,

]

b

b

v =b P

b

F, G

q(b)

Simultaneous Offers Independent private value model: s and b are

independent random variables

Ex post efficiency: trade if and only if s < b

Game: Each player simultaneously names a price;if p q then trade occurs at the price P = (p + q)/2;if p > q then no trade (each player gets zero)

79

Simultaneous Offers Payoffs:

Seller

Buyer

where the trading price is P = (p + q)/2

80

u(p,q,s,b) =P - s if p q

0 if p > qRST

v(p,q,s,b) =b - P if p q

0 if p > qRST

Let F and G be independent uniform distributions on [0,1]

Equilibrium conditions:

81

Example

(1) s [s, s],p(s) argmaxE {u(p,q,s,b)|s,q( )}

(2) b [b,b],q(b) argmaxE {v(p,q,s,b)|b,p( )p

b

qs

}

Assume p and q are strictly increasing Let x() = p-1() and y() = q-1() Optimization in (1) can be stated as

First-order condition-y'(p)[p - s] + [1 - y(p)]/2 = 0,

since q(y(p)) = p

82

Sellers Problem

max [(p q(b)) / 2 s]dbp y(p)

1

+ XZY

Optimization in (2) can be stated as

First-order condition

x'(q)[b - q] - x(q)/2 = 0,since p(x(q)) = q

83

Buyers Problem

max [b - (p(s) q) / 2]dsq

x(q)

+XZY0

Equilibrium Equilibrium condition:

s=x(p) and b=y(q)

Equilibrium first-order conditions:

(1') -2y'(p)[p - x(p)] + [1 - y(p)] = 0,(2') 2x'(q)[y(q) - q] - x(q) = 0

84

Solving (2') for y(q) and replacing q with p yields

Substituting into (1') then yields

85

Solution

(2 ) y(p) p 12

x(p)x (p)

, so y (p)= 32

12

x(p)x (p)[x (p)]2

= +

(1 ) [x(p) -p] 3 - x(p)x (p)[x (p)]

1 p 12

x(p)x (p)

=02

LNM

OQP+

LNM

OQP

Linear Solution:x(p) = p +

with = 3/2 and = -3/8

Using (2") yieldsy(q) = 3/2 q - 1/8

Inverting these functions results inp(s) = 2/3 s + 1/4 and q(b) = 2/3 b + 1/12

86

Analytical Solution

Figure 1

87

0.0

1.0

Valuations s, b

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

Offers p, q

p(s) q(b)

Trade occurs if and only if

p(s) q(b), or b - s 1/4

The gains from trade must be at least 1/4 or no tradetakes place

The outcome is inefficient

88

Outcome

Mechanism designGeneral understanding of incentives in decision settings

(bargaining, auctions, exchange, public goods, )

89

A General Model(Myerson and Satterthwaite, JET 1983)

Direct Revelation Game: Bilateral Exchange with independent private value s F with positive pdf f on b G with positive pdf g on F and G are common knowledge

In the DRG, the traders report their valuations and then an outcome is selected. Given the reports (s,b), an outcome specifies a probability of trade (p) and the terms of trade (x)

90

[s, s][b,b]

DefinitionDirect MechanismA direct mechanism is a pair of outcome functions p,x, where:

p(s,b) is the probability of trade given the reports (s,b), and x(s,b) is the expected payment from the buyer to the seller

91

PayoffsEx post utilities: Seller's ex post utility:

u(s,b) = x(s,b) - sp(s,b)

Buyer's ex post utility:v(s,b) = bp(s,b) - x(s,b)

Both traders are risk neutral (quasi-linear utility)

92

PayoffsDefine:

X(s) is the seller's expected revenue given sY(b) is the buyer's expected payment given bP(s) is the seller's probability of tradeQ(b) is the buyer's probability of trade

93

X(s) x(s,b)g(b)db Y(b) x(s,b)f(s)ds

P(s) p(s,b)g(b)db Q(b) p(s,b)f(s)ds.

b

b

s

s

b

b

s

s

= XZY =XZY

= XZY =XZY

Payoffs Interim Utilities:

U(s) = X(s) - sP(s) V(b) = bQ(b) - Y(b)

The mechanism p,x is incentive compatible if for all s, b, s, and b:(IC) U(s) X(s') - sP(s') V(b) bQ(b') - Y(b')

The mechanism p,x is individually rational if for alland

(IR) U(s) 0 V(b) 0

94

s [s, s] b [b, b]

Lemma 1(Mirrlees, Myerson)

The mechanism p,x is IC if and only if P() is decreasing, Q() is increasing, and

(IC)

95

U(s) U(s) P(t)dt

V(b) V(b) Q(t)dt

s

s

b

b

= +XZY

= +XZY

Lemma 1: ProofOnly if: By definition, U(s) = X(s) - sP(s) and

U(s') = X(s') - s'P(s'). This and (IC) imply

U(s) X(s') - sP(s') = U(s') + (s' - s)P(s'), andU(s') X(s) - s'P(s) = U(s) + (s - s')P(s)

Putting these inequalities together yields

(s' - s)P(s) U(s) - U(s') (s' - s)P(s')

96

Lemma 1: Proof

Taking s' > s implies that P() is decreasing Dividing by (s' - s) and letting s' s, then yields

dU(s)/ds = -P(s) Integrating produces (IC') The same is true for the buyer

97

Lemma 1: ProofIf: To prove (IC) for the seller, note that it suffices to show

that

s[P(s) - P(s')] + [X(s') - X(s)] 0 for all s, s

Substituting for X(s') and X(s) using (IC') and the definition of U(s) yields

98

[s, s]

X(s) sP(s) U(s) P(t)dts

s

= + +XZY .

