Convergence of a Dynamic Matching andBargaining Market with Two-sided Incomplete

Information to Perfect Competition

Mark Satterthwaite and Artyom Shneyerov∗†

October 5, 2003

Preliminary and incomplete. Please do not quotewithoutpermission.

Abstract

Consider a decentralized, dynamic market with an infinite horizon inwhich both buyers and sellers have private information concerning theirvalues for the indivisible traded good. Time is discrete, each period haslength δ, and each unit of time a large number of new buyers and sell-ers enter the market to trade. Within a period each buyer is matchedwith a seller and each seller is matched with zero, one, or more buyers.Every seller runs a first price auction with a reservation price and, if tradeoccurs, both the seller and winning buyer exit the market with their real-ized utility. Traders who fail to trade either continue in the market to berematched or become discouraged with probability βδ (β is the discour-agement rate) and exit with zero utility. We characterize the steady-state,perfect Bayesian equilibria as δ becomes small and the market–in effect–becomes large. We show that, as δ converges to zero, equilibrium pricesat which trades occur converge to the Walrasian price and the realizedallocations converge to the competitive allocation.

∗Kellogg School of Management, Northwestern University and Department of Economics,University of British Columbia respectively.

†We owe special thanks to Zvika Neeman who originaly devised a proof showing the strictmonotonicity of strategies. We also thank Paul Milgrom, Dale Mortensen, Asher Wolinsky,Jianjun Wu; the participants at the “Electronic Market Design Meeting” (June 2002, SchlossDagstuhl), the 13th Annual International Conference on Game Theory at Stony Brook, the2003 NSF Decentralization Conference held at Purdue University, the 2003 General Equilib-rium Conference held at Washington University, and the 2003 Summer Econometric SocietyMeeting held at Northwestern University; seminar participants at Carnegie-Mellon, Washing-ton University, and Northwestern University; and the members of the collaborative researchgroup on “Foundations of Electronic Marketplaces” for their constructive comments. Finally,both of us acknowledge gratefully that this material is based on work supported by the Na-tional Science Foundation under Grant IIS-0121541.

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1 Introduction

Asymmetric information and strategic behavior interferes with efficient trade.Nevertheless economists have long believed that for private goods’ economiesthe presence of many traders overcomes both these imperfections and resultsin convergence to perfect competition. This paper contributes to a burgeon-ing literature that shows the robust ability of simple market mechanisms toelicit almost accurate cost and value information from buyers and sellers evenas at it allocates the available supply almost efficiently. It shows how a com-pletely decentralized market with two-sided incomplete information convergesto a competitive outcome as each trader’s ability to contact other traders seriallyincreases. Thus a market that for each trader is big over time–as opposed tobig at a moment in time–overcomes the difficulties of asymmetric informationand strategic behavior. This is another step towards a full understanding ofwhy price theory with its assumptions of complete information and price-takingworks as well as it does even in markets where the validity of neither of theseassumptions is self-evident.These ideas may be made concrete by considering a bilateral bargaining

situation in which the single buyer has a value vi ∈ [0, 1] for an indivisiblegood and the single seller has a cost cj ∈ [0, 1]. They should trade only if vi ≥cj , but neither knows the other’s value/cost. Instead each regards the other’svalue/cost as drawn from [0, 1] in accordance with a distribution G (·) . Myersonand Satterthwaite (1983) showed that no individually rational, budget balancedmechanism exists that both respects the incentive constraints the asymmetricinformation imposes and prescribes trade only if vi ≥ cj . Bilateral trade withtwo-sided incomplete information is intrinsically inefficient.An instructive example of this phenomenon is the linear equilibrium Chat-

terjee and Samuelson (1983) derived for the bilateral 12 -double auction when Gis the uniform distribution on [0, 1]. The rules of this double auction are thatbuyer and seller simultaneously announce a bid B (vi) and offer S (cj) and theytrade at price p = 1

2 (B (vi) + S (cj)) only if the buyer’s bid is greater than theseller’s offer. In their linear equilibrium trade occurs only if vi−cj ≥ 1

4 , i.e., theasymmetric information and resulting misrepresentation of value/cost inserts aninefficient “wedge” of thickness 14 into the double auction’s outcome. Moreoverthe magnitude of this wedge is irreducible. Myerson and Satterthwaite (1983)showed that subject to budget balance, individual rationality, and incentive con-straints this equilibrium this equilibrium maximizes the ex ante expected gainsfrom trade and therefore is ex ante efficient:.A sequence of papers on the static, multi-lateral k-double auction in the in-

dependent private values environment have confirmed economists’ intuition thatincreasing the number of traders causes this wedge to shrink and ultimately van-ish in the limit. In the multilateral double auction there are n sellers each sup-plying one unit and n buyers each demanding one unit.. Each trader’s cost/valueis private and, from the viewpoint of every other trader, independently drawnfrom [0,1] with distribution G. Sellers and buyers submit offers/bids simulta-neously, a market clearing price p is computed, and the n units of supply are

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allocated at price p to those n traders who revealed through their offers/bidsthat they most value the available supply. Satterthwaite and Williams (1989)and Rustichini, Satterthwaite, and Williams (1994) established that as n in-creases the thickness of the wedge and the relative inefficiency associated witheach equilibrium are O (1/n) and O

¡1/n2

¢respectively. Relative inefficiency is

the expected gains that the traders would realize if the market were perfectlycompetitive divided into the expected gains that the traders fail to realize inthe equilibrium of the double auction market.Thus, quite quickly, the static double auction market with independent pri-

vate values converges to ex post efficiency–that is, perfect competition–as thenumber of traders grows.1 This is despite dispensing with the technically im-portant, but often unrealistic assumption of auction theory that the seller’s costis common knowledge among all participants. These results acknowledge thatthe effectiveness of a trading institution depends as much on its ability to elicitaccurate information from both sides of the market as it does on its ability toallocate goods efficiently in accordance with that information.All these results, however, are derived under two restrictive assumptions:

costs/values are independently drawn private signals and the timing of themarket is a one-shot static game. Papers by Fudenberg, Mobius, and Szeidl(2003), Cripps and Swinkels (2003), and Reny and Perry (2003) relax the for-mer assumption. Specifically, Fudenberg, Mobius, and Szeidl show that for largemarkets in an environment with correlated private costs/values an equilibriumto the static double auction exists and traders misrepresentation of their truevalues is O

¡1n

¢. Cripps and Swinkels, using a somewhat more general model of

correlated private values establish existence and that the relative inefficiency isO¡

1n2−ε

¢where ε is arbitrarily small. Reny and Perry consider the static double

auction market when traders cost/values have a common value component andtheir private signals are affiliated. They show, if the market is large enough, thatan equilibrium exists, is almost ex post efficient, and almost fully aggregates thetraders’ private information, i.e., the double auction equilibrium is almost thelimit market’s unique, fully revealing rational expectations equilibrium.This paper relaxes the latter restrictive assumption that the traders’ are

playing a one-shot game in which, if they fail to trade now, they never have alater opportunity to trade. We accomplish this with a dynamic matching andbargaining model in which trades are consummated in a decentralized matterand traders who do not trade in the current period are rematched in the nextperiod and try again. Gale (1987) and Mortensen and Wright (2002) studymodels of this type and show that, as the friction that impedes traders’ abil-ities to search for a favorable price goes to zero, the outcome approaches thecompetitive outcome. Their models, however, assume complete information andtherefore can not test if such markets can succeed in eliciting traders’ privateinformation even as it allocates supply.A reasonably complete, though somewhat informal description of our model

1 Indeed Satterthwaite and Williams (2001) show that for this environment converges asfast as possible in the sense of worst case asymptotic optimality.

