price formation in double auctions - economics · price formation in double auctions 3 processing...

Ž .GAMES AND ECONOMIC BEHAVIOR 22, 1]29 1998ARTICLE NO. GA970576

Price Formation in Double AuctionsU

Steven Gjerstad†

Hewlett-Packard Laboratories, Palo Alto, California 94304

and

John Dickhaut

Carlson School of Management, Uni ersity of Minnesota, Minneapolis, Minnesota 55455

Received November 30, 1995

We develop a model of information processing and strategy choice for partici-pants in a double auction. Sellers in this model form beliefs that an offer will beaccepted by some buyer. Similarly, buyers form beliefs that a bid will be accepted.These beliefs are formed on the basis of observed market data, including frequen-cies of asks, bids, accepted asks, and accepted bids. Then traders choose an actionthat maximizes their own expected surplus. The trading activity resulting fromthese beliefs and strategies is sufficient to achieve transaction prices at competi-tive equilibrium and complete market efficiency after several periods of trading.Journal of Economic Literature Classification Numbers: D41, D44, D8 Q 1998

Academic Press

1. INTRODUCTION

Ž .The double auction DA is one of the most common exchange institu-tions, used extensively in stock markets such as the New York StockExchange, commodity markets such as the Chicago Mercantile Exchange,and in markets for financial instruments, including options and futures.The prevalence of this institution can be traced to its operational simplic-ity, efficiency, and its capacity to respond quickly to changing marketconditions. Nevertheless, the DA is a persistent puzzle in economic theory.

* We thank Vernon Smith and Arlington Williams for providing data for comparison withthe model in this paper. We thank Dan Friedman, Leo Hurwicz, John Ledyard, KevinMcCabe, Charlie Plott, Jason Shachat, Vernon Smith, Arlington Williams, and Steve Williamsfor helpful discussions. We also would like to thank two referees for several useful sugges-tions. We especially thank Jim Jordan for his help with the formulation of this model as aninformationally decentralized decision process.

† E-mail: [email protected].

10899-8256r98 $25.00

Copyright Q 1998 by Academic PressAll rights of reproduction in any form reserved.

GJERSTAD AND DICKHAUT2

How is information that is held separately by many market participants}inthe form of privately known reservation values and marginal costs}quicklyand accurately coordinated through the trading process to reach the

Ž .competitive equilibrium CE price and allocation?ŽIn the double auction, any seller may at any time during a specified

.trading period submit an offer that is then observed simultaneously by allbuyers and sellers. Similarly, any buyer may submit a bid that is observedby the other buyers and by the sellers. When a buyer’s bid is acceptable tosome seller, that seller may then accept the buyer’s bid, and a trade isexecuted between the buyer whose bid was accepted and the seller whoaccepted this bid. Similarly, buyers may accept a seller’s offer at any time.

Market experiments have established that transaction prices convergequickly to a competitive equilibrium price in the DA for a wide variety ofmarket environments. Experimental investigation of trader behavior and

Ž .market performance in the DA began with Smith 1962 . Smith inducedsupply and demand conditions by giving buyers a redemption value foreach unit of an abstract commodity purchased, and by giving sellers a costfor each unit of this abstract commodity sold. Buyers receive a surplusequal to the difference between their redemption value and the purchaseprice negotiated with a seller, and sellers receive a surplus equal to thedifference between the purchase price paid by the buyer and their unitcost. Since reservation prices}and therefore supply and demand condi-tions}are known to the experimenter when this procedure is employed,the procedure makes possible a comparison between experimental out-comes and theoretical predictions. The basic result observed in theseexperiments is that prices do converge quickly to within a few cents ofcompetitive equilibrium prices in markets with stationary supply anddemand. Smith and many other economists in the 35 years since his initialstudies have also documented features of the path of convergence toequilibrium in a variety of market environments. For surveys and interpre-

Ž . Ž .tation of these experimental results, see Plott 1982 and Smith 1982 .Models of trader behavior in the DA have been constructed by several

Ž . Ž .authors, including Easley and Ledyard 1993 , Friedman 1991 , Gode andŽ . Ž .Sunder 1993 , and Wilson 1987 . Although these models have furthered

understanding of the interaction of individual behavior and institution inthe DA, we provide a model that accounts for several important regulari-ties of double auction data that no one of the previous models predicts orreplicates in simulations. For comparison of the predictions of the lastthree models with properties of experimental data, see Cason and Fried-

Ž .man 1993, 1996 .We model individual behavior of sellers and buyers in a continuous DA

and demonstrate that the persistent puzzle of convergence to CE pricesand allocations in the DA can be resolved with traders whose information

PRICE FORMATION IN DOUBLE AUCTIONS 3

processing and strategy choices are simple and intuitive. For each possiblebid, each buyer forms a subjective belief that some some seller will acceptthe bid. The buyer then determines which bid will maximize her ownexpected surplus. Similarly, each seller determines which offer will maxi-mize his expected surplus. Subjective beliefs are formed using only ob-served market activity, including bids, offers, and accepts of bids and

Žoffers. This procedure does not require any knowledge of the types costs.and valuations of other buyers and sellers; in fact, traders in this model do

not even have beliefs about the types of others. Nevertheless, this behaviorresults in efficient allocations, and convergence of transaction prices towithin a few cents of CE prices within several periods of trading. Inaddition, these beliefs respond quickly to changes in market conditions,such as shifts in market demand or supply.

The organization of the paper is as follows. The model is formulated inSection 2. Simulations of the model are shown and some importantstatistical properties of these simulations are reported in Section 3. Section4, the conclusion, summarizes the relationship between our model andexperimental data.

2. THE MODEL

Like most forms of market organization, the double auction is aninformationally decentralized system. Our model emphasizes this structureto give a more compelling answer to Hayek’s question: How is privatelyheld information coordinated through the market process? Hurwicz et al.Ž .1975 have shown that in general equilibrium environments, even withnonconvexities, there are simple and intuitive forms of market organiza-

Ž .tion and bidding behavior which they call the B process that lead toŽ .Pareto optimal outcomes. Gjerstad and Shachat 1996 construct a map

between partial equilibrium environments of standard market experimentsand general equilibrium economies. In this paper, we develop a model ofinformationally decentralized bargaining for these environments that re-sults not only in Pareto optimal outcomes, but also in substantial stabilityof transaction prices. Our bargaining model, together with the generalequilibrium interpretation of the environments considered here, results ina model of learning competitive equilibrium in a class of general equilib-rium environments.

In this section, we describe the elements of a microeconomic system,interpret the double auction environment and institution within thisframework, and construct an informationally decentralized model of traderbehavior for these environments in the double auction institution.


The double auction is an example of a microeconomic system, as inŽ . Ž .Hurwicz 1972 and Smith 1982 . The primary features of a microeco-

nomic system are the en¨ironment e, consisting of the characteristics of theeconomic agents, and the institution I, which includes the messages thattraders may send to one another, the allocation rules, and the adjustment

( )process rules. A microeconomy is an economic system S s e, I , togetherwith behavioral actions b i for market participants, as shown in Fig. 1.

� 4The environment e consists of a set AA s 1, 2, . . . , n of agents, and foreach agent i characteristics ei consisting of that agent’s preferences,technology, and endowment. The environment is then e s Ł ei. Theig AA

institution I consists of a message space M i for each agent, an adjustmentprocess rule specifying the sequence of agent messages, and an outcome

Ž . Ž 1Ž . 2Ž . nŽ ..function or allocation function h m s h m , h m , . . . , h m ,t t t tŽ 1 2 n. iwhere m s m , m , . . . , m g M s Ł M is the vector of agents’t t t t t ig AA t

messages.Ž .According to Smith 1982, p. 930 ,

We want to measure messages because we want to be able to identify theiŽ i .behavioral modes, b e , I , revealed by the agents and test hypotheses derived

from theories about agent behavior.