Lemma 1: ProofIf: Then it suffices to show for every s,s that

which holds because P() is decreasing

The proof for the buyer is similar

99

[s, s]

0 s[P(s) P(s )] + s P(s ) + P(t)dt sP(s) P(t)dt

(s s)P(s ) P(t)dt [P(t) P(s )]dt ,

s

s

s

s

s

s

s

s

XZY XZY

= +XZY = XZY

Lemma 2

An incentive compatible mechanism p,x is individually rational if and only if

(IR') and V( ) 0

Proof Clearly, (IR') is necessary for p,x to be IR

By Lemma 1, U() is decreasing; hence, (IR') is sufficient aswell

100

U(s) 0 b_

eq \O(b,_)

An incentive-compatible, individually rational mechanism p,x satisfies

101

Theorem:Characterization of IC & IR mechanisms

(*) U(s) + V(b)

b 1 G(b)g(b)

s F(s)f(s)

p(s,b)f(s)g(b)dsdb 0b s

sb

=

LNM

OQP

XZY

z .

Proof Using (IC') and the definition of U(s) yields

102


X(s) sP(s) U(s) P(t)dt.s

s= + + z

Taking the expectation with respect to s (and substituting in the definitions of X(s) and P(s)) shows that

103

b

b

s

s

b

b

s

s

b

b

s

s

x(s,b)f(s)g(b)dsdb

U(s) sp(s,b)f(s)g(b)dsdb

p(s,b)F(s)g(b)dsdb.

XZY

XZY =

+XZYXZY

+XZYXZY


The third term in the right hand side follows, since

104


s

s

s

s

s

s

s

t

s

s

p(t,b)f(s)dtds

p(t,b)f(s)dsdt p(s,b)F(s)ds .

XZY

XZY

= XZYXZY =

XZY

Preceding analogously for the buyer yields

Equating the right-hand sides of the last two equations andapplying (IR') completes the proof

105


b

b

s

s

b

b

s

s

b

b

s

s

x(s,b)f(s)g(b)dsdb

= -V(b) bp(s,b)f(s)g(b)dsdb

- p(s,b)f(s)[1- G(b)]dsdb.

XZY

XZY

+XZYXZY

XZY

XZY

Corollary:Impossibility of efficient trade

106

If it is not common knowledge that gains exist (the supports of the traders' valuations have non-empty intersection), then no incentive-compatible, individually rational trading mechanism can be ex-post efficient

A mechanism is ex-post efficient if and only if tradeoccurs whenever s b:

107

Proof

p(s,b)1 if s b0 if s b.

=>

RST

Proof To prove that ex-post efficiency cannot be attained, it

suffices to show that the inequality (*) in the Corollary fails when evaluated at this p(s,b). Hence,

108

b s

min{b,s}b

b 1 G(b)g(b)

s F(s)f(s)

f(s)g(b)dsdbz LNM OQPXZY

Proof

109

min{b,s} min{b,s}b b

b bs s

b b

b b

b b

b s

b

[bg(b)+G(b) 1]f(s)dsdb [sf(s)+F(s)]dsg(b)db

[bg(b)+G(b) 1]F(b)db min{bF(b),s}g(b)db (by parts)

[1 G(b)]F(b)db (b s)g(b)db

[1 G(b)]F(b)db

=

=

= +

= b b

s

s

b

[1 G(b)]db (by parts)

[1 G(t)]F(t)dt 0, since b s.

+

= <

Proof

The second term in the second line follows, since by integrating by parts

Since ex-post efficiency is unattainable, we need a weaker efficiency criterion with which to measure a mechanism's performance

110

[sf(s) F(s)]ds xF(x)s

x

+ =XZY .

What IC & IR mechanism maximizes expected gains from trade?

Ex ante efficient mechanism maximizes the expected gains from trade:

subject to IC & IR

111

U(s)f(s)ds V(b)g(b)dbs

s

b

bXZY +

XZY

Optimal trading game

Myerson and Satterthwaite show that the ex ante efficient decision rule (probability of trade) is:

where

and is chosen so that U( ) = V( ) = 0

112

p s,bif c(s, ) d(b, )if c(s, ) d(b, )

( ) =>

RST10

c( ,s) = s + F(s)f(s)

d( ,b) = b 1 G bg b

( )( )

s b

Remarks

If = 0, then p is ex post efficient (all the weight on the objective function)

If = 1, p maximizes the expression in (*); a constrained maximization

The ex ante efficient trading rule has the property that, given the reports, trade either occurs with probability one or not at all

113

Example

Valuations are uniformly distributed on [0,1] Ex ante efficient mechanism: linear equilibrium in

which trade occurs if and only if the gains from trade are at least 1/4 (Chatterjee & Samuelson)

If the traders cannot commit to walking away from gains from trade, then they would be unable to implement this mechanism

So long as it is not common knowledge that gains exist, the traders will, with positive probability, make incompatible demands in situations where gains from trade exist

114

Accomplishments

Characterization of the set of all BE of all bargaining games in which the players' strategies map their private valuations into a probability of trade and a payment from buyer to seller

Proof that ex post efficiency is unattainable if it is uncertain that gains from trade exist

Determination of the set of ex ante efficient mechanisms

Proof that ex ante efficiency is incompatible with sequential rationality

115

Dissolving a Partnership(Cramton, Gibbons, and Klemperer, 1987)

n traders. Each trader i {1,...,n} owns a shareri 0 of the asset, where r1 + ... + rn = 1

As in MS, player i's valuation for the entire good is vi The utility from owning a share ri is rivi Private values, vis are iid F() on A partnership (r,F) is fully described by the vector of

ownership rights r = {r1,...,rn} and the traders' beliefs F about valuations

116

[v, v]