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and result is this. An indivisible good is traded in a market in which timeprogresses in discrete periods of length δ and generations of traders overlap. Theparameter δ is the exogenous friction in our model that we take to zero. Everyactive buyer is randomly matched with an active seller each period. Dependingon the luck of the draw, a seller may end up being matched with several buyers,a single buyer, or even no buyers. Each seller solicits a bid from each buyer withwhom she is matched and, if the highest of the bids is satisfactory to her, shesells her single unit of the good, and both she and the successful buyer exit themarket. A buyer or seller who fails to trade remains in the market, is rematchedthe next period, and tries again to trade unless he should become discouragedand decide spontaneously to exit the market without trading.Each unit of time a large number of potential sellers enters the market along

with a large number of potential buyers. Each potential seller independentlydraws a cost c in the unit interval from a distribution GS and each potentialbuyer draws independently a value v in the unit interval from a distribution GB.Individuals’ costs and values are private to them. A potential trader only entersthe market if, conditional on his private cost or value, his equilibrium expectedutility is positive. Potential traders who have zero probability of profitable tradein equilibrium elect not to participate.If trade occurs between a buyer and seller at price p, then they exit with

utilities v − p and p − c respectively. As in McAfee (1993) unsuccessful activetraders may become discouraged and exit. This occurs for each trader eachperiod with probability δβ > 0 where β is the discouragement rate per unit oftime. If δ is large (i.e., periods are long), then a trader who enters the market isimpatient, seeking to consummate a trade and realize positive utility amongstthe first few matches he realizes. Otherwise he is likely to become discouragedand exit with zero utility. If, however, δ is small (i.e., periods are short), thena trader can wait through many matches looking for a good price with littleconcern about first becoming discouraged and exiting with no gain.Buyers with higher values find it worthwhile to submit higher bids than

buyers with lower values. At the extreme, a buyer with a value 0.1 will certainlynot submit a bid greater than 0.1 while a buyer with a value 0.95 certainly might.The same logic applies to sellers: low cost sellers may be willing to accept lowerbids than are higher cost buyers. This means high value buyers and low costsellers tend quickly to realize a match that results in trade and exit. Low valuebuyers and high cost sellers may take a much longer time on average to tradeand are likely to exit through discouragement rather than trade. Consequently,among the buyers and sellers who are active in the market in a given period t,low value buyers and high cost sellers may be overrepresented relative to theentering distributions GB and GS.We characterize subgame perfect Bayesian equilibria for the steady state of

this market and show that, as the period length goes to zero, all equilibria ofthe market converge to the Walrasian price and the competitive allocation. TheWalrasian price pW in this market is the solution to the equation

GS (pW ) = a (1−GB (pW )) , (1)

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i.e., it is the price at which the measure of entering sellers with costs less than pWequals the measure of entering buyers with values greater than pW . If the marketwere completely centralized with every active buyer and seller participating inan enormous exchange that cleared each period’s bids and offers simultaneously,then pW would be the market clearing price each period. Our result is this.Given a δ > 0, then each equilibria induces a trading range

hpδ, pδi. It is the

range of offers that sellers of different types make, the range of bids that buyersmake, and the range of prices at which trades are actually consummated inthis equilibrium. We show that limδ→0 pδ = pW and limδ→0 pδ = pW , i.e., thetrading range converges to the competitive price. That the resulting allocationsgive traders the expected utility they would realize in a perfectly competitivemarket follows as a corollary.This result, both intuitively and in its proof, is driven by two phenomena:

local market size and global market clearing.2 By local market size we meanthe number of other traders with whom each individual trader interacts. Thiscontrasts with global market size–the total number of traders active in theentire market–which is always large in our model. Thus as the time periodδ shrinks each trader expects to match an increasing number of times beforebecoming discouraged and exiting. Each trader’s local market becomes big overtime as opposed to big at a point in time as is the case in the standard model ofperfect competition or in the centralized k-double auction. This creates a strongoption value effect for every trader. Even if a buyer has a high value, he hasan increasing incentive as δ decreases to bid low and hold out for an offer nearthe low end of the offer distribution. Therefore all serious buyers bid withinan increasingly narrow range just above the minimum offer any seller makes.A parallel argument applies to sellers, with the net effect being, as δ becomessmall, that all bids and offers concentrate within an interval of decreasing length,i.e., the trading range converges to a single price.Local market size only forces the market to converge to a single price, not

necessarily to the Walrasian price. It is global market clearing that forces con-vergence to the Walrasian price. To see this, suppose the market converges toa price p that is less than the Walrasian price. At this price more buyers wantto buy than there are sellers who want to sell. Buyers are rationed through dis-couragement, for even a high value buyer may fail to be matched with a sellerwho wants to sell at p before he becomes discouraged and exits. This, however,is inconsistent with equilibrium: the high value seller can increase his bid abovep and guarantee that he will trade and not be rationed out. This increases hisexpected utility and contradicts the hypothesis that the equilibrium convergesto the price p rather than the Walrasian price.A substantial literature exists that investigates the non-cooperative founda-

tions of perfect competition using dynamic matching and bargaining games.3

2De Fraja and Sákovics (2001) introduced these concepts.3There is a related literature that we do not discuss here concerning is the micro-structure

of intermediaries in markets, e.g., Spulber (1999) and Rust and Hall (2002). These modelsallow entry of an intermediary who posts fixed ask and offer prices and is assumed to be

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Most of the work of which we are aware has assumed complete informationin the sense that each participant knows every other participant’s values (orcosts) for the traded good. The books of Osborne and Rubinstein (1990) andGale (2000) contain excellent discussions of both their own and others’ contri-butions to this literature. Papers that have been particularly influential includeMortensen (1982), Rubinstein and Wolinsky (1985, 1990), Gale (1986, 1987)and Mortensen and Wright (2002). Of these, our paper is most closely relatedto Gale (1987) and Mortensen and Wright (2002). The two main differences be-tween their work and our’s is that (i) when two traders meet they reciprocallyobserve the other’s cost/value and (ii) each trader pays a small participationfee.4 The first difference–full vs. incomplete information–is fundamental, forthe purpose of our paper is to determine if a decentralized market can elicitprivate valuation information at the same time it uses that information to as-sign the available supply efficiently.5 The second difference prevents traders whohave low or zero probabilities of successfully trading from entering. In our modelthe assumption that potential traders who have zero probability of trading electnot to become active serves much the same purpose.Butters (circa 1979), Wolinsky (1988), and De Fraja and Sákovics (2001) are

the most important dynamic bargaining and matching models that incorporateincomplete information, albeit one-sided in the cases of Wolinsky and of DeFraja and Sákovics . In an incomplete manuscript Butters analyzes almost theidentical two-sided incomplete information model that we analyze and makes agreat deal of progress towards proving a variant of the theorem that we prove inthis paper.6 Wolinsky considers the steady state of a market in which each sellersets a reservation price on his single unit of an indivisible differentiated good andthe randomly matched buyers bid for his unit supply. The cost of each seller’sgood is zero. Each buyer each period independently draws his idiosyncraticvalue of the good for which he is bidding from a distribution G. Traders timediscount their expected gains so they are impatient. Wolinsky shows that if thediscount rate approaches zero, which implies that buyers can almost costlesslysearch a very large number of sellers, then the equilibrium price in the marketconverges to zero even if the ratio of buyers to sellers in the market is much largerthan one. The reason for this non-Walrasian result is the differentiated productassumption that is central to his model. For small discount rates each buyerpatiently waits until he is matched with a seller whose good is almost a perfect

large enough to honor any size buy or sell order without exhausting its inventory or financialresources.

4Mortenson and Wright assume a small per period participation fee.5Gale (1987, p. 31) comments that within a limit theorem such as the one he proves

there and the one we prove here the complete information assumpiton is “restrictive.” This,somewhat surprisingly is not true within “theorems in the limit.” Then, he observes, “there isno scope for inferring an agent’s type from his willingness to delay. The only effect of assumingincomplete information is to force an agent to treat all other agents symmetrically. Forexample, in [Gale (1985)] it is strictly easier to show the bargaining equilibrium is Walrasianunder incomplete information than complete information.”

6We thank Asher Wolinsky for bringing Butters’ manuscript to our attention in April 2003after we had completed an earlier version of this paper.

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match for him, i.e., he waits until he realizes a value from the extreme rightof G’s tail. When he does obtain such a draw he, somewhat paradoxically, canbid close to zero even if he is bidding against several other buyers. The reasonis that the other buyers almost certainly are not well matched with the seller’sgood; consequently they bid essentially zero and wait for a better match. Colesand Muthoo (1998) have extended Wolinsky’s model and results to a market inwhich buyers do not randomly search, but rather preferentially seek out newlyentering traders.Wolinsky’s result illustrates the importance of competition within the local

market in insuring competitive outcomes. As the discount rate goes to zero themarket becomes large in the sense that each seller and each buyer can matchmany times at low cost, but it remains small with each seller only infrequentlyencountering even a single serious bidder. The monopsony equilibrium price thatresults is a variant of Diamond’s classic observation (1971) that even arbitrarilysmall search costs can give one side of the market complete market power.The major difference between the models of Wolinsky and De Fraja and

Sákovics (2001) is that the good being traded is homogeneous, not differentiated.Therefore in the latter model (as in our model) each buyer’s value remainsconstant when he fails to trade and is rematched in the next period to a newseller. A degree of competition in the local market is guaranteed by assumingthat each seller each period has positive probability that two buyers will bematched with her and have to compete with each other. The matching dynamicsare that if a buyer of value v succeeds in trading, then both the buyer andseller–who always has cost zero–exit and are immediately replaced with abuyer of the same cost v and a seller. This means that the distribution of buyers’types in the market is always the exogenously given distribution G. Given themanner in which they define the Walrasian price, their conclusion is that theprice distribution only converges to the Walrasian price as the market frictionvanishes if the parameters of matching process are chosen appropriately.7

The next section formally states the model and our main result. Section3 derives basic properties of equilibria and section 4 contains an example thatillustrates our result. Section 5 proves our result and section 6 concludes.