When an environment e and an institution I are specified in a marketexperiment, and an outcome X is observed, the only elements remaining to

� iŽ i .4be specified are the behavioral actions b H ¬ e , I where H is thet ig AA thistory of activity observed by agents through time t. The focus of the

� iŽ i .4research in this paper is to specify forms of behavior b H ¬ e , I thatt ig AA

are consistent with observations X from exchange environments e whenthe institution I is the double auction. We now describe a representationof the double auction in terms of this framework.

FIGURE 1


2.1. En¨ironment

In the double auction market environments we consider, there are twogoods: an experimental currency X and a fictitious commodity Y. Ourtheory addresses the case of markets with the set of traders partitionedinto a group I of sellers and a group J of buyers. Both types of agentsŽ .sellers and buyers are assumed to have preferences over monetaryrewards that are monotonically increasing.

Ž 1 2 m i.Seller i g I has a vector of induced unit costs c s c , c , . . . , c fori i i i

production of an abstract commodity. here c1 is the cost to seller i of thei

first unit sold, c2 is the cost of the second unit, and so on. We index thei

cost of each unit because the model of trader behavior is developed forŽ .traders who frequently sell or purchase multiple units. The gain to seller

k Ž k . ki on his kth unit sold is the difference p p , c s p y c between thes, i k i k i

price p received from a buyer for that unit, and the cost ck at which thek i

unit is produced. If seller i sells m F m units at prices p , p , . . . , p ,i i 1 2 m i

Ž m i Ž k .. Ž .then the utility to this seller is U Ý p y c , where U ? is mono-s, i ks1 k i s, i

tonically increasing.1

Ž . Ž .Gjerstad and Shachat 1996 GS96 show that the cost vector of eachseller i g I is dual to a technology that is described by a production

m iiŽ .function f x . Let x G Ý c . A seller with an endowment v si s, i is1 i s, iŽ . Ž .x , 0 will have sufficient currency the input good to produce each ofs, i

the m units for which he has finite cost. Then characteristics of seller iis, i Ž .are described by the vector e s f , v . In Example 1 below, we carryi s, i

out their construction for one seller in a market experiment.Ž 1 2 n j.Buyer j g J has a vector of unit valuations ¨ s ¨ , ¨ , . . . , ¨ ,j j j j

where ¨ 1 is the redemption value for the first unit acquired, ¨ 2 is thej j

redemption value for the second, and so forth. Buyer j has an endowmentx of trading currency that is sufficient to purchase each unit at a priceb, j

n 1jup to the redemption value of the unit, i.e., x G Ý ¨ . Monetaryb, j ls1 jl Ž l. lrewards for buyers are the difference p p , ¨ s ¨ y p between theb, j l j j l

redemption values of units purchased and the price p paid to a seller. Ifl

buyer j purchases n F n units at prices p , p , . . . , p the monetary gainj j 1 2 n jn j Ž l .from trading for buyer j is Ý ¨ y p , and the utility of this monetaryls1 j l

Ž n j Ž l .. Ž .gain is U Ý ¨ y p , where U ? is monotonically increasing.b, j ls1 j l b, j

For any vector of valuations ¨ , GS96 show that there is a quasi-linearj

Ž . Ž . Ž .utility function u x, y s x q ¨ y y x and an endowment v s x , 0j j j j j

1 In Section 2.4.6, where sellers’ strategies are formulated, we assume that seller i attemptsto maximize surplus on each of his m units separately and in sequence.i


such that the demand for good X by this buyer is ¨ . The characteristics ofjb, j Ž .buyer j are then e s u , v . In Example 1, we carry out theirj b, j

construction for one buyer in a market experiment.We describe the environment of an induced cost and valuation experi-

ment by the collection

e s f , v j u , v .� 4Ž . Ž .� 4i s , i j b , jigI jgJ

EXAMPLE 1. Figure 2 shows supply and demand conditions for markettrading experiment 3pda01 run in the experimental lab at the University ofArizona by Vernon Smith and Arlington Williams. In this market there arefour buyers, each with positive valuations for three units, and four sellers,each with finite costs for three units. The vector of buyers’ valuations is

� 4 � 4f s 3.30, 2.25, 2.10 , 2.80, 2.35, 2.20 ,�� 4 � 42.60, 2.40, 2.15 , 3.05, 2.35, 2.30 .4

The vector of sellers’ cost is

� 4 � 4c s 1.90, 2.35, 2.50 , 1.40, 2.45, 2.60 ,�� 4 � 42.10, 2.30, 2.55 , 1.65, 2.35, 2.40 .4

Since buyer j with redemption value ¨ l makes a monetary gain at anyjpurchase price p - ¨ l, and since buyers’ preferences are assumed to bejmonotonically increasing in monetary gain, this buyer is willing to pay anyprice up to ¨ l for the lth unit purchased. Therefore, the demand shown injFig. 2 is determined by arraying the buyers’ redemption value vectors.Supply is obtained analogously.

FIGURE 2


Ž .The diagram on the left of Fig. 3 shows the production function f x1� 4that is dual to the cost vector c s 1.90, 2.35, 2.50 for seller 1 in this1

market, where

xr1.90 0 F s F 1.90¡x q 0.45 r2.35 1.90 - x F 4.25Ž .~f x s 1Ž . Ž .1 x q 0.75 r2.50 4.25 - x F 6.75Ž .¢

3 6.75 - x .

Ž .We assume that seller 1 has an endowment v s x , 0 , where x iss, 1 s, 1 s, 1large enough to produce all three units. It is easy to verify that the supplyof a cost-minimizing producer with the production function f will be 01units if the price is below 1.90, 1 unit if the price is between 1.90 and 2.35,and so on, so the seller’s vector of unit costs is dual to the production

Ž . 1 Ž .function in Eq. 1 . The characteristics of seller 1 are then e s f , v .1 s, 1The diagram on the right of Fig. 3 shows two indifference curves for

� 4buyer 1, whose vector of unit valuations is ¨ s 3.30, 2.25, 2.10 . For buyer11, define the valuation function ¨ by1

3.30 y 0 F y F 1¡2.25y q 1.05 1 - y F 2~¨ y s 2Ž . Ž .1 2.10 y q 1.35 2 - y F 3¢7.65 3 - y.

Ž . Ž .If we assume that buyer 1 has the endowment v s x , 0 s 7.65, 0b, 1 b, 1

Ž . Ž .and that the utility function of buyer 1 is u x, y s x q ¨ y y 7.65, then1 1the lower indifference curve in Fig. 3 corresponds to u s 0 and the upper1one to u s 1.1

If the production functions and endowments of the remaining threesellers and the utility functions and endowments of the other 3 buyer are

FIGURE 3


�Ž .44constructed in this way, then the environment is e s f , v ji s, i is1�Ž .44u , v . In what follows, we refer to this market as the symmetricj b, j js1

market design.