Dissolving a PartnershipMS Case: n = 2 and r = {1,0} There does not exist a BE of the trading game such

that:(1) is (interim) individually rational and(2) is ex post efficient

CGK Case: If the ownership shares are not too unequally

distributed, then it is possible to satisfy both (1) and (2), (satisfying IC, IR, EE and BB)

117

Dissolving a Partnership

A partnership (r,F) can be dissolved efficiently if there exists a Bayesian Equilibrium of a Bayesian trading game such that is interim individually rational and ex post efficient

118

Theorem

The partnership (r,F) can be dissolved efficiently if and only if

(*)

where vi* solves F(vi)n-1 = ri and G(u) = F(u)n-1

119

[1 F(u)]udG(u) F(u)udG(u)v

v

v

v

i 1

n

i*

i*

LNMOQP z z= 0

Examples F(vi) = vi n=2, then (*)

n=3, then

(*)

120

r 3 4i3 2

i 1

3

=

=1

2

2 23 0 1

0.21 0.79

Proposition

For any distribution F, the one-owner partnershipr = {1,0,0,...,0} cannot be dissolved efficiently

The one-owner partnership can be interpreted as an auction Ex post efficiency is unattainable because the seller's value

v1 is private information: the seller finds it in her best interest to set a reserve price above her value v1

An optimal auction maximizes the seller's expected revenue over the set of feasible (ex post inefficient) mechanisms

121

Theorem If a partnership (r,F) can be dissolved efficiently, then

the unique symmetric equilibrium of the followingbidding game is interim individually rational andachieves ex-post efficiency: given an arbitrary mini-mum bid b,

each player receives a side-payment, independent of thebidding,

the players choose bids bi [b,) the good goes to the highest bidder each bidder i pays

122

p (b , ,b ) b 1n 1

bi 1 n i jj i

n =

c (r , , r ) udG(u) udG(u).i 1 nv

v1n

v

v

j 1

ni*

j*

= XZY XZY=

Auctioning manysimilar items

Lawrence M. Ausubel, Peter Cramton,Marek Pycia, Marzena Rostek, and Marek Weretka (2014)

"Demand Reduction and Inefficiency in Multi-Unit Auctions,"Review of Economic Studies, 81:4, 1366-1400

123

http://www.cramton.umd.edu/papers2010-2014/acprw-demand-reduction.pdf

124

Examples of auctioningsimilar items Treasury bills Stock repurchases and IPOs Telecommunications spectrum Electric power Emission allowances

125

Ways to auction many similar items Sealed-bid: bidders submit demand schedules

Pay-as-bid auction (traditional Treasury practice) Uniform-price auction (Milton Friedman 1959) Vickrey auction (William Vickrey 1961)

Bidder 1 Bidder 2

+

AggregateDemand

=

P

Q1 Q2 Q

P P

126

Pay-as-bid Auction:All bids above P0 win and pay bid

Price

Quantity

Supply

Demand(Bids)

QS

P0Clearing price

127

Uniform-Price Auction:All bids above P0 win and pay P0

Price

Quantity

Supply

Demand(Bids)

QS

P0Clearing price

128

Vickrey Auction:All bids above P0 win and pay opportunity cost

Price

Quantity

Residual SupplyQS ji Qj(p)

DemandQi(p)

Qi(p0)

p0

129

Vickrey Auction: m Discrete Items

Allocate m items efficiently: m highest marginal values

Winning bidder pays kth highest losing bid of others on kth item won

Payment = social opportunity cost of items won

3 bidders, 3 itemsmarginal values

A B C

1st 10 8 4

2nd 6 7 2

3rd 3 5 1

5 6

4

130

Payment rule affects behavior

Price

Quantity


DemandQi(p)

Qi(p0)

p0

Pay-as-bid

Uniform-Price

Vickrey

131

Optimal bids by payment rule

Price

Quantity

True demandQi(p)

p0

Uniform-Price

Vickrey

132

More ways to auction many similar items

Ascending-bid: Clock indicates price; bidders submit quantity demanded at each price until no excess demand

Clock auction (single price) Clock auction with Vickrey prices (Ausubel 1997)

133

Ascending clock:All bids at P0 win and pay P0

Price

Quantity

Supply

Demand

QS

P0

ClockExcessDemand

134

Ascending clock with Vickrey pricesAll bids at P0 win and pay price at which clinched

Price

Quantity


DemandQi(p)

Qi(p0)

p0

Clock

Excess Demand

135

More ways to auction many similar items Ascending-bid

Simultaneous ascending auction (FCC spectrum) Sequential

Sequence of English auctions (auction house) Sequence of Dutch auctions (fish, flowers)

Optimal auction Maskin & Riley 1989

136

Research ProgramHow do standard auctions compare? Efficiency

FCC: those with highest values win Revenue maximization

Treasury: sell debt at least cost

137

Efficiency(not pure common value; capacities differ)

Uniform-price and standard ascending-bid Inefficient due to demand reduction

Pay-as-bid Inefficient due to different shading

Vickrey Efficient in private value setting Strategically simple: dominant strategy to bid true demand Inefficient with affiliated information

Dynamic Vickrey (Ausubel 1997) Same as Vickrey with private values Efficient with affiliated information

138

Inefficiency TheoremIn any equilibrium of uniform-price auction, with

positive probability objects are won by bidders other than those with highest values

Winning bidder influences price with positive probability Creates incentive to shade bid Incentive to shade increases with additional units Differential shading implies inefficiency

139

Inefficiency from differential shading

P0

Large Bidder Small Bidder

Q1 Q2

mv1

mv2

Large bidder makes room for smaller rival

D1 D2b1 b2

140

Vickrey inefficient with affiliation Winners Curse in single-item auctions

Winning is bad news about value Winners Curse in multi-unit auctions

Winning more is worse news about value Must bid less for larger quantity Differential shading creates inefficiency in Vickrey

141

What about seller revenues?