2 Model and theorem

We study the steady state of a market with two-sided incomplete informationand an infinite horizon. In it heterogeneous buyers and sellers meet once perperiod (t = . . . ,−1, 0, 1, . . .) and trade an indivisible, homogeneous good. Everyseller is endowed with one unit of the traded good that she is willing to trade

7More specifically, because each pair of traders who consummates a trade is replaced byan identical pair, if there is a single price in the limit, then this price, by construction, clearsthe market of entering traders. Thus, if Gale’s (1987) concept of flow equilibrium is adopted(and this is the concept we use), then the market by construction converges to the Walrasianprice. If, on the other hand, one follows De Fraja and Sákovics in defining the Walrasian pricein terms of the steady state “stock” of traders in the market rather than the “flow” of tradersthrough it, then they show that the limiting price may or may not be Walrasian.

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if the price she can obtain is at least her cost ci ∈ [0, 1]. This cost is privateinformation to her; to other traders it is an independent random variable withdistribution GS and density gS. Similarly, every buyer seeks to purchase oneunit of the good if the price he can obtain is at most his value vj ∈ [0, 1]. Thisvalue is private; to others it is an independent random variable with distributionGB and density gB. Our model is therefore the standard independent privatevalues model. We assume that the two densities are bounded away from zero:a g > 0 exists such that, for all c, v ∈ [0, 1], gS(c) > g and gB(v) > g.The length of each period is δ. Each unit of time a large number of potential

sellers and a large number of potential buyers consider entering the market; for-mally each unit of time measure 1 of potential sellers and measure a of potentialbuyers consider entry where a > 0. This means that each period measure δ ofpotential sellers and measure aδ of potential buyers consider entry. Only thosepotential traders whose expected utility from entry is positive actually elect toenter and become active traders.. Active buyers and sellers who did not leavethe market the previous period carry over. Let ζ be the endogenous steady stateratio of active buyers to active sellers in the market. A period consists of threesteps:

1. Every buyer is matched with one seller. His match is equally likely to bewith any seller and independent of the matches other buyers realize. Sincethere are a continuum of buyers and sellers the matching probabilities arePoisson: the probability that a seller is matched with k = 0, 1, 2, . . .buyersis8

ξk =ζk

k! eζ. (2)

Consequently a seller may end up being matched with zero buyers, onebuyer, two buyers, etc. Each seller, at the time she decides which if anybids she accepts, knows how many buyers with whom she is matched.Each buyer, at the time he submits his bid, only knows the endogenoussteady-state probability distribution of how many buyers with whom heis competing.

2. The matched traders bargain in accordance with the rules of the buyers’bid double auction: each buyer simultaneously announces a take-it-or-leave-it offer B (v) to the seller with whom they are bargaining. The selleraccepts the highest offer she receives provided it is at least as large as herreservation value S (c) .9 If two or more buyers tie with the highest bid,then the seller uses a fair lottery to choose between them. If a type vbuyer trades in period t, then he leaves the market with utility v −B(v).If a type c seller trades at price p,then she leaves the market with utility

8 In a market with M sellers and ζM buyers, the probability that a seller is matched withk buyers is ξMk =

¡ζMk

¢ ¡1M

¢k ¡1− 1

M

¢ζM−k. Poisson’s theorem (see, for example, Shiryaev,

1995) shows that limM→∞ ξMk = ξk.9 See Satterthwaite and Williams (1989) and Williams (1991) for a full analysis of the

properties of the buyers-bid double auction.

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p − c where p is the bid she accepts. Each seller, thus, runs an optimalauction; moreover their commitment to this auction is fully credible sincethe reservation value each sets stems from their dynamic optimization.10

3. Every active buyer and active seller who fails to trade decides for exoge-nous reasons if he will remain in the market the next period. Let β > 0 bethe discouragement parameter. Each period with probability δβ > 0 eachactive trader’s situation may change sufficiently that he becomes discour-aged and decides to exit the market to pursue other opportunities.

Traders do not discount their expected utility; it is each trader’s likelihood ofbecoming discouraged and exiting that induces impatience. Discounting couldalso be incorporated in the model as in McAfee (1993), though it appears doingso will not have significant effect on the analysis because, as the period lengthδ becomes short, all trade in equilibrium occurs quite quickly.Step 3’s assumption that a trader every period has probability δβ of becom-

ing discouraged, turning to another sort of opportunity, and spontaneously ex-iting contrasts with the alternative assumption that Gale (1987) and Mortensen(2002) employ in their full information models. They assume that each traderboth discounts utility and incurs a small participation cost for being activewithin the market. This causes any potential trader who has a low or zeroprobability of trading to refuse entry while traders who have non-negligibleprobabilities of trade only exit through trade, no matter how many periods ittakes. We do not adopt this approach for two reasons. First, casual empiricismsuggests that discouragement–or distraction by alternative opportunities–isa real phenomenon. In any case, it is a venerable assumption to make withintheories of bargaining. See, for example, Binmore, Rubinstein, and Wolinsky’s(1986) careful discussion of the distinction between impatience stemming froman exogenous probability of bargaining breakdown and impatience stemmingfrom time preference. Also, see the model of McAfee (1993) that incorporatesan exogenous probability of exit into a similar model that has one-sided incom-plete information. Second, equilibria of double auctions with participation costsare not yet well understood, so tractability argues for incorporating discourage-ment rather than participation costs into the model.11

As explained in the introduction, a seller who has low cost tends to tradewithin a short number of periods of her entry because most buyers with whomshe might be matched have a value higher than her cost and therefore agree totrade. A high cost seller, on the other hand, tends not to trade as quickly or notat all. As a consequence, in the steady state, among the population of sellerswho are active high cost sellers are relatively common and low cost seller arerelatively uncommon. Exactly parallel logic implies that, in the steady state,low value buyers are relatively common and high value buyers are relatively

10We do not know if these auctions are the equilibrium mechanism that would result if wetried to replicate McAfee’s analysis (1993) within our model.

11Jianjun Wu, a Ph.D. student at Northwestern University, is exploring the properties ofstatic double auctions as part of his dissertation.

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uncommon. Moreover, this tendency of traders to wait several periods beforetrading or exiting implies that the total measure of traders active within themarket may be larger–perhaps much larger–than the total measure (1 + a)δof potential traders who enter each period.To formalize the fact that the distribution of trader types within the market’s

steady state is endogenous, let TS be the measure of active sellers in the marketat the beginning of each period, TB be the measure of active buyers, FS bethe distribution of active seller types, and FB be the distribution of activebuyer types. The corresponding densities are fS and fB and, establishing usefulnotation, the right-hand distributions are FS ≡ 1− FS and FB ≡ 1− FB. Let,in the steady state, the probability that in a given period a type c seller tradesbe ρS [S(c)] and the let the probability that a type v buyer trades be ρB [B(v)] .DefineWS (c) andWB (v) to be the beginning-of-period steady-state net payoffsto a seller of type c and the buyer of type v, respectively. Define

c ≡ S (0) , (3)

c ≡ supc{c |WS (c) > 0}, (4)

v ≡ infv{v |WB (v) > 0} , and

v ≡ B (1) .

Since a trader only becomes active in the market if his expected utility fromparticipating is positive, no seller enters whose cost ci exceeds c and no buyerenters whose value is less than vi. We show in the next section that activesellers’ equilibrium bids all fall in the interval [c, c], active buyers’ equilibriumoffers all fall in [v, v] ,and that these intervals equal the trading range [c, c] =[v, v] = [p, p].Our goal is to establish sufficient conditions for symmetric, steady state

equilibria to converge to the Walrasian price and competitive allocation as theperiod length in the market goes to zero. By a symmetric, steady state equi-librium we mean one in which every seller in every period plays an anonymous,time invariant strategy S (·) , every buyer plays an anonymous, time invariantstrategy B (·) , and both these strategies are always optimal. Formally, giventhe friction δ, a perfect Bayesian equilibria consists of strategies {B,S}, ratioζ, and distributions {FB, FS} such that (i) {B,S}, ζ, and {FB, FS} generate ζand {FB, FS} as their steady state and (ii) no type of trader can increase his orher expected utility by a unilateral deviation from the strategies {B,S}.12 Weassume that equilibria satisfy two conditions:

A1. Sellers always bid their full dynamic opportunity cost.