Trading Periods

A typical laboratory market experiment involves trading over severalperiods. Each seller has costs induced for the trading period, and eachbuyer has valuations induced. A buyer’s valuation for a unit remains ineffect throughout the trading period or until the buyer transacts that unit.After a unit is transacted, the seller’s cost and the buyer’s valuation for theunit just transacted are removed from the supply and demand schedules,trading continues, and this process proceeds until there are no moresurplus enhancing trades remaining, or until time expires in the tradingperiod. At the conclusion of a trading period, the costs and valuations arereinitialized}possibly at different amounts}in the subsequent period. Inmarket experiment 3pda01, the supply and demand conditions in Fig. 2were employed in each of nine trading periods, each lasting 300 sec.

2.2. Institution

In the double auction, sellers post ask prices and buyers post bids. Themessage space defines the set of allowable messages for each agent. In thispaper we consider the double auction with a bid]ask spread reduction ruleŽ .defined below . In effect, this produces restrictions on agents’ messages asa function of some previous messages of other agents. In a microeconomicsystem, adjustment process rules specify the set of allowable messages foreach trader, the time when exchange of message begins, a transition rulegoverning the sequencing and exchange of messages, and a stopping rule.The DA imposes no restrictions on the sequencing of messages: any tradercan send a message at any time during the trading period. Allocation ofunits is by mutual consent between any buyer and seller. If a seller’s ask isacceptable to a buyer, then a transaction is completed when the buyer

Ž .takes accepts the seller’s ask. Similarly, a buyer’s bid may be accepted bya seller.

˜Ž . � 4DEFINITION 1 MESSAGE SPACE . Let N s x : x s nr100 for n g N .s, i s, i ˜� 4 � 4Seller i at time t has a message space M , where M ; i = 0 = N.t t

b, j b, j ˜� 4 � 4Buyer j has a message space M , where M ; 0 = j = N.t t

Ž .DEFINITION 2 ASKS . An ask a by seller i is an amount that seller i iswilling to accept from a buyer as payment for a unit of the commodity

Ž .being traded. To submit an ask of a seller i sends the message i, 0, a .


Ž .DEFINITION 3 BIDS . A bid b by buyer j is an amount that buyer j iswilling to pay to some seller for a unit. Buyer j submits this bid by sending

Ž .the message 0, j, b .

Ž .DEFINITION 4 SPREAD REDUCTION RULE . The lowest ask in the mar-ket at any time is called the outstanding ask and is denoted oa. At any

˜time sellers may place an ask a g N with a - oa. The highest bid is called˜the outstanding bid ob. At any time, buyers may make a bid b g N above

the outstanding bid. The outstanding ask oa and outstanding bid ob definew xthe bid]ask spread ob, oa . In markets with a spread reduction rule, all

bids and asks must fall within the bid]ask spread.Note that in a double auction with the bid]ask spread reduction rule,

any ask that is permissible must be lower than the current outstanding ask,so each new ask either results in a trade or it becomes the new outstandingask. A similar remark applies to bids.

Ž . Ž .DEFINITION 5 ACCEPTANCE . If seller i sends the message i, 0, a andholds the outstanding ask oa s a, then a take of oa by buyer j is anagreement by j to purchase a unit from seller i at the transaction pricep s oa. Buyer j accepts the outstanding ask oa by sending the messageŽ .0, j, b , where b G oa. Similarly, if the outstanding bid ob is held by buyerjX, then a take of ob by seller iX is an agreement by iX to sell a unit to buyerjX at the transaction price p s ob.

Ž .2.3. Obser ed History Outcome

Ž .EXAMPLE 2 MESSAGES AND HISTORIES . Consider the market of Exam-ple 1 depicted in Fig. 2. If the first action in the period is an offer of 3.00

Ž .by seller 3, we indicate this with the message m s 3, 0, 3.00 . Suppose1that the next action is a bid of 3.00 by buyer 1, which we indicate with the

Ž .message m s 0, 1, 3.00 . At this point a trade is completed between seller23 and buyer 1 at the price 3.00. We indicate the history of these twomessages by the list

� 4H s h , h2 1 2

s 3, 0, 3.00 , 3, 1, 3.00 .� 4Ž . Ž .

Note that in H the triple h unambiguously denotes an accept by buyer 12 2of the offer of 3.00 made with message 1 by seller 3, because seller 3 holds

Žthe outstanding offer of 3.00, which is the transaction price. See Accept of.oa in Definition 6.

Ž .DEFINITION 6 HISTORIES . After n messages have been sent, there willbe a history H consisting of n ordered triples. For any message m sn nq1Ž .m , m , m that is sent, one of six cases will hold:nq1, 1 nq1, 2 nq1, 3


Ž .Inälid ask or bid. A message m s i, 0, a is not valid if a G oa.nq1An invalid ask will not be included in the history. In effect, the institutionignores messages that violate the spread reduction rule. Similarly, a

Ž .message m s 0, j, b is not valid if b F ob.nq1

No ask outstanding. If no ask has been made since the last transaction,˜then there is no outstanding ask, and any ask a g N is valid. If in addition,

m ) ob, then h s m .nq1, 3 nq1 nq1

No bid outstanding. Similarly, if no bid has been made since the last˜transaction, then there is no outstanding bid, and any bid b g N is valid. If

m - oa, then h s m .nq1, 3 nq1 nq1

Accept of ob. If m / 0 and m F ob, then seller m isnq1, 1 nq1, 3 nq1, 1making an offer at or below ob, so m is an accept of ob. The buyer’snq1identity is found by looking back in H and finding the last h withn k

U � 4 Ž .h / 0, that is, k s max k : h / 0 . Then h , h , h sk , 2 k , 2 nq1, 1 nq1, 2 nq1, 3Ž .Um , h , ob .nq1, 1 k , 2

Accept of oa. If m / 0 and m G oa, then m is an acceptnq1, 2 nq1, 3 nq1Ž .of oa by buyer m . The seller’s identity is found by looking back innq1, 2

U � 4 Ž .H and finding k s max k : h / 0 . Then h , h , h sn k , 1 nq1, 1 nq1, 2 nq1, 3Ž .Uh , m , oa .k , 1 nq1, 2

Ž .Improïng ask or bid. If m g ob, oa then m is either annq1, 3 nq1improving ask, or an improving bid, and h s m .nq1 nq1

2.4. Behaïor

2.4.1. Frequencies of Takesk Ž k .As noted in Section 2.1, sellers attempt to maximize p p , c ands, i k i

l Ž l.buyers attempt to maximize p p , ¨ . Since asks or bids must beb, j l jaccepted for a transaction to take place, we take the point of view thatsellers will maximize expected surplus myopically, where the expectation is

Ž .taken relative to beliefs p a that an ask a will be accepted by some buyer.ŽThese beliefs are formed on the basis of observed market data as

.described in Section 2.4.2 . Similarly, buyers are assumed to maximizeexpected surplus myopically, where the expectation is taken relative to

Ž .beliefs q b that a bid b will be accepted by some seller.The history that traders consider in forming beliefs is restricted to those

� 4messages leading up to the last L transactions, where L g 0, 1, 2, . . . .The parameter L is the memory length of traders. Note that the numbersof messages remembered and the clock time elapsed within the traders’memory may vary, while the number of trades completed within the

Ž .traders’ memory does not vary once L trades have occurred . The next


definition provides a procedure for truncating the history, so that beliefscan be constructed using the data within the traders’ memories. The

Ž .procedure for constructing beliefs using this truncated history is de-scribed in Section 2.4.2.