Price

Quantity


DemandQi(p)

Qi(p0)

p0

Pay-as-bid

Uniform-Price

Vickrey

Exercise 2 bidders (L and R), 2 identical items L has a value of $100 for 1 and $200 for both R has a value of $90 for 1 and $180 for both Uniform-price auction

Submit bid for each item Highest 2 bids get items 3rd highest bid determines price paid

Ascending clock auction Price starts at 0 and increases in small increments Bidders express how many they want at current price Bidders can only lower quantity as price rises Auction ends when no excess demand (i.e. just two

demanded); winners pay clock price

142

143

Uniform price may perform poorly Independent private values uniform on [0,1] 2 bidders, 2 units; L wants 2; S wants 1 Uniform-price: unique equilibrium

S bids value L bids value for first and 0 for second Zero revenue; poor efficiency

Vickrey price = v(2) on one unit, zero on other

144

Standard ascending-bid may be worse 2 bidders, 2 units; L wants 2; S wants 2 Uniform-price: two equilibria

Poor equilibrium: both L and S bid value for 1 Zero revenue; poor efficiency

Good equilibrium: both L and S bid value for 2 Get v(2) for each (max revenue) and efficient

Standard ascending-bid: unique equilibrium Both L and S bid value for 1

Ss demand reduction forces L to reduce demand Zero revenue; poor efficiency

145

Efficient auctions tend to yield high revenues

Theorem. With flat demands drawn independently from the same regular distribution, sellers revenue is maximized by awarding good to those with highest values

Generalizes to non-private-value model with independent signals:vi = u(si,s-i)

Award good to those with highest signals if downward sloping MR and symmetry

146

Downward-sloping demand:pi(qi) = vi gi(qi)

Theorem. If intercept drawn independently from the same distribution, sellers revenue is maximized by

awarding good to those with highest values if constant hazard rate

shifting quantity toward high demanders if increasing hazard rate

Note: uniform-price shifts quantity toward lowdemanders

147

But uniform price has advantages

Participation Encourages participation by small bidders

(since quantity is shifted toward them and less strategy) May stimulate competition

Post-bid competition More diverse set of winners may stimulate competition

in post-auction market

148

Bidding behavior in electricity markets Marginal cost bidding is a useful benchmark, but

not a norm of behavior Profit maximization is an appropriate norm of

behavior in markets Profit maximization should be expected and

encouraged Market rules should be based on this norm

149

Uniform-price auction:All bids below p0 win and get paid p0

Price

Quantity

Supply(as bid)

Demand

q0

p0(clearing price)

150

Residual demand removessupply of other bidders

Demand Supplyof others

Residual demand

=

q qi qi

p p

q0

Supplyfirm i

Supply

p

p0

q q

151

Price

Quantity

Di(p) = D(p) ji Sj(p)

As-bid supply Si(p)

Residual demand curve

qi

p0

Residual demand

152

Price

Quantity

Di

As-bid supplySi = MCi

Bidding strategywith perfect competition

qi

p0 Residual demand

Loss

153

Incentive to bid above marginal cost:tradeoff higher price with reduced quantity

p

q

Residual demandDi

As-bid supplySi

qi

p0

MCi

Loss

Gain

154

Optimal bid balances marginal gain and loss

p

q

Residual demandDi

As-bid supplySi

qi

p0 MCi

Loss

Gain

155

p

0

Still bid above marginal cost when others bid marginal cost

q-i

p-i

10GW

Otherbidders

p

0 qi

p0

1GW

Firm i

D

S-i=MC-i

Di

MCi

Si

p0

156

Residual demand response reduces incentive to inflate bids

p

q

Residual demandDi

As-bid supplySi

qi

p0

MCi

LossGain

157

p

0

Residual demand is steeper for large bidders

ql

p0

10GW

Large bidder p

0 qs

p0

1GW

Small bidder

Dl

Sl

Ds

Ss

158

p

0

Large bidder makes roomfor its smaller rivals

ql

p0

10GW

Large bidder p

0 qs

p0

1GW

Small bidder

Dl

Sl

Ds

Ss

MCl

MCs

159

Economic vs. Physical Withholding

p

q

Di

Si

qiqi

p0MCiMCi

qe

p

q

Di

SiSi

qiqi

p0MCiMCi

qe

160

Forward contracts mitigate incentive to bid above marginal cost

p

q

Residual demandDi

Si no forward

qi

p MCi

Si with forward

p0

qiqF

qS

Forward sale

Revenue equivalence and optimal auction in general model

Lawrence M. Ausubel and Peter Cramton (2001) The Optimality of Being Efficient, Working Paper,

University of Maryland.

161

http://www.cramton.umd.edu/papers1995-1999/98wp-optimality-of-being-efficient.pdf

162

Identical items model Seller has quantity 1 of divisible good (value = 0) n bidders; i can consume qi [0,i]

q = (q1,,qn) Q = {q | qi [0,i] & iqi 1} ti is is type; t = (t1,,tn); ti ~ Fi w/ pos. density fi Types are independent Marginal value vi(t,qi) is payoff if gets qi and pays xi:

v t y dy xi iqi ( , ) z0

163

Identical items model (cont.)

Marginal value vi(t,qi) satisfies: Value monotonicity

non-negative increasing in ti weakly increasing in tj weakly decreasing in qi

Value regularity: for all i, j, qi, qj, ti, ti > ti,vi(ti,ti,qi) > vj(ti,ti,qj) vi(ti,ti,qi) > vj(ti,ti,qj)

164

Identical items model (cont.)