A2. For each δ > 0 an equilibrium satisfying A1 exists in which each potentialtrader’s ex ante probability of trade is positive.

12Requirement (ii) that each period each active trader must solve his dynamic optimizationproblem guarantees that all equilibriia are subgame perfect.

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Assumption A1 is natural given that in the buyer’s bid double auction the sellerscan never affect the price; it is also needed to prove that the sellers’ strategyS is increasing, a feature that is both intuitive and necessary in the proofs.Assumption A2 states that well behaved equilibria exist in which trade occurs.This is necessary for two reasons. First, a no-trade equilibrium always exists inwhich neither buyers nor sellers enter the market. Second, it is an open questionwhether such non-trivial equilibria always exist, though numerical experiments(see section 4) suggest that they do for well behaved distribution GS and GB.In order to state our theorem we must define admissible sequences of equi-

libria. An admissible equilibrium sequence rules out sequences in which thebuyer-seller ratio ζδ goes to either 0 or ∞ as δ goes to 0. Consider for exam-ple a sequence of equilibria indexed by δ such that δ1, δ2, . . . , δn, . . . → 0 andζδ → ∞. Such a sequence is uninteresting because it violates the spirit of as-sumption A2’s requirement that each trader’s ex ante probability of trading ispositive. Specifically, the number of buyers with which each seller is matchedgrows unboundedly. Therefore each seller is sure to sell to a buyer whose valuevi is arbitrarily close to 1 and each buyer whose value is significantly less than1 is certain not to trade. In fact, if PEABδ denotes the ex ante probability that apotential buyer will trade, then in such sequences limδ→∞ PEABδ = 0 because theprobability of a buyer drawing value vi = 1 is zero.

Definition 1 A sequence of equilibria indexed by δ such that δ1, δ2, . . . , δn, . . .→0 is admissible if a ζ > 0 exists such that the equilibrium for each δn satisfiesA1, A2, and ζδ ∈ (1/ζ, ζ).We are now ready to state the paper’s main theorem.

Theorem 2 Fix any admissible sequence of equilibria and let {Sδ, Bδ} be thestrategies associated with the equilibrium that δ indexes, let [cδ, cδ] = [vδ, vδ] bethe offer/bid ranges and buyer seller ratios respectively those strategies imply,and let WSδ (c) and WBδ (v) be the resulting expected utilities of the sellers andbuyers respectively. Then both the bidding and offering ranges converge to pW :

limδ→0

cδ = limδ→0

cδ = limδ→0

vδ = limδ→0

vδ = pW . (5)

In addition, each trader’s expected utility converges to the utility he would realizeif the market were perfectly competitive:

limδ→0

WSδ (c) = max [0, pW − c] (6)

andlimδ→0

WBδ (v) = max [0, v − pW ] . (7)

A word of explanation may be helpful here concerning the theorem’s secondhalf. If the market were completely centralized and cleared each period at theWalrasian price–that is, if it were perfectly competitive–then each buyer oftype v who traded would realize utility v − pW and each seller who traded of

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type c would realize utility pW − c. Participants who failed to trade would exitwith zero utility. .The proof of the theorem is contained in section 5. In the appendix we modify

the proof to show a variant of this theorem in which we no longer assume thatpotential traders whose expected utility is zero choose not to enter the market.Instead we allow such traders to enter. These traders never successfully tradeand they exit only as the result of discouragement. The reason for including thisvariant of theorem 2 is that potential traders may make errors in their entrydecision and we want to demonstrate that our result is not sensitive to sucherrors.

3 Basic properties of equilibria

In this section we derive basic properties that equilibria of our model satisfy.These properties–strategies are strictly increasing, formulas for probabilities oftrade, and necessary conditions for a strategy pair (S,B) to be an equilibrium–enable us to compute the examples of equilibria in section 4 and provide thefoundations for the proof of our main result in section 5. We assume throughoutboth this section and section 5 that that δ and the equilibrium it indexes isan element of an admissible equilibrium sequence. We also assume that δ issufficiently small so that, δβ < 1. This states that active traders have a positiveprobability of continuing in the market if they fail to trade.

3.1 Strategies are strictly increasing

This subsection demonstrates the most basic property that our equilibria sat-isfy: equilibrium strategies are strictly increasing. As a preliminary, we firstcharacterize the set of traders that are active in the market. We then turn tothe monotonicity results.

Claim 3 In any pure-strategy equilibrium, v < 1 and c > 0, and

(v, 1] ⊆ {v|WB (v) > 0}, (8)

[0, c) ⊆ {c|WS (c) > 0}. (9)

Proof. The fact that v < 1 and c > 0 follows from the fact that in any equi-librium with trade, TB

R 1vρB [B (v)] fB(v)dv > 0 and TS

R c0ρS [S (c)] fS(c)dc >

0. By bidding B(v) in every period, a buyer gets an equilibrium payoffWB(v) =vPB [B (v)]−DB (B (v)) where PB [B (v)] is his ultimate probability of tradingwhen his bid is B (v) and DB (v) is his expected equilibrium payment. ByMilgrom and Segal’s (2002) theorem 2,

WB(v) =WB (v) +

Z v

v

PB [B (x)] dx,

soWB(·) is non-decreasing on (v, 1]. Assume, contrary to (8), thatWB [B (v0)] =

0 for some v0 ∈ (v, 1]. It then follows by the monotonicity of WB (·) that

12

WB(v) = 0 for all v ∈ (v, v0). But this implies WB [B (v)] = 0 on (v, v0),contradicting the definition of v. Therefore WB [B (v)] > 0 for all v ∈ (v, 1],establishing (8). The proof of (9) is exactly parallel and is omitted.¥

Claim 4 B is strictly increasing on (v, 1].

Proof. WB(v) = supλ≥0(v − λ)PB(λ) = (v − B (v))PB(B (v)) is the up-per envelope of a set of affine functions . It follows that WB (·) is a continu-ous, increasing, and convex function that is differentiable almost everywhere.13

Convexity implies that W 0B (·) is non-decreasing on [v, 1]. By the envelope the-

orem W 0B(·) = PB [B (·)] ; PB [B (·)] is therefore non-decreasing on [v, 1] at all

differentiable points. Milgrom and Segal’s (2002) theorem 1 implies that atnon-differentiable points v0 ∈ [v, 1]

limv→v0−

W 0B (v) ≤ PB (B (v

0)) ≤ limv→v0+

W 0B (v) .

Thus PB [B (·)] is everywhere non-decreasing on [v, 1].Pick any v, v0 ∈ (v, 1] such that v < v0. Since PB [B (·)] is everywhere

non-decreasing, PB [B (v)] ≤ PB [B (v0)] necessarily. We first show that B is

non-decreasing on (v, 1]. Suppose, to the contrary, that B(v) > B(v0). Therules of the buyer’s bid double auction imply that PB (·) is non-decreasing;therefore PB [B (v)] ≥ PB [B (v

0)]. Consequently PB [B (v)] = PB [B (v0)] . But

this gives v0 incentive to lower his bid to B(v0), since by doing so he will buywith the same positive probability but pay a lower price. This contradicts Bbeing a buyer’s optimal strategy and establishes that B is non-decreasing. IfB(v0) = B(v) (= λ) because B is not strictly increasing, then any buyer withv00 ∈ (v, v0) will raise his bid infinitesimally from λ to λ0 > λ to avoid therationing that results from a tie. This proves that B is strictly increasing.14¥

Claim 5 S is continuous and strictly increasing on [0, c).

Proof. Assumption A1 states that since sellers in the market do not affectprice, they bid their total opportunity cost:

S(c) = c+ (1− δβ)WS(c) (10)

for all c ∈ [0, c) where WS(c) is the equilibrium payoff to a seller with cost c.In a stationary equilibrium WS(c) = D(S (c)) − cPS(S(c)) where PS [S (c)] ishis ultimate probability of trading when his offer is S (c) and D(S (c)) is theexpected equilibrium payment to the seller with cost c. Milgrom and Segal’stheorem 2 implies that WS (·) is continuous and can be written as

WS (c) =WS (c) +

Z c

c

PS(S(x))dx =

Z c

c

PS(S(x))dx (11)

13An increasing function is differentiable almost everywhere.14Alternatively, one can use Theorem 2.2 in Satterthwaite and Williams (1989) with only

trivial adaptations.