Note 1. We will work with the vector H , although traders do not havenaccess to all of the information in H . Traders know their own asks ornbids, but do not know the identity of the trader making other bids or asks.Information about identities is not used in the formation of beliefs orstrategies, so use of H is made only to avoid complicating notation.n

Ž .DEFINITION 7 REMEMBERED HISTORY . Let HH be the space of possi-nble history vectors of length n. Given H g HH , we make the followingn ndefinitions:

� 4nTrade function. For a vector H , define a function T : HH ¬ 0, 1 byn nŽ . Ž .setting T H s I h . Then each component T of T indicatesk n �h ?h ) 04 k kk , 1 k , 2

whether a trade occurred in the k th element of the history.

Ž .Number of trades. Let x s x , x , . . . , x . For each n, define S :1 2 n n� 4n Ž . n Ž Ž ..0, 1 ¬ N by S x s Ý x . Then S T H is the number of tradesn ks1 k n nresulting from the first n messages.

Remembered history. Let L be the memory length of a given trader.Ž Ž .. XFor fixed n and H , to simplify notation, let S s S T H . Let n be then n n

X ˜ŽL.position of trade S y L if S ) L, and let n s 0 if S F L. Define H byn˜ŽL. � 4X XH s h , h , . . . , h .n n q1 n q2 n

Y � Ž . 4Deletion of oa and ob from history. Let n s max k : T H s 1 , i.e.,k nnY is the index of the most recent trade. Let nU be the index of the lowestŽ . � 4Ymost recent ask in the vector h , . . . , h . Let n# be the index of then q1 n

� 4 Ž .Yhighest bid in the vector h , . . . , h . Note that if T H s 0, then hn q1 n n n n, 3is either the outstanding ask or the outstanding bid, as a consequence of

Ž .the spread reduction rule, and if T H s 1, then there is no outstandingn nŽ .bid and no outstanding ask. If T H s 0 and h s 0, then h is then n n, 1 n, 3

� Y 4 Uoutstanding bid. If h / 0 for some k g n q 1, . . . , n y 1 , then n / Bk , 1ŽL. ŽL. � 4X X U Uand we define H by H ' h , h , . . . , h , h , . . . , h .n n n q1 n q2 n y1 n q1 ny1

ŽL. ˜ŽL.UThat is, H is H with h and h removed. This is done because it isn n n n#

not known at time n if the outstanding ask or bid will be accepted. Theother case}where h is the outstanding ask}is treated similarly. Thenn, 3H ŽL. is the history remembered by traders with memory length L whonobserve the history H .n

Set of asks and bids. Let DŽL. be the set of all asks and bids that havenŽL. ŽL. � 4X Ubeen made in H , i.e., D ' D h .n n k g �n q1, . . . , n4_�n#, n 4 k , 3


Ž . ŽL. Ž .DEFINITION 8 ASK FREQUENCIES . For each d g D , let A d be thenŽ .total number of asks that have been made at d, and let TA d be the total

Ž . Ž . Ž .number of these that have been accepted. Let RA d ' A d y TA d bethe rejected asks at d.

Ž . � XFor A d , the counting procedure is as follows. For each k g n q4 � U4 Ž .1, . . . , n _ n#, n , if h s d, h / 0, and h s 0, then A d is incre-k , 3 k , 1 k , 2

Ž .mented by 1. If h s d and T h s 1, then h is either a taken ask or ak , 3 k k kU �taken bid. To determine which is the case, find m s min m G 1 : hkym , 3

4 Ž . Ž .Us h . If h / 0, then A d and TA d are incremented by 1. Thek , 3 kym , 1

Ž . Ž . Ž .rejected asks at d are given by RA d ' A d y TA d .

Ž . ŽL. Ž .DEFINITION 9 BID FREQUENCIES . For each d g D , let B d be thenŽ .total number of bids that have been made at d, and let TB d be the total

Ž . Ž . Ž .number of these that have been accepted. Let RB d ' B d y TB d .Ž . Ž . Ž .The interpretations and counting procedures for B d , TB d , and RB d

are analogous to those described in Definition 8 for asks.At each time during a market, the proportion of asks at a g D that have

been accepted is

TA aŽ .p a sŽ .ˇ

A aŽ .

Ž .whenever A a ) 0. The proportion of bids at b g D that have beenaccepted is

TB bŽ .q b sŽ .ˇ

B bŽ .

Ž .whenever B b ) 0.In stationary market environments, these empirical frequencies show

Ž . Ž .substantial regularity: p a tends to be a decreasing function of a and q bˇ ˇtends to be an increasing function of b.

Note 2. In what follows, the sets of asks and bids are frequentlydenoted D, with the subscripts and superscripts omitted. When tradershave finite memory, that will be noted. After n messages have been sent,the relevant set of asks and bids is DŽL. and the relevant history is H ŽL..n n

2.4.2 Beliefs

Ž . Ž .While the frequencies p a and q b tend to be monotonic when theˇ ˇnumber of asks and bids is large, there is more variability in small samples.For this reason, it is useful to work with a modification of these summarystatistics.


Ž .Modification of p a is made by taking the point of view that if an askˇaX - a is rejected, then had that ask been made at a it would also havebeen rejected. This assumption is made because a ) aX and is thereforeless appealing to buyers than aX, which was rejected. Similarly, if ask aX ) awas made and taken, then that ask would also have been taken if it weremade at a. Furthermore, if a bid bX ) a is made, then an ask aX s bX would

Žhave been taken if it had been made the assumption being that this ask ofX X.a would be acceptable to the buyer who bid b . This heuristic}and an

analogous one for buyers’ beliefs}are formalized in the next two defini-tions.

Ž .DEFINITION 10 SELLERS’ BELIEFS . For each potential as a g D, define

Ý TA d q Ý B dŽ . Ž .dG a dG ap a s . 3Ž . Ž .ˆ

Ý TA d q Ý B d q Ý RA dŽ . Ž . Ž .dG a dG a dF a

Ž .Then p a is the seller’s belief that an ask amount a will be acceptableˆto some buyer. We assume that sellers always believe that an ask ata s 0.00 will be accepted with certainty, and that there is some value

Ž .M ) 0 such that p M s 0.ˆŽ .The notation of Eq. 3 is simplified by the following definitions. Let

Ž . Ž . Ž . Ž . Ž . Ž .TAG a s Ý TA d ,BG a s Ý B d , and RAL a s Ý RA d .dG a dG a dF aThese are the taken asks greater than or equal to a, the bids greater thanor equal to a, and the rejected asks less than or equal to a, respectively.

Ž .Then Eq. 3 may be rewritten as

TAG a q BG aŽ . Ž .p a s . 4Ž . Ž .ˆ

TAG a q BG a q RAL aŽ . Ž . Ž .

Ž .DEFINITION 11 BUYERS’ BELIEFS . For each possible bid b g D, define

Ý TB d q Ý A dŽ . Ž .dF b dF bq b s . 5Ž . Ž .ˆ

Ý TB d q Ý A d q Ý RB dŽ . Ž . Ž .dF b dF b dG b

Ž .We assume that buyers always believe that q 0.00 s 0 and that there isˆŽ .some value M ) 0 such that q M s 1.ˆ

Ž .As in Definition 10, to simplify notation in Eq. 5 we introduceŽ . Ž . Ž . Ž . Ž .functions TBL b s Ý TB d , AL b s Ý A d , and RBG b sdF b dF b

Ž .Ý RB d . These are the taken bids less than or equal to b, the asks lessdG bthan or equal to b, and the rejected bids greater than or equal to b. Then

TBL b q AL bŽ . Ž .q b s . 6Ž . Ž .ˆ

TBL b q AL b q RBG bŽ . Ž . Ž .