Bidder is marginal revenue:marginal revenue seller gets from awarding additional quantity to bidder i

MR t q v t q F tf t

v t qti i i i

i i

i i

i i

i( , ) ( , ) ( )

( )( , )

=

1

165

Revenue Equivalence

Theorem 1. In any equilibrium of any auction game in which the lowest-type bidders receive an expected payoff of zero, the sellers expected revenue equals

E MR t y dyt iq t

i

ni ( , )( )

01z

=

LNM

OQP

166

Optimal Auction MR monotonicity

increasing in ti weakly increasing in tj weakly decreasing in qi

MR regularity: for all i, j, qi, qj, ti, ti > ti,MRi(t,qi) > MRj(t,qj) MRi(ti,ti,qi) > MRj(ti,ti,qj)

Theorem 2. Suppose MR is monotone and regular. Sellers revenue is maximized by awarding the good to those with the highest marginal revenues, until the good is exhausted or marginal revenue becomes negative.

167

Optimal Auction is Inefficient

Assign goods to wrong parties High MR does not mean high value

Assign too little of the good MR turns negative before values do

168

Three Seller Programs

1. Unconstrained optimal auction(standard auction literature)Select assignment rule and pricing rule to

max E[Seller Revenue]s.t. Incentive Compatibility

Individual Rationality

169


2. Resale-constrained optimal auction(Coase Theorem critique)Select assignment rule and pricing rule to


Individual RationalityEfficient resale among bidders

170


3. Efficiency-constrained optimal auction(Coase Conjecture critique)Select assignment rule and pricing rule to


Individual RationalityEfficient resale among bidders

and seller

171

1. Unconstrained optimal auction

Select assignment rule to

All feasible assignment rules.}

q t

E MR t y dy

Qq t Q t i

q t

i

ni

( )

max ( , )

{( )

( )

=zLNM

OQP

=

01

172

1. Unconstrained optimal auction(two bidders)

quantity

price

0

d2 d1 MR1 MR2

D

q

S

1

p2

MR

173

3. Efficiency-constrained optimal auction


Ex post efficient assignment rules.}

q t

E MR t y dy

Q

R

q t Qt i

q t

i

n

R

R

i

( )

max ( , )

{

( )

( )

=zLNM

OQP

=

01

174

3. Efficiency-constrained optimal auction (two bidders)

quantity

price

0

d2 d1 MR1 MR2

D

q=1

S p1

175

2. Resale-constrained optimal auction


Resale - efficient assignment rules.}

q t

E MR t y dy

Q

R

q t Qt i

q t

i

n

R

R

i

( )

max ( , )

{

( )

( )

=zLNM

OQP

=

01

176

2. Resale-constrained optimal auction(two bidders)

quantity

price

0

d2 d1 MR1 MR2

D MRR

qR

S

1

p1

177

Theorem. In the two-stage game (auction followed by perfect resale), the seller can do no better than the resale-constrained optimal auction.

Proof. Let a(t) denote the probability measure on allocations at end of resale round, given reports t. Observe that, viewed as a static mechanism, a(t) must satisfy IC & IR. In addition, a(t) must be resale-efficient.

178

Can we obtain the upper bound on revenue?resale process is coalitionally-rational against

individual bidders if bidder i obtains no more surplus si than i brings to the table: si v(N | q,t) v(N ~ i | q,t).

That is, each bidder receives no more than 100% of the gains from trade it brings to the table.

179

Vickrey auction with reserve pricingSeller sets monotonic aggregate quantity that will

be assigned to the bidders, an efficient assignment q*(t) of this aggregate quantity, and the payments x*(t) to be made to the seller as a function of the reports t where

Bidders simultaneously and independently report their types t to the seller.

{ }

* ( )*

0

*

( ) ( ( , ), , ) , where

( , ) inf | ( , ) .

i

i

q t

i i i i i

i i i i i it

x t v t t y t y dy

t t y t q t t y

=

=

180

Can we attain the upper bound on revenue?Theorem 5 (Ausubel and Cramton 1999). Consider

the two-stage game consisting of the Vickrey auction with reserve pricing followed by a resale process that is coalitionally-rational against individual bidders. Given any monotonic aggregate assignment rule , sincere bidding followed by no resale is an ex post equilibrium of the two-stage game.

ApplicationsElectricity market design

Climate policySpectrum auction design

Future of mobile communicationsFuture of transportation

Future of financial markets181

Electricity

182

Goals of electricity markets

Short-run efficiency Least-cost operation of existing resources

Long-run efficiency Right quantity and mix of resources

183

Key idea: open access

Challenges of electricity markets

Must balance supply and demandat every instantat every location

Physical constraints of network Absence of demand response Climate policy

184

Three Markets Short term (5 to 60 minutes)

Spot energy market Medium term (1 month to 3 years)

Bilateral contracts Forward energy market

Long term (4 to 20 years) Capacity market (thermal system) Firm energy market (hydro system)

Address risk, market power, and investment

185

Long-term market:Buy enough in advance

186

Product

What is load buying? Energy during scarcity period (capacity)

Enhance substitution Technology neutral where possible Separate zones only as needed in response to binding

constraints Long-term commitment for new resources to reduce risk

187

Pay for Performance

Strong performance incentives Obligation to supply during scarcity events

Deviations settled at price > $5000/MWh Penalties for underperformance Rewards for overperformance

Tend to be too weak in practice, leading to Contract defaults Unreliable resources

But not in best markets: ISO New England, PJM

188

David MacKay, Peter Cramton, Axel Ockenfels and Steven Stoft), Nature, 526, 315-316, 15 October 2015.