13

for c ∈ [0, c] since WS (c) = 0 from the definition of c. This immediately impliesthat WS (·) is strictly decreasing (and therefore almost everywhere differen-tiable) because the definition of c implies that PS (S (c)) > 0 for all c ∈ [0, c).It, when combined with equation (10) , also implies that S (·) is continuous.Therefore, for almost all c ∈ [0, c),

S0(c) = 1− (1− δβ)PS[S(c)] > 0

because W 0S (c) = −PS[S(c)]. Since S (·) is continuous, this is sufficient to es-

tablish that S (·) is strictly increasing for all c ∈ [0, c).¥

Claim 6 S (c) = c = v = B(1) and B (v) = v = c = S(0).

Proof. Given that S is strictly increasing, S (0) = c is the lowest offer anyseller ever makes. A buyer with valuation v < c does not enter the market sincehe can only hope to trade by submitting a bid at or above c, i.e. above hisvaluation. S is continuous by claim 5, so a buyer with valuation v > c will enterthe market with a bid B (v) ∈ (c, v) since he can make profit with with positiveprobability. Therefore limv→c+B (v) = v = c.By definition c ≡ supc{c |WS (c) > 0}. Eq. (10) therefore implies that

S (c) = c. A seller with cost c > v ≡ B(1) will not enter the market, so c ≤ B(1).If c = S(c) < v ≡ B(1), then a seller with cost c0 ∈ (c, B (1)) can enter and,with positive probability, earn a profit with an offer S (c0) ∈ (c0,B (1)) . This,however, is a contradiction:

supc{c |WS (c) > 0} ≥ c0 > c ≡ sup

c{c |WS (c) > 0}.

Therefore S(c) = c = v = B(1).¥These findings are summarized as follows.

Proposition 7 Suppose that {B,S} is a stationary equilibrium satisfying A1and A2. Then B and S are strictly increasing over their domains. They alsosatisfy the boundary conditions c = v = B(1) and v = c = S(0).

Note that strict monotonicity of B and S allows us to define their inverses,V and C: V (λ) = inf {v : B(v) > λ} and C(λ) = inf {c : S(c) > λ}. Thesefunctions are used frequently below.

3.2 Formulas for the probabilities of trading

Focus on a particular seller of type c who has in equilibrium has a positiveprobability of trade. In a given period she is matched with zero buyers withprobability ξ0 and with one or more buyers with probability ξ0 = 1−ξ0. Supposeshe is matched and v∗ is the highest type buyer with whom she is matched. Sinceeach buyer’s bid function B (·) is increasing by Proposition 7, she accepts hisbid if and only if B (v∗) ≥ a, where a is her offer. The density from whichv∗ is drawn is f∗B (·) ; it is generated by the steady state density of buyer types

14

fB (·) and the distribution {ξ0, ξ1, ξ2, . . .} specifying the probabilities with whichthe each seller is matched with zero, one, two, or more buyers. Formally, thedistribution F∗B is conditional on the seller being matched and is defined as, forv ∈ [v, 1],

F ∗B(v) =1

ξ0

∞Xi=1

ξi [FB (v)]i

=e−ζFB(v) − e−ζ

1− e−ζ.

Therefore, if a seller offers λ, then

ρS (λ) =1− e−ζFB(V (λ))

1− e−ζ,

conditional on being matched with at least one buyer. The density f∗B is thederivative of F ∗B. Notice that F

∗B exhibits first order stochastic dominance with

respect to FB, i.e., for all v ∈ [v, 1], F ∗B(v) ≤ FB (v) .A similar expression obtains for ρB (λ), the probability that a buyer sub-

mitting bid λ successfully trades in any given period. Focus on a particularbuyer in the steady state and let ω1 be the probability that no other buyersbe matched with the same seller as he, ω2 be the probability that one otherbuyer is matched with the same seller as he , ω3 is the probability that twoother buyers are matched with the same seller as he, etc. Recall that ξk is theprobability that a seller will be matched with k buyers in a given period. Then

ωj =j ξjX∞k=1

k ξk=

j ξjζ;

that this correct can be seen by considering a large number of sellers and, giventhe probabilities {ξ0, ξ1, ξ2, . . .} , counting what proportion of buyers have nocompetition, what proportion have one competitor, and so forth.A buyer who bids λ and is the highest bidder has probability FS(C (λ)) of

having his bid accepted; this is just the probability that the seller with whomthe buyer is matched will have a low enough reservation value so as to accept hisoffer. Similarly if j buyers are matched with the seller with whom he is matched,then he has j − 1 competitors and the probability that all j − 1 will bid lessthan λ is [FB (V (λ))]

j−1 . Therefore the probability the bid λ is successful in aparticular period is

ρB (λ) = FS (C (λ))X∞

j=1ωj [FB (V (λ))]

j−1

= FS (C (λ)) e−ζFB(V (λ)),

where the second equality follows by direct calculation.We now turn to the ultimate trading probabilities PS (λ) and PB(λ). In a

stationary equilibrium, these probabilities must satisfy the recursion

PS (λ) = ξ0ρS (λ) + (ξ0 + ξ0ρS (λ))(1− δβ)PS (λ) ,

15

where ξ0 = 1− ξ0 and ρS (λ) = 1− ρS (λ). It follows that

PS (λ) =ξ0ρS (λ)

δβ + ξ0(1− δβ)ρS (λ). (12)

A similar recursion leads to

PB (λ) =ρB (λ)

δβ + (1− δβ)ρB (λ). (13)

3.3 Necessary conditions for strategies and steady statedistributions

In this subsection the goal is to write down a set of necessary conditions everyequilibrium that are sufficiently complete so as to form a basis for calculatingthe section 4’s examples .and, also, to create a foundation for the section 5’sproof of theorem 1. We first derive fixed point conditions that traders’ strategiesmust satisfy. Consider sellers first. Substituting (11),

WS (c) =

Z c

c

PS(S(x))dx (14)

into (10) gives a fixed point condition sellers’ strategies must satisfy:

S (c) = c+ (1− δβ)

Z c

c

PS(S(x))dx. (15)

The parallel expression for a buyer’s expected utility is15

WB(v) =

Z v

v

PB [B(x)]dx (16)

for v ∈ [v, 1]. Alternatively,WB(v) = max

λ∈[0,1](v − λ)PB(λ) = (v −B(v))PB (B (v)) .

Substituting (16) into this and solving gives the fixed point condition buyers’strategies must satisfy:

B(v) = v − 1

PB [B(v)]

Z v

v

PB [B(x)] dx for v > v. (17)

for v ∈ [v, 1] .In our model, the distributions {FB, FS} are endogenously determined by

traders’ strategies. In any steady state, the numbers of entering and leavingtraders must be equal. Therefore the buyers’ density fB must satisfy, for v ∈[v, 1] ,

aδgB(v) = TBfB (v) {ρB [B (v)] + (1− ρB [B (v)])δβ} (18)

15Formally, theorem 2 of Milgrom and Segal (2002) justifies this standard expression.

16

where the left-hand side is the measure of type v buyers of who enter each periodand the right-hand side, is the measure of type v buyers who exit each period.Note that it takes into account that within each period successful traders exitbefore discouraged traders. The analogous mass balance equation for sellers is

δgS (c) = TSfS (c) {ρS [S (c)] + (1− ρS [S (c)])δβ} . (19)

Together the fixed point conditions (15 and 17),the expected utility formulas(14 and 16), and the steady state conditions (18 and 19) form a useful set ofnecessary conditions for equilibria of our model.

4 Examples (to be done)

5 Proof of the theorem

5.1 Some preliminary results

Our purpose in this subsection is to show that buyers’ equilibrium strategiesB (·) must be within δ1/3 of either v or v except within some interval containedin [v, 1] that has length no greater than δ1/3. The first claim we establish is apreliminary restriction on the shape of B (·) .

Claim 8 In equilibrium, for all c ∈ [0, c],Z 1

V (S(c))

[B (x)− S (c)] fB (x) dx ≤ 2ζβδ. (20)

Proof. WS(c),a seller’s expected utility can be written recursively as thesum of the seller’s expected gains from trade in the current period plus hisexpected continuation value if he fails to trade in the current period:

WS (c) = ξ0

Z 1

V (S(c))

B (x) f∗B (x) dx− ξ0F∗B (V (S (c)))

+©ξ0 + ξ0F

∗B (V (S (c)))

ª(1− δβ)WS (c) .

where F ∗B (V (S (c))) is the probability that, conditional on at least one buyerbeing matched with him, he fails to trade in the current period. Move allterms involving WS (c) to the left-hand-side (LHS) and insert the expression−S(c) + c+ (1− βδ)WS (c) = 0, which is eq. (10) rewritten, into its RHS:

WS (c)©1− (1− δβ)ξ0 − (1− δβ)ξ0F

∗B (V (S (c)))

ª=

ξ0

Z 1

V (S(c))

B (x) f∗B (x) dx− ξ0cF∗B (V (S (c)))

+ ξ0F∗B (V (S (c))) {−S (c) + c+ (1− δβ)WS (c)} .