With the specification of seller’s beliefs in Definition 10, the beliefŽ . Ž .function p a is a monotonically decreasing function of a Proposition 1 .ˆ

Ž . Ž .The argument of p ? is the price that the seller asks, and the value of p ?Ž .at a represents the seller’s assessment of the probability belief that an

offer at a will be accepted by some buyer. It is reasonable to expect thatseller’s beliefs are monotonic: this captures the intuition that a trader whohas seen an ask of a rejected should decrease the belief that a will beaccepted later, and decrease the belief that an ask at any value greaterthan a will be accepted. The buyers’ belief function has an analogous

Ž .property: q b is a monotonically increasing function of the bid b.ˆ

2.4.3 Spread Reduction Rule and Beliefs

The spread reduction rule has the effect of making the probability of aŽtake for an ask a G oa equal to 0 where oa is the outstanding offer from

. Ž . Ž . Ž .Definition 4 . We denote this modification of p a by p a , where p a sˆ ˜ ˜Ž . Ž . Ž .p a if a - oa, and p a s 0 if a G oa. Similarly, q b s 0 for all b withˆ ˜ ˜

b F ob. These facts are incorporated into traders’ beliefs in the followingdefinition.

Ž . Ž . Ž .DEFINITION 12. Let p a s p a ? I a for each a g D. That is,˜ ˆ w0, o a.Ž . Ž . Ž . Ž .p a s p a if a - oa and p a s 0 if a G oa. For all b g D, let q b s˜ ˆ ˆ ˜Ž . Ž . Ž .q b s q b ? I b .˜ ˆ Žob, M x

2.4.4. Cubic Spline Interpolation

The belief functions in Definition 12 are defined on the set D of alloffers and bids within the traders’ memory. These beliefs are extended tothe positive reals using cubic spline interpolation. For each successive pair

Ž Ž .. Ž Ž ..of data points a , p a and a , p a , we construct a cubic equa-˜ ˜k k kq1 kq1Ž . 3 2tion p a s a a q a a q a a q a passing through these two points3 2 1 0

with the following four properties:

Ž . Ž .1. p a s p a˜k k

Ž . Ž .2. p a s p a˜kq1 kq1XŽ .3. p a s 0kXŽ .4. p a s 0kq1

These four conditions generate the four equations represented in matrixŽ .Eq. 7 below. The coefficients a are obtained as the solution to thej


equation

3 2a a a 1k k k a p aŽ .3 ˜ k3 2a a a 1 akq1 kq1 kq1 p a2 Ž .˜ kq1s . 7Ž .2 a3a 2 a 1 0 01k k

a2 003a 2 a 1 0kq1 kq1

Ž . Ž Ž ..The function q b is defined similarly, using the pairs b , q b and˜k kŽ Ž ..b , q b .˜kq1 kq1

2.4.5. Monotonicity of Beliefs

Ž .The function p a defined in Section 2.4.2 is monotonically nonincreas-ˆŽ .ing. That is, as the ask a is increased, p a }the belief that an ask a willˆ

Ž .be accepted}is nonincreasing in a. Similarly, q b is nondecreasing in b:ˆbuyers believe higher bids are more likely to be accepted. These results areproven in Propositions 1 and 2. This monotonicity property is extended

Ž . Ž . Ž . Ž .successively to p a , q b , p a , and q b in Propositions 3]6.˜ ˜Ž . Ž .PROPOSITION 1. For all a g D, a g D, with a - a , p a G p a .ˆ ˆ1 2 1 2 1 2

Ž . Ž . Ž . Ž . Ž .Proof. Let G a s TAG a q BG a . Note that G a F G a and2 1Ž . Ž .RAL a F RAL a because a - a . Multiplying these two inequalities1 2 1 2

results in

G a RAL a - G a RAL a . 8Ž . Ž . Ž . Ž . Ž .2 1 1 2

Ž . Ž . Ž .Now add G a G a to both sides of inequality 8 and from this sum2 1Ž . Ž .factor out G a from the left side of the equation and factor G a out of2 1

w Ž .the right side, then divide both sides of the resulting inequality by G a1Ž .xw Ž . Ž .x Ž . Ž .q RAL a G a q RAL a to get p a G p a . Bˆ ˆ1 2 2 1 2

Ž . Ž .PROPOSITION 2. For all b g D, b g D with b - b , q b F q b .ˆ ˆ1 2 1 2 1 2

Proof. The proof is similar to the proof of Proposition 1. BŽ .PROPOSITION 3. The function p a is nonincreasing.˜

Ž . Ž . Ž . Ž .Proof. Since p a s p a I a , and since p a is nonincreasing,˜ ˆ ˆw0, o a.Ž .p a is also nonincreasing. B˜

Ž .PROPOSITION 4. The function q b is nondecreasing.˜Proof. The proof is similar to the proof of Proposition 3. B

Ž .PROPOSITION 5. Let a g D and a g D, with a - a . Then p a is1 2 1 2Ž .nonincreasing on a , a .1 2


Ž . Ž . 3 2Proof. The belief function p a is given by p a s a a q a a q a a3 2 1Ž .q a on the interval a , a , where the coefficients a are given by the0 1 2 j

Ž . Ž .solution to Eq. 7 . The slope of p a on this interval is

pX a s 3a a2 q 2a a q a . 9Ž . Ž .3 2 2

Ž .The coefficients a , a and a can be obtained from Eq. 7 by Cramer’s3 2 1Ž .rule. Substitution of these values into Eq. 9 results in

6 a y a p a y p aŽ . Ž . Ž .Ž .2 1 1 2Xp a s a y a a y a .Ž . Ž . Ž .1 24a y aŽ .2 1

Ž . Ž . Ž .4Note that a y a , p a y p a , and a y a are all nonnegative.2 1 1 2 2 1Ž . Ž .Since a y a ) 0 and a y a - 0 on a , a , it follows that p a is1 2 1 2

Ž .nonincreasing on a , a . B1 2

Ž .PROPOSITION 6. The belief function q b is nondecreasing.

Proof. The proof is similar to the proof of Proposition 5. B

2.4.6. Expected Surplus Maximization

When attempting to sell his kth unit, seller i with cost ck - oa mayiw . w k Ž k .xmake an offer a g 0, oa that results in expected surplus E p a, c ss, i i

Ž k . Ž .a y c ? p a . The maximum expected surplus of seller i for the sale ofithis unit 2 is

k k kS s max max E p a, c , 0 . 10Ž .� 4Ž .s , i ag Žob , o a. s , i i

l Ž .Similary, if buyer j with valuation ¨ for her lth unit bids b g ob, ` ,j

w l Ž l.x Ž l . Ž .this results in an expected surplus E p b, ¨ s ¨ y b ? q b . Theb, j j j

maximum expected surplus on this unit for buyer j is

l l lS s max max E p b , ¨ , 0 . 11Ž .Ž .½ 5b , j ag Žob , o a. b , j j

2.4.7. Timing of Messages

Let t be the parameter for time within a trading period. Let T be thew .length of the trading period and let t g 0, T be the time of the k thk

offer, bid, or acceptance of an offer or bid. At time t let T k be thek s, i

2 � 1 2 m i4Seller i with cost vector c s c , c , . . . , c , faces the problem of choosing a sequencei i i im i Ž k k . kof asks or accepts to maximize Ý p y c , where p is the purchase price received forks1 i i i

unit k. We assume that the seller will attempt to maximize the surplus of each unit insequence, independently other units. In addition to simplifying the strategy choice, this isconsistent with the myopic formulation of strategy choice. A similar remark applies to buyers.


random variable that specifies the time that seller i would allow to elapsebefore sending a message; let T k be the random variable that specifiesb, jthe time that buyer j would allow to elapse before sending a message.