Symposium: International Climate NegotiationsCramton, Ockenfels, Stoft (eds.), Gollier, Stiglitz, Tirole, WeitzmanEconomics of Energy & Environmental Policy, 4:2, September 2015

Price CarbonI Will If You WillMacKay, Cramton, Ockenfels & StoftNature, 15 October 2015

Global Carbon PricingWe Will If You WillCramton, MacKay, Ockenfels & StoftMIT Press, under review, 2016carbon-price.com

190

http://www.cramton.umd.edu/papers2015-2019/eeep-symposium-international-climate-negotiations.pdfhttp://www.cramton.umd.edu/papers2015-2019/mackay-cramton-ockenfels-stoft-price-carbon.pdfhttp://www.cramton.umd.edu/papers2015-2019/global-carbon-pricing-cramton-mackay-okenfels-stoft.pdfhttp://www.cramton.umd.edu/papers2015-2019/mackay-cramton-ockenfels-stoft-price-carbon.pdfhttp://www.cramton.umd.edu/papers2015-2019/global-carbon-pricing-cramton-mackay-okenfels-stoft.pdf

Consensus Aspiration: 2C goal

192

IPCC, SYR Figure SPM.10

1.5

How to bridge gulf between goal and intentions?

Paris Agreement

193

Until 2020max political power

no excusesnational policy

Until 2030moderate power

some excusesnational plans

Until 2100no power or blame

many excusesglobal aspiration

Abat

emen

t effo

rt

No down-payment

No payments first 15 years

Non-binding jumbo

payments

Economics:Price carbon

194

Direct Efficient Transparent Promotes international cooperation

195

Remarkably, price never appears in 31 page COP21 Final Agreement

Treaty Design:Promoting cooperation in international negotiations

196

197

Individual commitments(intended nationally determined contributions)

cannot promote cooperation

Individual commitments cannot promote cooperation 10 players; individual endowment = $10 Each $ pledged will be doubled and distributed evenly

to all players Voluntary pledges are enforced Result: Zero cooperation, all pledge $0

Pledge $10$0

Uniqueequilibrium No cooperation

198

Dynamics of individual commitments: Upward spiral of ambition?

History: Japan, Russia, Canada, and New Zealand left the Kyoto agreement

Ostrom (2010), based on hundreds of field studies: insufficient reciprocity leads to a downward cascade

Supported by theory and laboratory experiments

199

Common commitment: I will if you will

200

Trump: I wont cause you wont

201

I will if you will promotes cooperation

10 players; Individual endowment = $10 Each $ pledged will be doubled and distributed evenly

to all players Pledge is commitment to reciprocally match the

minimum pledge of others Voluntary pledges are enforced Result: Full cooperation, all pledge $10

Pledge $10$0

Uniqueequilibrium

Full cooperation

202

Price is focalcommon commitment

203

Direct, efficient, common intensity of effort Consistent with tax or cap & trade

(flexible at country level) Consensus that price should be uniform reduces

dimensionality problem:

PCountry = Pglobal

(No such consensus exists for quantity commitment)

204

Price commitment reduces risk

Countries keep carbon revenues

Eliminates the risk of needing to buy credits

205

But are quantity commitments equivalent?

China, you will be safe, if you accept a Business-as-Usual target for 2008 2012.

Jeffery Frankel, 1998

Business as Usual means what experts think 1999 US Dept. of Energy: 7.5 Gt of CO2 Reality in 2008 2012: 36.6 Gt of CO2

206

Cap targets P = $30, but then P $45

Chinas unexpected costs > $1 trillion $817 B Payments to US, EU, India ??

207

$0

$15

$30

$45

0 10 20 30 40

Carb

on P

rice

EmissionsDOE prediction Actual

Unexpectedabatement cost under Global Cap

$225 B

$817 BUnexpected Trading Cost

underCap & Trade

Gt

Prediction-Error Trading Costs for China, 2008 2012

Carbon Pricing: P = $30

Chinas unexpected costs = $88 B Payments clean up Chinas pollution

208

$0

$15

$30

$45

0 10 20 30 40

Carb

on P

rice

EmissionsDOE prediction Actual

Unexpectedabatement cost under

Global Pricing$88 B

Gt

Prediction-Error Pricing Costs for China, 2008 2012

Sharing the burden

209

Use Green Fund to maximize abatement As before, reduce dimensionality

Carbon price = intensity of cooperation Generosity parameter = intensity of Green Fund

Last resort enforcement with trade sanctions

210

Designing the Green Fund

Excess emissions = deviation from world per capital average (+ for US, - for India)

This addresses differentiated responsibilities Rich, high-emission countries pay into fund Poor, low-emission countries receive from fund

211

Payment into Green Fund =

Generosity parameter

Excess emissions

Global carbon price

Maximizing treaty strength

If G is high, rich countries will want P* low If G is low, poor countries will want P* low Some moderate G maximizes the P* that a super-

majority will accept

212

A mechanism for the willing (G20?) Countries with little stake in the Green Fund (near

average emissions) first determine G G will be determined so that both rich and poor countries

benefit from an effective agreement

Then countries vote for P*; low price wins No country i commits to a P* > Pi, so any country could

protect itself by naming a low Pi if G were unacceptable

Mechanism promotes a strong agreement

213

China USIndia

7.2 440.1

QatarRwanda

1.6 17.2Emissions per capita (tons/year)

Summary

Keys to a strong climate treaty I will if you will (common commitment) Two parameters

Carbon price (common intensity of effort) Green fund intensity (addresses asymmetries)

Further research Equilibrium simulations using standard climate models

to identify best climate club Develop details of treaty (e.g. voting mechanism)