17

Cancel two terms on the RHS and move terms to the LHS to get

WS (c)

½1− (1− βδ)ξ0−(1− δβ)ξ0

£F∗B (V (S (c))) + F ∗B (V (S (c)))

¤ ¾ =ξ0

Z 1

V (S(c))

B (x) f∗B (x) dx− ξ0F∗B (V (S (c)))S (c) .

Recall that F ∗B(v) + F∗B (v) = 1 and ξ0 + ξ0 = 1. Then

δβWS (c) = ξ0

Z 1

V (S(c))

[B (x)− S (c)] f∗B (x) dx, (21)

i.e., in equilibrium, for a type c seller, the expected marginal cost of waiting anadditional period to trade is equal to the expected marginal expected gain fromwaiting.Rearranging (21) givesZ 1

V (S(c))

[B (x)− S (c)] f∗B (x) dx =βδ

ξ0WS (c) ≤ βδ

ξ0

because WS(c) ≤ 1. First order stochastic dominance implies thatZ 1

V (S(c))

[B (x)− S (c)] fB (x) dx ≤Z 1

V (S(c))

[B (x)− S (c)] f∗B (x) dx;

Therefore Z 1

V (S(c))

[B (x)− S (c)] fB (x) dx ≤ βδ

ξ0(22)

for all c ∈ [0, c). The probability that a seller will not be matched with anybuyer is

ξ0 =ζ0

0! eζ=1

eζ< e−1/ζ (23)

because the equilibrium is an element of an admissible sequence and thereforeζ ∈ [1/ζ, ζ]. A bound on the the complementary probability is

ξ0 > 1− e−1/ζ >1

2ζ

because 1 − e−x = x − x2

2 + . . . and 12ζis both small and positive. Using this

observation, we conclude from (22):Z 1

V (S(c))

[B (x)− S (c)] fB (x) dx ≤ 2ζβδ.¥ (24)

The bound (24) does not have any bite if fB (x) becomes small as δ becomessmall. Therefore in the next claim we establish a lower bound on fB (v) that isindependent of δ.

18

Claim 9 For all v ∈ [v, 1], fB (v) ≥ g (c−B (v)) .

Proof. Consider the highest type buyer, v = 1. In equilibrium he bids B (1)instead of some λ < B (1) . His expected gain from following this strategy isPB (B(1)) (1−B (1)) . If he bids λ < B (1), then his expected gain is PB (λ) (1−λ). Revealed preference implies PB (B(1)) (1−B (1)) ≥ PB (λ) (1−λ). Therefore

PB (λ) ≤ PB (B(1)) (1−B (1))

1− λ=

PB (c) (1− c)

1− λ. (25)

Note also that, for λ < B (1) , ρB [B (1)] ≥ ρB (λ) because ρB is a non-decreasingfunction for buyers.Inequality (25) permits us to bound ρB (λ) from above. It and formula (13)

imply the following sequence of inequalities

PB (λ) =ρB (λ)

ρB (λ) + δβ [1− ρB (λ)]≤ PB (c) (1− c)

1− λ, (26)

ρB (λ)

ρB (λ) + βδ≤ PB (c) (1− c)

1− λ,

ρB (λ) ≤βδ

1−λPB(c)(1−c) − 1

,

ρB (λ) ≤βδ

1−λ(1−c) − 1

, and

ρB (λ) ≤βδ (1− c)

c− λ

where the second line follows from dropping the less than unity factor(1− ρB (λ)) ,the third line from solving the inequality, the fourth line from PB (c) ≤ 1, andthe fifth line from simplifying the fourth line.Inequality (26) allows us to establish the desired lower bound on fB (v)

provided we have an upper bound on TB,the mass of buyers active in the market.Suppose all potential buyers (measure a each period) entered and became active,none successfully traded, and all left the market only due to discouragement.The total mass of active buyers in the market would then be TB = a/β. Sincemany buyers leave as a result of successful trade an upper bound on the massof sellers in the market is TB ≤ a/β.Solve (18) for fB (v) and substitute in the

19

c

v

1/ 3c − δ

1/ 3v + δ

*v **v

( )B v

,c v

λ

1

1

Figure 1: 3√δ band that confines B(·)

bound on ρB (λ) to get

fB (v) ≥ 1

TB

aδ gB (v)

(1− βδ) βδ(1−c)c−λ + βδ(27)

≥ gB (v)

(1− βδ) 1−cc−λ + 1

≥ g

(1− βδ) 1−cc−λ + 1

≥ g(1−c)c−λ + 1

=g

λ(c− λ)

≥ g (c−B (v))

where TB < a/β implies the second line, g being the lower bound on the densitiesgB and gS implies the third line, (1− βδ) ≤ 1 implies the fourth line, and λ < 1implies the fifth line.¥We now use the bounds established in claims 8 and 9 to place a strong

restriction on the shape of B. Figure 1 shows the construction used in the nextclaim and shows how the claim’s conclusion confines B (·) to a narrow band ofwidth proportional to δ1/3.

Claim 10 Suppose c− v ≥ 2δ1/3. For given δ > 0, let v∗ = V³v + δ1/3

´and

20

v∗∗ = V³c− δ1/3

´. Then

v∗∗ − v∗ ≤ 2ζg

βδ1/3 (28)

Proof. Substituting in (24) inequality (27), a lower bound on the densityfB , gives

g

Z 1

V (S(c))

(B (x)− S (c)) (c−B (x)) dx ≤ 2ζβδ.

The special case of this inequality in which c = 0 gives the restriction on thebuyers’ strategy B (·):Z 1

v

(B (x)− v) (c−B (x)) dx ≤ 2ζβδg

(29)

because S (0) = c = v and V (v) = v.Note that, for x ∈ [v∗, v∗∗], the following inequalities are true: B (x)− v ≥

δ1/3 and (c−B (x)) ≥ δ1/3. ThereforeZ v∗∗

v∗(B (x)− v) (c−B (x)) dx ≥ δ2/3 (v∗∗ − v∗) .

This inequality together with the observation that the integrand of (29) is pos-itive for the whole interval of integration [v, 1] implies

aβδ

g≥

Z 1

v

(B (x)− v) (c−B (x)) dx

≥Z v∗∗

v∗(B (x)− v) (c−B (x)) dx

≥ δ2/3 (v∗∗ − v∗) .

The first and last terms of this sequence of inequalities together imply (28).¥

5.2 The law of one price

In this section, we demonstrate that limδ→0(cδ − vδ) = 0. Since all tradesoccur at prices within the interval [vδ, cδ] this means that as the period lengthapproaches zero all trades occur at essentially one price. Intuitively this is drivenby increasing local market size and the resulting option value: as δ becomes smalleach trader can safely wait for a very favorable offer/bid. In the next subsectionwe show that this one price must be the Walrasian price pW .

Proposition 11 Consider any ζ-admissible sequence of equilibria δn → 0. Then

limδ→0

(cδ − vδ) = 0.

21

v

v

1/ 3v − δ

1/ 3v + δ

*v

( )B v

,c v

λ

1

1

v

v v%

b

b′

v′%

**v

Figure 2: Construction of v∗, v∗∗, ev, ev0, b and b0.

Proof. Suppose a sequence of equilibria indexed by δ exists such thatδ1, δ2, . . . , δn, . . . → 0 and limδ→0(cδ − vδ) = η > 0. We show that this is acontradiction: therefore, necessarily, limδ→0(cδ − vδ) = 0. From now on, fix asubsequence such that limn→∞(cδ − vδ) = η.Pick a small δ from the subsequence and let the strategies {S,B}, proba-

bilities {ξ0, ξ1, ξ2, . . .}, and distributions [FS, FB] characterize the equilibriumassociated with it. Recall that S (0) = c = v = B(v) and B (1) = v = c = S (c) .Also recall above from above the definitions of v∗ and v∗∗. Define in addition

v = v − 14(v − v), b = B (v) , b0 = b+ δ1/3, and v0 = V (b0)

as shown in figure 2. We prove the proposition with a sequence of four claims,the last of which has the proposition as a corollary. The first of the claimsderives four intermediate inequalities.

Claim 12 Given the construction of v, v∗, v∗∗, b, and b0 and given that, byassumption, limδ→0(cδ − vδ) = η > 0, if δ is sufficiently small, then

v ≤ v∗, (30)

infv∈[v,v0]

(v −B (v)) ≥ 12η, (31)

v0 − v ≥ 18η. (32)

22

Proof. We begin with three observations:

O1 The assumption that v − v ≥ η for all δ in the sequence and the definitionv = v − 1

4(v − v) imply v + 14η < v.