We assume that T k is exponentially distributed, and that the parameters, ia in the distribution of T k depends only on the maximum expecteds, i s, i

k Ž Ž ..surplus S of seller i from Eq. 10 on the length T of the tradings, iperiod, and on t , the time elapsed in the trading period. We write thisk

Ž k .dependence as a s f S ; t , T . Similarly, for buyers, b ss, i s, i s, i k b, jŽ l . Xf S ; t , T . Then the probability that seller i will be the next trader tob, j b, j k

send a message in the market is

f X SkX ; t , TŽ .s , i s , i k

Xp ss , i k lÝ f S ; t , T q Ý f S ; t , TŽ . Ž .ig I s , i s , i k jg J b , j b , j k

a Xs , is . 12Ž .Ý a q Ý big I s , i jg J b , j

Ž . XEquation 12 indicates that the probability that seller i is the next to senda message is equal to the parameter a X of seller iX divided by the sum ofs, ithe parameters of all agents. This is shown in the following proposition.

PROPOSITION 7. If T k and T k are independent exponentially distributeds, i b, jw . Ž k .random ¨ariables on 0, ` with parameters a s f S ; t , T and b ss, i s, i s, i k b, j

Ž l .f S ; t , T , i.e,b, j b, j k

� 4 ya s , i? tPr T - t s 1 y e ,s , i

then the probability that seller iX will be the next trader to send a message isŽ .X Xp , where p is gi en by Eq. 12 .s, i s, i

Proof. Consider, for example, seller 1. Let

T s min T , T .� 4s , y1 s , i b , ji)1, jG1

This random variable is exponentially distributed with parameter

a s a q b .Ý Ýs , y1 s , i b , ji)1 jG1


Then the probability that seller 1 is the next seller to move will be theprobability that T - T , i.e.,s, 1 s, y1

� 4p s Pr T - Ts , 1 s , 1 s , y1

` uya ? t ya ?us , 1 s , y1s a ? e ? a ? e dt duH H s , 1 s , y1

0 0

a s , 1s . 13Ž .a q as , 1 s , y1

After substitution of the definitions of a and a , the expression ins, 1 s, y1Ž . Ž . XEq. 13 is the same as Eq. 12 for i s 1. B

There are two reasons for defining the timing of each trader’s messageas an exponential random variable. The first is an important conceptualissue: with this formulation, the mechanism is informationally decentral-ized, in that the information about each trader’s surplus is not held by any

Ž .agent. Each trader’s timing decision is independents of any unobservedcharacteristics }such as costs or valuations}of other traders. The secondissue is empirical. With this formulation, the timing of bids and asks istestable within the model, and it is possible to compare the timing data for

Ž k . Ž l .various specifications of the functions f S ; t , T and f S ; t , Ts, i s, i k b, j b, j kwith timing data from experiments. It should be noted that the arguments

Ž .of f ? may be any data observed by or known to seller i. In thes, iformulation above, the surplus Sk is a summary statistic derived from thes, iinformation privately held by and publicly observed by seller i.

Ž k . Ž l .The specifications of f S ; t , T and f S ; t , T used in thes, i s, i k b, j b, j k

simulations reported in Section 3 are

Tk kf S ; t , T s S ? 14Ž .Ž .s , i s , i k s , i T y atŽ .k

and

Tl lf S ; T , T s S ? , 15Ž .Ž .b , j b , j k b , j T y atŽ .k

Ž .where a g 0, 1 .This specification has been chosen to reflect two empirical observations

about timing of bids and offers in experimental markets. There is strongpositive rank-order correlation between buyers’ valuations and the order inwhich buyers purchase units, and strong negative rank-order correlationbetween sellers’ costs and the order in which units are sold. In theformulation above, buyers with high valuations will have higher maximum


expected surplus, and will therefore have a larger parameter b for theb, jtiming decision. Since the expected time until buyer j sends a message isproportional to the reciprocal of b , buyers with high valuations will tendb, jto send messages more frequently and will trade earlier. Similarly, sellerswith low costs will trade earlier. The second observation is that tradingactivity is typically concentrated at the beginning of the trading period,when many high surplus units are traded, and toward the end of the

Ž . Ž . Ž .period. The term Tr T y at in Eqs. 14 and 15 is consistent with thesek

observations, since high surplus units will trade earlier, but as t ª T , thisk

Ž .term approaches 1r 1 y a , and low surplus units will be traded towardthe close of each period if a is near 1.

Although the model is formulated so that timing data can be obtainedŽ .and examined, we have followed the reduced form in Eq. 12 in simula-

tions and generated messages without the time stamp.

Ž .EXAMPLE 3 BELIEFS, SURPLUS, AND TIMING . Consider again the mar-ket of Example 1. In Example 2 we discuss the first two messages sent in

�Ž .experiment 3pda01, which result in the history H s 3, 0, 3.00 ,2Ž .4 Ž . Ž .3, 1, 3.00 . The set of bids and offers D where the beliefs p a and q bˆ ˆ

� 4 � 4 Ž .are calculated is D s 3.00 j 0.00, 10.00 . The values of p a at theseˆŽ . Ž . Ž .three points are p 0.00 s 1.0, p 3.00 s 1.0, and p 10.00 s 0.0. Theˆ ˆ ˆ

Ž . Ž . Ž .values of q b at these three points are q 0.00 s 0.0, q 3.00 s 1.0, andˆ ˆ ˆŽ .q 10.00 s 1.0. Since there is no outstanding ask or bid, the spreadˆ

Ž . Ž . Ž . Ž .reduction rule has no effect, so p a s p a for all a g D and q b s q b˜ ˆ ˜ ˆŽ . Ž .for all b g D. Finally, the belief functions p a and q b , shown in Fig. 4,

are

1 0 F b F 3.00¡1

2 3~p a s 16Ž . Ž .100 q 180 x q 39 x q 2 x 3.00 F b F 1.00Ž .343¢0 10.00 F b ,

FIGURE 4


and

1¡ 2x 3 y 2 x 0.00 F b F 3.00Ž .~q b s 17Ž . Ž .27¢1 b ) 3.00.

After the trade between seller 3 and buyer 1 in this example, the lowestcost units of sellers 1]4 are 1.90, 1.40, 2.30, and 1.65, respectively. Thehighest unit values for buyers 1]4 are 2.25, 2.80, 2.60, and 3.05. Theexpected surplus maximizing bids and offers and the maximum expectedsurplus for each agent are easily determined from these values and costs

Ž . Ž .and the belief functions in Eqs. 16 and 17 . The values of the maximumexpected surplus for buyers 1]4 are 0.38, 0.66, 0.55, and 0.81. For sellers1]4, the values of maximum expected surplus are 2.55, 2.91, 2.27, and 2.73.

Ž .As a result, given the formulation of the timing of messages in Eqs. 14Ž .and 15 , in our model an offer will be more than 4 times as likely as a bid

with this history. Even if the next message is an offer, the expected surplusof sellers will continue to be greater than the expected surplus for buyers,and there is a high probability of a series of decreasing offers. As a result,the next transaction price is likely to move toward the market equilibriumof 2.35.