214

215

Price CarbonI will if you will

Spectrum

216

Spectrum auctions

Many items, heterogeneous but similar Competing technologies and business plans Complex structure of substitutes and complements

Government objective: Efficiency Make best use of scarce spectrum Address competition issues in downstream market

217

Key design issues

Establish term to promote investment Enhance substitution

Product design Auction design

Encourage price discovery Dynamic price process to focus valuation efforts

Encourage truthful bidding Pricing rule Activity rule

218

Simultaneous ascending auction

219

Prepare

220

Italy 4G Auction, September 2010470 rounds, 22 days, 3.95B Auction conducted on-site with pen and paper Auction procedures failed in first day No activity rule

221

Thailand 3G Auction, October 2012 3 incumbents bid 3 nearly identical licenses Auction ends at reserve price + 2.8%

222

223

Conflict and cooperation in Germanys spectrum auctionsIn Germany (1999) 10 spectrum blocks were simultaneously sold:

New bid for a particular block must exceed prior bid by at least 10 percent

Auction ends when no bidder is willing to bid higher on any block

What happened?

224

The auction begins 1 2 3 4 5 6 7 8 9 10

1

2

3

225

The offer of Mannesmann (=M)1 2 3 4 5 6 7 8 9 10

1 36.4M

36.4M

36.4M

36.4M

36.4M

40M

40M

40M

40M

56M

2

3

226

T-Mobile (=T) accepted1 2 3 4 5 6 7 8 9 10

1 36.4M

36.4M

36.4M

36.4M

36.4M

40M

40M

40M

40M

56M

2 40T

40T

40T

40T

40T

- - - - -

3

227

Perfect cooperation1 2 3 4 5 6 7 8 9 10

1 36.4M

36.4M

36.4M

36.4M

36.4M

40M

40M

40M

40M

56M

2 40T

40T

40T

40T

40T

- - - - -

3 - - - - - - - - - -

Combinatorial Clock Auction

228

Combinatorial clock auction

Auctioneer names prices; bidder names package

Price increased if excess demand Process repeated until no excess demand

Supplementary bids Improve clock bids Bid on other relevant packages

Optimization to determine assignment/prices

No exposure problem (package auction) Second pricing to encourage truthful bidding Activity rule to promote price discovery

229

CCA SucksLook at Switzerland!

230

No:Rules were poor

Bidding likely poorSetting was difficult

Pricing rule

231

Bidder-optimal core pricing

Minimize payments subject to core constraints Core = assignment and payments

such that Efficient: Value maximizing assignment Unblocked: No subset of bidders offered seller a better

deal

232

Optimization

Core point that minimizes payments readily calculated

Solve Winner Determination Problem Find Vickrey prices Constraint generation method

(Day and Raghavan 2007) Find most violated core constraint and add it Continue until no violation

Tie-breaking rule for prices is important Minimize distance from Vickrey prices

233

5 bidder example with bids on {A,B}

b1{A} = 28

b2{B} = 20

b3{AB} = 32

b4{A} = 14

b5{B} = 12

Winners

Vickrey prices:p1= 14

p2= 12

234

The Core

b4{A} = 14b3{AB} = 32

b5{B} = 12

b1{A} = 28

b2{B} = 20

Bidder 2Payment

Bidder 1Payment

14

12

3228

20

The Core

Efficient outcome

235

The Core

b4{A} = 14b3{AB} = 32

b5{B} = 12

b1{A} = 28

b2{B} = 20

Bidder 2Payment

Bidder 1Payment

Vickrey prices

14

12

3228

20

Vickrey prices: How much can each winners bid be reduced holding others fixed?

Problem: Bidder 3 can offer seller more (32 > 26)!

236

The Core

b4{A} = 14b3{AB} = 32

b5{B} = 12

b1{A} = 28

b2{B} = 20

Bidder 2Payment

Bidder 1Payment

Vickrey prices

14

12

3228

20

Bidder-optimal core prices: Jointly reduce winning bids as much as possible

Problem: bidder-optimal core prices are not unique!

237

Uniquecore prices

b4{A} = 14b3{AB} = 32

b5{B} = 12

b1{A} = 28

b2{B} = 20

Bidder 2Payment

Bidder 1Payment

Vickrey prices

14

12

3228

20

Core point closest to Vickrey prices

17

15

Each pays equal share above Vickrey

238

Activity rule

239

Clock stage performs wellProposition: With revealed preference w.r.t. final round

If clock stage ends with no excess supply,final assignment = clock assignment

Supplementary bids cant change assignment;but can change prices

May destroy incentive for truthful bidding in supplementary round

Supplementary round still needed to determine competitive prices

Possible solutions Do not reveal demand at end of clock stage; possibility of excess

supply motivates more truthful bidding (Canada 700 MHz) Do not impose final price cap (UK 4G)

240

Summary:CCA is an important tool Eliminates exposure Reduces gaming Enhances substitution Allows auction to determine band plan, technology Readily customized to a variety of settings Many other applications

241

242

Broadcast Incentive Auction

29 March 2016

Motivation

Year

Value per MHz

1985 1990 1995 2000 2005 2010 2015

Value of over-the-air broadcast TV

Value of mobile broadband

Gains from trade

243

What is the Incentive Auction?

The worlds most complicated auction!