O2 The definition B (v∗∗) = v − δ1/3 and the inequality (implied by A1)B (v∗∗) ≤ v∗∗ imply that v − δ1/3 ≤ v∗∗.

O3 Recall from (28) that v∗∗ ≤ v∗ + 2ζg βδ1/3.

To derive (30), note that O2 and O3 imply

v ≤ v∗ +µ1 +

2ζβ

g

¶δ1/3

Combining this with O1 gives

v ≤ v∗ − 14η +

µ1 +

2ζβ

g

¶δ1/3.

Thus, for small enough δ,v ≤ v∗. (33)

Turning to (31), that B (·) is increasing, v ≤ v∗, B (v) = b, B (v∗) = v+δ1/3,

b0 = b + δ1/3, and B (v∗∗) = b0 together imply that b ∈ [v, v + δ1/3] and b0 ∈[v + δ1/3, v + 2δ1/3]. Consequently, for sufficiently small δ,

infv∈[v,v0]

(v −B (v)) ≥ v − b0 (34)

≥ v − v − 2δ1/3≥ η − 2δ1/3

≥ 1

2η.

This proves (31). Finally, to establish (32), note that by construction v∗ < v0.Therefore

v0 − v > v∗ − v (35)

≥ v∗∗ − v − 2ζg

δ1/3

≥ v − v −µ1 +

2ζ

g

¶δ1/3

≥ 1

4η −

µ1 +

2ζ

g

¶δ1/3

≥ 1

8η

23

where line two follows from v∗∗ ≤ v∗ + 2ζg δ1/3, line three follows from v∗∗ >

v−δ1/3, line four follows from v− v ≥ 14η, and line five follows if δ is sufficiently

small.¥

Claim 13 Given limδ→0(cδ − vδ) = η > 0, if δ is sufficiently small, then aγ > 0 exists such that

ρB (b0)

ρB (b)> 1 + γ.

Proof. Since V (b0) = v0 and V (b) = v, the ratio of ρB(b0) and ρB(b) is

ρB(b0)

ρB (b)=

FS (C (b0)) e−ζFB(v0)

FS (C (b)) e−ζFB(v)

≥ eζ[FB(v0)−FB(v)]

≥ 1 + ζ [FB(v0)− FB(v)]

≥ 1 +1

ζ

Z v0

v

fB (x) dx,

where the second line follows from b0 > b and both FS and C being increasing,the third line follows by ex ≥ 1 + x (x ≥ 0), and the last line follows fromζ ≥ 1/ζ. Recall from (27) that fB (v) ≥ g (v −B (v)) . Therefore

ρB(b0)

ρB (b)≥ 1 +

g

ζ

Z v0

v

(v −B (v)) dx

≥ 1 +g

ζ(v0 − v) inf

v∈[v,v0](v −B (v))

≥ 1 +g

ζ

µ1

8η

¶µ1

2η

¶= 1 +

1

16

g

ζη2

= 1 + γ

where line three follows from (31) , (32), and γ = 116ζ

gη2.¥

Claim 14 Given limδ→0(cδ − vδ) = η > 0, if δ is sufficiently small, then

PB (b0)

PB (b)≥ 1 + γ∗

where γ∗ = 14γη > 0.

Proof. Direct calculation proves this. Recall from (13) the formula forPB (b) . Therefore

PB (b0)PB (b)

=ρB (b

0)ρB (b)

βδ + ρB (b)− βδρB (b)

βδ + ρB (b0)− βδρB (b

0).

24

Define x and y so that ρB (b0) = βδx and ρB (b) = βδy. Then, after some

manipulation,

PB (b0)

PB (b)=

1 + 1y − βδ

1 + 1x − βδ

≥ 1 + 1+γx − βδ

1 + 1x − βδ

= 1 +γx

1 + 1x − βδ

≥ 1 +γx

1 + 1x

= 1 +γβδ

ρB (b0) + βδ

≥ 1 +1

2γ (v − b0)

where line two follows from claim 13’s implication that 1y ≥ 1+γx , line four follows

from βδ ∈ (0, 1) , line five follows from the definition of x, and line six followsfrom inequality (26) and 1− b0 < 1. By construction b0 ∈ (v + δ1/3, v + 2δ1/3).Hence, for δ sufficiently small,

PB (b0)

PB (b)≥ 1 +

1

2γ(v − v − δ1/3)

≥ 1 +1

4γη

because v − v > η.¥

Claim 15 Given limδ→0(cδ− vδ) = η > 0, if δ is sufficiently small, then a typev buyer has an incentive to deviate from bidding B (v) = b to bidding b0 > b.

If we denote the expected utility of a type v buyer who bids λ as πB (λ, v) ,then to prove this we need to show that πB(b0, v)−πB (b, v) > 0 for δ sufficientlysmall. First note that by construction v = v − 1

4 (v − v) and b < v + δ1/3.Therefore

v − b ≥ v − 14(v − v)− v − δ1/3

=3

4(v − v)− δ1/3

≥ 3

4η − δ1/3

≥ 1

2η

25

for sufficiently small δ because v − v > η. Next observe that, for sufficientlysmall δ,

πB(b0, v)− πB (b, v) = (v − b0)PB (b0)− (v − b)PB (b)

≥ [(1 + γ∗) (v − b0)− (v − b)]PB (b)

=h(1 + γ∗)

³v − b− δ1/3

´− (v − b)

iPB (b)

=hγ∗(v − b)− (1 + γ∗) δ1/3

iPB (b)

≥·1

2η γ∗ − (1 + γ∗) δ1/3

¸PB (b)

> 0

where line 2 follows from claim 14. ¥Claim 15 directly implies proposition 11 because it contradicts the main-

tained hypothesis that δn → 0 admits a subsequence such that limδ→0(cδ−vδ) =η > 0.

5.3 Convergence of the bidding and offering ranges to theWalrasian price

Recall that the Walrasian price is the solution pW to the equation

GS (pW ) = a GB(pW ); (36)

it is just the price at which the measure of sellers entering the market with costc ≤ pW equals the measure of buyers entering the market with values v ≥ pW .This price would clear the market each period if there were a centralized market.In this section we prove our main result: as δ → 0 the bidding range [v, v]collapses to the Walrasian price. More formally, for any sequence of equilibriaindexed by δ such that δ1, δ2, . . . , δn, . . .→ 0, both

limδ→0

vδ = pW and limδ→0

vδ = pW . (37)

We show this through the proof of two claims. Each of these claims uses theidea that if the price is not converging to the Walrasian price, then the marketdoes not clear globally and an excess of traders builds up on one side or theother of the market. Traders on the long side then have an incentive to deviatefrom their prescribed bid/offer in order to trade before becoming discouraged.

Claim 16 limδ→0vδ ≥ pW .

Proof. Let v∗ = limδ→0vδ and assume, contrary to the statement in theclaim, that v∗ < pW . Let vδ = vδ+

√vδ − vδ (note that vδ ∈ (vδ, 1] for all small

enough δ, by Proposition 11). Revealed preference implies that

πB (Bδ (vδ) , vδ) ≥ πB (Bδ (1) , vδ)

[vδ −Bδ (vδ)] PBδ [Bδ (vδ)] ≥ [vδ −Bδ (1)] PBδ [Bδ (1)] .

26

Therefore

PBδ [Bδ (vδ)] ≥ vδ −Bδ (1)

vδ −Bδ (vδ)PBδ [Bδ (1)] (38)

≥ vδ −Bδ (1)

vδ − vδPBδ [Bδ (1)] ,

where the second inequality follows from the fact that Bδ (·) is strictly increasingand therefore Bδ (vδ) ≥ Bδ (vδ) = vδ. Continuing with the chain of inequalitiesfor PBδ [Bδ (vδ)], we get

vδ −Bδ (1)

vδ − vδ=

vδ − vδvδ − vδ

(39)

=

√vδ − vδ√

vδ − vδ + vδ − vδ,

where the inequality in the first line follows from Bδ (1) = vδ, and the equalityin the second line is just the substitution of the definition vδ = vδ +

√vδ − vδ.

Combining (38) and (39) we get

PBδ [Bδ (vδ)] ≥√vδ − vδ√

vδ − vδ + vδ − vδPBδ [Bδ (1)] ,

So in particular, PBδ [Bδ (1)] = 1 and, by Proposition 11, limδ→0(vδ − vδ) = 0imply16

limδ→0

PBδ [Bδ (vδ)] = 1.