3. SIMULATIONS

While the formation of traders’ beliefs, their choices of strategies, andthe timing of messages are simple and intuitive, the dynamics of the modelare complex because of the nonstationarity of beliefs and the probabilitydistribution over the timing of agents’ messages. As a result, analyticcharacterization of properties of the model is difficult to obtain. For thisreason, much of the evidence presented on the performance of the modelis from simulations. It may be possible to obtain analytic results onasymptotic convergence of prices to an approximate equilibrium, butasymptotic convergence alone does not provide information about thepath. For the important question of the path of convergence to equilib-rium, simulations are useful tool for investigating properties of this model.

3.1. Criteria

The primary objective of this section is to demonstrate that prices andallocations in our model converge to the competitive equilibrium price andallocation. To identify the effect on convergence to competitive equilib-rium of the belief formation and strategy choice defined in Section 2.4, we


compare the outcomes of simulations of our model to the ‘‘zero-intelli-Ž . Ž .gence trade’’ ZI model of Gode et al. 1993 , which has no belief

formation or learning.3 By convergence of the sequence of prices, we meanŽ .Nthat for some n G 1, each element of the sequence p of transac-0 n nsn0

tion prices is ‘‘close to’’ p , the competitive equilibrium price. This condi-etion is met if the mean absolute deviation of transaction price fromequilibrium price is small, so we measure convergence using this statistic.Of course, convergence to competitive equilibrium implies convergence toa Pareto optimum, so we also test this weaker condition. With efficiencymeasured as the ratio of surplus extracted by agents to total surpluspossible, we find that in simulations of the model, market allocations are

Žnearly efficient over 99.9% of possible surplus is extracted after several.periods of trading , and prices are close to competitive equilibrium prices.

Note that while we do not examine the market allocation directly, theoutcome is a competitive equilibrium if and only if the transaction pricesare the competitive equilibrium price and the allocation is Pareto optimal,

Ž .and we do establish that these two conditions hold approximately insimulations of our model.

3.2. En¨ironments

Throughout this section, we consider a market design with four sellers,each with finite unit costs for three units, and four buyers, each withpositive unit valuations for three units. The unit costs and valuations for

Ž .the design we consider are given in Example 1 of Section 2.1 and areŽ .depicted in Fig. 2 and in the left column of Fig. 5 . We reported statistics

from seven laboratory market experiments reported initially by KetchamŽ .et al. 1984 and statistics from 100 simulations of our model with the same

environment parameters. Finally, to demonstrate that our model is capableof responding to shifts in market conditions, we show the results of asimulation in which the market design described above is employed throughfive periods of trading, and then for the remaining five periods of thesimulation each unit cost and unit valuation are increased to amounts 0.50above those employed in periods 1]5.

3.3. E¨aluation

The only free parameter in the model is the memory length of traders.We report simulations with memory length L s 5. For short memory

3 In this model, sellers make offers which are random and uniformly distributed on thew xinterval c, M , where c is the seller’s cost, and M is some upper bound on their set of

w xpossible choices. Buyers make bids that are uniformly distributed on 0, ¨ where ¨ is thebuyer’s valuation.


FIGURE 5

Ž .lengths L F 3 the outcomes are unstable. For long memory lengthsŽ .L G 8 the outcomes are similar to those with intermediate memory

Ž .lengths 4 F L F 7 , but computation time increases significantly. It shouldalso be noted that beliefs change more slowly in markets with shifts insupply and demand if memory length is long, so traders with intermediateand short memory lengths will adapt to changes in market conditions morequickly.


TABLE IEfficiency Statistics from Simulations of Models and from Laboratory Data

Periods Symmetric Model ZI Modelevaluated markets simulations simulations

First two periods 0.907 0.9982 0.968Entire experiment 0.959 0.9991 0.968Last two periods 0.970 0.9992 0.967

Efficiency

The sum of consumers’ and producers’ surpluses provides a convenientmeasure of efficiency for these markets. We evaluate the surplus obtainedby traders in the market divided by the maximum surplus available todetermine the efficiency of trade. Table I summarizes efficiency statisticsfrom the seven laboratory markets, from 100 simulations of our model, andfrom 100 simulations of the ZI model. This table show that our modelattains higher efficiency than both the laboratory experiments and the ZImodel simulations.

Con¨ergence

The diagrams in the left column of Fig. 5 show graphs of the supply anddemand conditions for the symmetric market 3pda01. On the right side ofthat figure in the top row is a graph of the sequence of transaction pricethrough the nine periods of trading in this laboratory market. In thatfigure, the equilibrium price is shown as a solid line across the diagram.Prices from each of the nine trading periods are separated by a verticalline, and the number of transactions in each period is indicated at thebottom of the diagram below the vertical line that indicates the end of thetrading period. A simulation of our model under the same supply anddemand conditions is shown in the right side of Fig. 5 in the center row.

ŽPrice sequences from laboratory experiments with this design as in the top. Žof Fig. 5 and from simulations with this market design as in the center

.row of Fig. 5 both converge quickly to prices near the competitiveequilibrium price, and an equilibrium quantity of trade typically occurs ineach period.

Ž . Ž .The belief functions p a and q b shown in Fig. 6 are produced usingdata from the the end of the second period of the simulation of Fig. 5,using Definitions 10 and 11 of Section 2.4.2. In this graph, a seller’s beliefthat ask a will be accepted by a buyer is shown for each ask from 2.32 to


FIGURE 6

2.38; buyers’ beliefs are shown for bids from 2.30 to 2.36.4 These beliefŽ .functions are monotonic see Propositions 5 and 6 , so the value of the

Ž .sellers’ belief is p a s 1 for all a - 2.33 and it is 0 for all a ) 2.38. Withthis belief function, and with myopic surplus maximization, the optimal askis approximately 2.33 for any seller with unit cost below 2.30. The buyers’belief functions in this case have a similar property: the optimal bid for abuyer with valuation of 2.40 or greater is 2.35. At the beginning of eachperiod, the sellers’ costs are 1.90, 1.40, 2.10, and 1.65 and the buyers’valuations are 3.30, 2.80, 2.60, and 3.05, so the first action at the beginningof the third period will be either an ask of 2.33 or a bid at 2.35. We see inthe center row of Fig. 5 that the first transaction price is 2.35 in period 3 ofthis simulation: the belief functions and strategy choice described fre-quently produce transactions at equilibrium, even from the beginning ofthe trading period.

The figures in the bottom row of Fig. 5 show a simulation of the ZImodel in the symmetric market environment. While the ZI traders attainhigh efficiency in this market design, that model does not result in theformation of equilibrium prices. In the ZI model, there is no beliefformation process. As a result, there is no convergence of transactionprices to equilibrium, as the diagram on the right side of the bottom row inFig. 5 clearly shows.5 In Table II, the mean absolute deviation of transac-

4 Note that the range from the lowest cost to the highest valuation in this market is 1.40 to3.30, with an equilibrium price of 2.35; beliefs are focused in a narrow range around theequilibrium price.

5 w xGode and Sunder 1993, p. 129 argue: ‘‘By the end of a period, the price series in budgetconstrained ZI trader markets converges to the equilibrium level almost as precisely as theprice series from human trader markets does.’’ In the ZI simulation of figure 5, final trades in8 of 10 periods are within 0.10 of equilibrium. We apply a definition of convergence that ismore demanding and conforms more closely to the intuitive notion of convergence of marketprices. We argue that prices in a stable market environment converge if, after several periods,the mean deviation of all trades from equilibrium is small. By this criterion, the ZI modeldoes not converge to equilibrium.