Goal: clear broadcast spectrum, and repurpose for wireless use Consumer demand for broadcast TV has waned, while

demand for mobile broadband keeps increasing Spectrum in use for broadcast television is very

desirable Excellent propagation characteristics

Convert a set of 6MHz channels into 5+5MHz paired wireless blocks

Ex: 21 channels = 126MHz of spectrum = 10 paired blocks + guard bands

What is the Incentive Auction? Auction has been in the works for a long time

Part of the FCCs National Broadband Plan released in 2010 Initiated by Spectrum Act legislation passed in 2012 Initial design released in 2014; final design released June 2015 Began 29 March 2016; ends late 2016 or early 2017 Stage 1, reverse auction ended 29 June 2016; revenue requirement

$88 billion

Auction involves many novel challenges Combines both a reverse and a forward auction into a single

process Clearing television spectrum is a hard problem Interplay between forward and reverse auctions introduces

additional complexity

Auction summary Oct 16: FCC announces opening price for each station Mar 29: Station decides whether to accept opening price (participate) Apr 29: FCC optimization (min impairment) to determine clearing target

Largest target with impairment < equivalent of 1 nationwide block (10% at 126MHz)

Stage 1: initial clearing target (126MHz, 10 blocks) May 31-Jun 29: Reverse auction to determine clearing cost ($88 billion) Aug 16-30: Forward auction to determine auction proceeds ($23 billion)

Stage 2: reduce clearing target (114MHz, 9 blocks) Sep 10-Oct 11: Reverse continues with lower target (need fewer volunteers) Forward continues with fewer blocks; End if proceeds > cost

Stage End: FCC optimization to determine channel assignments

Auction progress

Quantity(5+5 MHz blocks)

Price (billion $/block)

Stage 1

Stage 2(example)

Stage 3(example)

7 8 9 10

247

88

23

54

303337

Reverse clearing cost

Forward auction proceeds

8.8

2.3Done!

Keys features of good design

Simple expression of supply and demand Station: at this lower price are you still willing to clear? Carrier: at this higher price how many blocks do you

want?

Monotonicity Reverse: prices only go down (excess supply) Forward: prices only go up (excess demand)

Activity rule Reverse: station exit is irreversible Forward: 95% activity requirement

Reverse auction(broadcasters: supply side) Identifies prices and stations to be cleared to achieve

target

There are many different ways for broadcasters to participate

Relinquish their broadcast license entirely Move to a lower band (change from UHF to VHF) Share a channel and facilities with another broadcaster

Decision to participate is voluntary Any broadcaster that does not participate can continue

broadcasting with their current coverage, but channel may change

Forward auction(carriers: demand side) Ascending clock, rather than simultaneous multiple round

Issue: participation in reverse auction determines licenses on offer

Nonparticipating broadcasters must be assigned a channel FCC needs to impair some licenses if too many nonparticipants

Issue: closing requires forward auction to generate enough revenue

Must pay for reverse auction payments + relocation costs + FCC costs

If closing conditions are not met, FCC holds an extended round to increase bidding in top-40 markets, or continues process with a lower clearing target

What is Repurposing broadcast spectrum?CURRENT CHANNEL ALLOCATIONS (sample large market) (29 stations total; 15 above channel 30)

15 Occupied Channels

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV

High V's Ch 7-13 UHF Broadcast UHF Broadcast

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV TV

High V's Ch 7-13 UHF Broadcast UHF Broadcast

POST AUCTION ALLOCATIONS

TV TV TV TV TV TV TV TV TV

Available for mobile broadband6 stations from Ch. 31 51 moved into Ch. 14 - 30

VHF unchanged

REQUIRED AUCTION VOLUNTEERS: (9 stations from channels 31-51 have no available channel)

Repack Stations

Sheet1

High V's Ch 7-13UHF BroadcastUHF Broadcast

789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV

Sheet2

Sheet3

Sheet1


789101112131415161718192021222324252627282930313233343536373839404142434445464748495051



789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV

TVTVTVTVTVTVTVTVTV

6 added

9 without home in UHF

Sheet2

Sheet3

Sheet1


789101112131415161718192021222324252627282930313233343536373839404142434445464748495051



789101112131415161718192021222324252627282930313233343536373839404142434445464748495051

TVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVTV

TVTVTVTVTVTVTVTVTV

6 added

9 without home in UHF

Sheet2

Sheet3

Possible band plans

Notes:1. Numbers in the table represent current broadcast television channels2. Letters in the table represent paired blocks for mobile wireless services3. Grey blocks represent guard bands and their width (in MHz)4. Impaired markets will have a band plan from the table, but not at the nationwide clearing target5. Band plan above stops at UHF channel 21 (does not show UHF channels 14 to 20)

Number of Paired

Blocks

BroadcastSpectrumCleared (MHz)

Current Broadcast Television Band

Proposed Mobile Wireless ServicesRepacked Broadcasters

Why is this complicated? Have a supply side (broadcasters) and demand side (wireless carriers),

so why not just determine where supply meets demand?

Issue: the supply side is extremely complicated Buy stations until # stations = # channels only works at local scale Two stations can only use the same channel if they do not cause interference

Interference between stations creates a complex set of constraints Depends on physical factors: distance, transmitting power, terrain, antenna If stations interfere strongly enough, cannot even be on adjacent channels Large markets can interfere with each other, e.g. New York and Philadelphia

Nationwide interference constraints

Finding channel assignments Finding a nationwide channel assignment is extremely

complicated Hundreds of thousands of interference constraints between

stations Border regions must respect treaties with Canada and Mexico Some channels reserved for emergency communications in many

markets

Checking if a set of stations can be repacked is computationally hard

Equivalent to classic Satisfiability (SAT) problem from computer science

No generally efficient solution known; widely believed not to exist The auction needs to make hundreds of thousands of these checks Fortunately, this can be made (mostly) efficient in practice

Design of the reverse auctionThe reverse auction is a scored descending clock auction: Each television station is assigned a score (starting price) A single clock ticks down from 100% to 0%; at each time

step: Every station is offe

auctions and market design - peter...

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