Steady-state equilibrium implies global market clearing, soZ 1

vδ

agB(x)PBδ [Bδ (x)] dx =

Z vδ

0

gS(x)PSδ [Sδ (x)] dx. (40)

Equation (40) is a simple statement of the fact that, in a steady state, themeasures of entering buyers and sellers who ultimately trade must be equal.Given that PBδ [Bδ (·)] is increasing and vδ > v,Z 1

vδ

agB(x)PBδ [Bδ (x)] ≥ PBδ [Bδ (vδ)]

Z 1

vδ

agB(x)dx ≥ PBδ [Bδ (vδ)] aGB(vδ)

and Z vδ

0

gS(x)PSδ [Sδ (x)] dx ≤ GS [vδ] .

Therefore it follows from (40) that

PBδ [Bδ (vδ)] aGB(vδ) ≤ GS [vδ]

16A type 1 buyer always trades immediately because B (1) = v = c = S (c) .

27

or, since PBδ [Bδ (vδ)] ≤ 1,

aGB(vδ) ≤ GS [vδ] . (41)

By taking limits in (41) as δ → 0 and invoking continuity of GS and GB, weobtain

aGB [limδ→0vδ] ≤ GS [limδ→0vδ] . (42)

Since by proposition 11 limδ→0vδ = limδ→0 [vδ +√vδ − vδ] = v∗ and by hy-

pothesis limδ→0vδ = v∗ , we obtain from (42):

aGB(v∗) ≤ GS (v∗) .

This, however, is a contradiction since, by the maintained assumption thatv∗ < pW , aGB(v∗) > aGB(pW ) = GS (pW ) > GS (v∗).¥

Claim 17 limδ→0cδ ≤ pW .

Proof. Verification of this claim follows the same logic as that of Claim 16.Define c∗ = limδ→0cδ and suppose, contrary to the statement in the claim, thatc∗ > pW . Let ecδ = cδ +

√cδ − cδ noting that proposition 11 implies ecδ ∈ [0, cδ)

for all small enough δ. A seller who offers and succeeds in trading does notrealize Sδ (v) as her revenue. She realizes something more because the bid sheaccepts is at least as great as Sδ (v) . Therefore, for each δ sufficiently small, afunction φδ (·) : [0, cδ]→ [cδ, cδ] exists that maps, conditional on consummatinga trade, the seller’s offer into her expected revenue from the sale. Thus φδ [Sδ (c)]is a type c seller’s expected revenue given that she offers Sδ (c) . Take note thatφδ [Sδ (c)] ∈ [Sδ (c) , cδ] because the expected revenue can not be less than theseller’s offer Sδ (c) . Revealed preference implies that

πS (Sδ (cδ) , cδ) ≥ πS (Sδ (0) , cδ)

[φδ [Sδ (cδ)]− cδ] PSδ [Sδ (cδ)] ≥ [φδ [Sδ(0)]− cδ] PSδ [Sδ(0)] . (43)

Solving,

PSδ [Sδ (cδ)] ≥ φδ [Sδ(0)]− cδφδ [Sδ (cδ)]− cδ

PSδ [Sδ(0)] (44)

≥ cδ − cδcδ − cδ

PSδ [Sδ(0)]

=

√cδ − cδ

cδ − cδ +√cδ − cδ

PSδ [Sδ(0)]

where the second line follows from the fact that φδ [Sδ(0)] ≥ Sδ (0) = cδ andthe third line follows by substitution of the definition for ecδ. So in particular,PSδ [Sδ(0)] = 1 and, by Proposition 11, limδ→0(cδ − cδ) = 0 together imply

limδ→0

PSδ [Sδ (cδ)] = 1. (45)

28

As in the proof of Claim 16, the market-clearing equation (40) must hold:Z 1

cδ

agB(x)PBδ [Bδ (x)] dx =

Z cδ

0

gS(x)PSδ [Sδ (x)] dx. (46)

Since Z cδ

0

gS(x)PSδ [Sδ (x)] dx ≥ PSδ [Sδ (cδ)]GS(cδ)

and Z 1

cδ

agB(x)PBδ [Bδ (x)] dx ≤ aGB(cδ),

it follows from (46) that

aGB(cδ) ≥ PSδ [Sδ (cδ)]GS(cδ)

or, since PSδ [Sδ (cδ)] ≤ 1,aGB(cδ) ≥ GS(cδ). (47)

By taking limits in (47) as δ → 0 and invoking continuity of GS and GB, weobtain

aGB(limδ→0cδ) ≥ GS

£limδ→0cδ

¤. (48)

Since limδ→0cδ = c∗ and limδ→0cδ = c∗ by Proposition 11, (48) implies

aGB(c∗) ≥ GS (c∗) .

This, however, is a contradiction since, by the maintained assumption c∗ > pW ,aGB(c∗) < aGB(pW ) = GS (pW ) < GS (c∗).¥Proof of the main theorem. Claims 16 and 17 together with infδ→0(cδ−

vδ) = 0 establishes (37): prices realized in the market converge to the Walrasianprice. The proofs of those two claims together show that an arbitrarily smalldeviation upward in a buyer’s equilibrium bid or an arbitrarily small deviationdownward in a seller’s equilibrium offer can guarantee trade, provided δ is suf-ficiently small. This, together with the result that realized prices converge tothe Walrasian price, establishes (6) and (7): equilibrium expected utility forboth buyers and sellers approaches what they would expect if the market werecompetitive.¥

6 Conclusions

In this paper we have proven that essentially all equilibria of a simple, dynamicmatching and bargaining market in which both sellers and buyers have incom-plete information converge to the Walrasian price and competitive allocationas the model’s friction–the length of the matching period–goes to zero. Thisconvergence is driven by the interaction of two forces within the model: local

29

market size and global market clearing. The significance of our result is to showthat in the presence of private information a fully decentralized market suchas the one we model can deliver the same economic performance as a central-ized market such as the k-double auction that Rustichini, Satterthwaite, andWilliams (1994) studied. This is an important extension of the full informa-tion dynamic matching and bargaining models, for it shows that a decentralizedmarket can handle the elicitation of private values and costs even as it allocatesthe market supply to the traders who most highly value that supply.The limitations of our model and results immediately raise further questions.

Six particularly stand out for future investigation:

• How fast does the market converge to the competitive allocation as δ de-creases? The numerical experiments reported in Section 4 suggest that thek-double auction’s quadratic rate of convergence towards efficiency doesnot carry over to this dynamic model: convergence in this model is appar-ently only linear because the matching market leads to δβ proportion ofeach trader type each period to leave the market discouraged, irrespectiveof their potential gains from trade. The quadratic convergence of the k-double auction is achieved because the “wedge” in that market excludesfrom trade only those traders who have the smallest gains from trade torealize.

• We only considered the buyer’s bid double auction mechanism. Does con-vergence robustly hold for a broader class of trading mechanisms, e.g., the0.5-double auction?

• Our model assumes independent private values and costs. We would liketo know if our results generalize to both correlated costs/values and tointerdependent values with a common component and affiliated privatesignals.

• It would be attractive to incorporate into our model the alternative as-sumption that every active trader pays a small participation cost as inthe full information bargaining and matching models of Gale (1987) andMortensen and Wright (2002).

• An attractive feature of the centralized k-double auction as studied byRustichini, Satterthwaite, and Williams (1994) is that if the number oftraders on both sides is large, then a trader who reports his cost or valuetruthfully as opposed to following his equilibrium strategy realizes almostno reduction in expected utility. In other words, for large markets thek-double auction is almost a dominant strategy mechanism.17 Buyers arenot so fortunate in our bargaining and matching market even if the periodlength is very short. To play optimally–or even acceptably–they musthave good knowledge of the distribution of prices at which trades are oc-curring. Nevertheless the following conjecture appears worth exploring: if

17 See Satterthwaite (2001) for a development of this idea.

30

our matching and bargaining market were populated with non-optimizingbuyers and sellers, then a class of simple learning rules exists such thatthe market converges to the Walrasian price and competitive allocation.If this is true, then those rules would be almost-dominant strategies andthe market as a whole would be almost strategy-proof.

• Kultti (2000) studies a complete information model in which each buyer,as part of his optimization, decides whether to be a searcher or waiter.Sellers do the same. Searching buyers are then randomly matched withwaiting sellers and searching sellers are randomly matched with waitingbuyers. Giving every trader the choice to be a searcher or waiter would bean elegant way of endogenizing the model’s market structure rather thanimposing that all buyers are searchers and all sellers are waiters.

If these and other questions can be answered adequately in future work, thenthis theory may become useful in designing and regulating decentralized marketswith incomplete information in much the same way auction theory has becomeuseful in designing auctions. The ubiquity of the Internet with its capability forincreasing local market size and reducing period length makes pursuit of thisgoal attractive.

7 Appendix (to be completed)

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