TABLE IIConvergence Statistics from Simulations of Models and from Laboratory Data

Periods Symmetric Model ZI Modelevaluated markets simulations simulations

First two periods 0.101 0.077 0.276Entire experiment 0.050 0.045 0.237Last two periods 0.088 0.040 0.209

tion price from equilibrium price is shown for 100 simulations of the ZImodel in the symmetric market design.6 This statistic is also shown for 100simulations of our model7 for the seven laboratory markets. These dataŽ .and the ZI simulation graph at the bottom of Fig. 5 show that the ZImodel does not result in convergence to competitive equilibrium. Thebehavior in our model does produce convergence to approximate equilib-rium prices after several periods of trading. The contrast between theoutcome of the simulations of the ZI model and that of our model identifythe effect on market convergence of belief formation and myopic surplusmaximization in our model. From the graphs and the statistics, it is clearthat this effect is substantial. Moreover, data on mean absolute deviationin Table II show not only that our model converges to within a few cents ofthe equilibrium price, but that the rate of convergence is similar to}al-though initially slightly faster than}that found in laboratory experiments.

Shifting Conditions

The diagram on the left of Fig. 7 shows two sets of supply and demandconditions. The lower set, shown with thicker lines labeled S and D, isidentical to the supply and demand conditions in Fig. 2 and Fig. 5. If afterseveral periods of trading, buyers have each valuation increased by 0.50and sellers have the cost of each unit increased by 0.50, then the newsupply and demand are those shown with thinner lines in Fig. 7. Theequilibrium quantity of trade and the total surplus are unaffected by thisshift, but the equilibrium price increases from 2.35 to 2.85. Since expecta-tions focus near the original equilibrium after several periods of tradingŽ .see Fig. 6 , the dynamics of movement to the new equilibrium can beexamined by considering this type of market. The sequence of transaction

6 Although the mean absolute deviation in the last two periods of the ZI model simulationsis less than in the first two periods, this is not the result of convergence. The price sequencein each period constitutes a draw from the same distribution.

7 Ž . Ž .The timing specification employed in the simulations is given in equations 14 and 15 .In one alternative tested, all agents with positive surplus were equally likely to send amessage. This resulted in higher variance in transaction prices and in more instability in theoutcomes.


FIGURE 7

prices from a simulation of our model in this type of market}with theshift occurring after five periods of trading}is shown on the right side ofFig. 7. From periods 6 through 10 in this market, the equilibrium price is2.85. In the simulation shown, convergence to the original equilibriumoccurs by the end of period 2. Beginning in period 6, the equilibrium priceshifts up 0.50. By the end of the period 7, transaction prices establishesnear the new equilibrium price. This simulation shows that in the modeldeveloped here, traders respond to shifting market parameters, and pricesquickly adjust to a new equilibrium.

3.4. Boundaries on Performance

Although we have developed a model that simultaneously converges tocompetitive equilibrium prices, produces efficient outcomes, and respondsquickly to altered market conditions, we want to indicate a direction forimprovement in the outcomes of the bargaining behavior developed in ourmodel. We do so by describing a distinctive feature of price sequencesfrom experimental markets.

Ž .NWe split the sequence p into two subsequences, one consisting ofn ns1Ž .those exchange prices p that were initially proposed by the sellingn ng Ns

side of the market, and the other consisting of the transaction pricesŽ .p that were proposed initially by the buying side. We providen ng Nb

Ž .evidence that in laboratory experiments, the mean of p is greatern ng Ns

Ž .than the mean of p . Yet in simulation of our model, this pattern isn ng Nb

reversed. This observation allows us to evaluate behavior in the model anduncover a key difference between the behavior of laboratory subjects andbehavior in the model, providing direction for further research on bargain-ing behavior in DA markets.

In laboratory trading experiments, there is a clear difference betweenmean prices of trades initiated by sellers and those initiated by buyers. For


example, in the seven symmetric markets we consider, Table III shows themarket equilibrium price for each of these experiments in column 2 andthe mean difference of transaction price from the equilibrium in column 3.Column 4 shows the mean difference between the transaction price andequilibrium price for all trades initially proposed by sellers. Column 5shows the same statistics for all trades initially proposed by buyers. Notethat in each of these seven experiments, the transactions initiated bysellers have a higher mean price than those initiated by buyers. Twoadditional data sets}one involving over 11, 000 trades}are examined by

Ž .Gjerstad 1995 and this price difference is a feature of all markets in bothdata sets considered there. In simulations of our model we find that thisfeature is reversed, and as a result, we are able to discern a differentbetween the bargaining behavior of laboratory subjects and behavior in themodel.

Ž . Ž .Consider again the belief functions p a and q b shown in Fig. 6. Inthe description of the beliefs in these graphs in Section 3.3, we note that atthe beginning of a period, sellers in this market with these beliefs will allhave an optimal offer of 2.33 and buyers’ optimal bids will all be 2.35. Atthe beginning of the third period, the first action will be either an ask of2.33 or a bid at 2.35. Suppose that the first action is a bid of 2.35. As aresult of the spread reduction rule, buyers’ bids must be greater than 2.35.

Ž 2 .A bid of 2.36 would result in an expected surplus ¨ y 2.36 ? 1 for eachjof the four buyers in this market. Since the distribution of costs andvaluations at the beginning of each period in this market is symmetric, andsince the probability of each trader being the next to send a message is

Ž .equal to that trader’s proportion of total surplus see Proposition 7 , theprobability that a buyer will send the next message is approximately 0.50,so that the probability of two consecutive bids is approximately 0.25, and inthis case the price will be above the equilibrium price. In general, the

TABLE IIIPrice Sequence and Subsequence Means in Symmetric Markets

Market p p y p p y p p y pe e a e b e

3pda01 $2.35 $0.04 $0.12 $0.012pda17 $6.20 $0.03 $0.07 $0.002pda20 $6.20 $0.06 $0.12 $0.012pda21 $5.30 y$0.03 y$0.01 y$0.042pda24 $4.70 $0.02 $0.04 y$0.022pda47 $6.35 y$0.03 y$0.01 y$0.052pda53 $7.55 $0.00 $0.01 y$0.02Mean $0.01 $0.05 y$0.02


distribution of absolute deviations from equilibrium is approximately geo-metric, since a low price results from a sequence of asks and a high priceresults from a sequence of bids. Recall that in Table III, the transactionsinitiated by the selling side typically are above equilibrium, so the simula-tions and the laboratory markets lead to the opposite result in this respect.

While this is a subtle feature of DA data, a model that eliminates thisdifference would capture fine aspects of bargaining behavior and wouldrepresent progress in modeling behavior in this institution.

4. CONCLUSIONS

In this paper we have defined beliefs for agents in a double auctionmarket that are generated endogenously on the basis of observed marketactivity. An agent’s choice of an action depends only on these beliefs andon that agent’s private information about his or her own costs or valua-tions. Agents who adopt the simple bargaining strategy of myopic expectedsurplus maximization and employ these beliefs trade at prices that con-verge quickly and accurately to within several cents of the market equilib-rium price and reach the competitive allocation. These beliefs and strate-gies are flexible enough to respond quickly to changes in supply anddemand conditions.

Laboratory market experiments dating back 35 years have demonstratedthat human subjects quickly and reliably reach competitive equilibriumoutcomes. The model developed in this paper demonstrates that thiscapability of double auction market participants to reach competitiveequilibrium outcomes may result from simple, intuitive information pro-cessing and strategy choice. Since we know that laboratory subjects oper-ate with limited information-processing capabilities and boundedly rationalstrategy choices, this finding resolves, at least for the class of environmentsconsidered here, the puzzle Hayek posed.

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