Double Auction with a Small Participation Cost

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1 Double Auction with a Small Participation Cost Jianjun Wu University of Arizona July 30, 2005 Abstract This paper studies a simple market with N potential buyers and N potential sellers where each trader needs to incur a small participation cost to enter the trading for an indivisible good. In this case, the realized market might be asymmetric in two senses: first the numbers of traders in each side are not equal; second, the entering buyers and sellers will have different supports, either partially overlapped or completely separated. This two aspects might cause the equilibrium strategies to become kinked. However, it canbeshownthatthepostentryefficiency loss in this simple market will be at most O 1 N 2 even with the presence of those complexities. 1 Introduction It is well known that both buyers and sellers with private information over their preferences will misrepresent their values and costs to achieve better bargain terms. On the other hand, Department of Economics, University of Arizona. jwu@eller.arizona.edu. The author gratefully acknowledges many helpful conversations with Mark Satterthwaite who suggested this problem. The author also thanks Jim Dana, Peter Eso, Kathleen Hagerty, Ming-yang Kao, Ehud Kalai, Francisco Ruiz-Aliseda, Rakesh Vohra and participants of 2003 NSF Foundation of Eletronic Market workshop in Santa Barbara for helpful comments. 1

2 it is commonly believed that if the market size becomes large, all traders will tend to tell the truth since they are less likely to influence the market price. This result is presented in a line of research that use double auctions to model market trading activities 1.Inthose models, it is shown that in a Bayesian Nash equilibrium, both buyers and sellers strategies will converge to truth telling as market gets large. However, to my knowledge all models in those literatures assume that there is no participation cost. While in reality, this is not the case. In the internet market such as Ebay, a seller needs to pay a listing fee regardless whether she sells the item or not. On the other side, to surf the internet market, a buyers at least has to incur some time cost connecting to the Internet Service Provider regardless whether he purchases the good or not. Thus buyers and selllers will enter the market if and only if they can earn a positive payoff at least as large as the participation cost. On the other hand, those literatures on double auctions without participation cost usually assume that zero probability traders will enter the market and tell the truth. It is innocuous in the sense that there is no cost for them to stay in the market. Nevertheless, their existence is important in the sense that the truth telling behavior of those zero probability traders forms a pressure over those traders with positive probability of trading. That is, it forces the latter to restrain their overshooting when making bids and offers. Whenever there is a participation cost, those zero probability traders will not enter the market for sure. This presumablly would cause an indirect strategic effect on those traders in the market. An immediate question is: without the pressure from those zero probability traders, will those entrants strategy converge to truth telling when the market gets large? If the strategies do not converge, the market might not converge to efficiency. 1 For example, see Chatterjee and Samuelson (1983), Myerson and Satterthwaite (1983), Hagerty and Rogerson(1985), Wilson(1985) Gresik and Satterthwaite (1989), Satterthwaite and Williams (1989, 2002) and Rustichini, Satterthwaite and Williams (1994). 2

3 Specifically, I study a market with N potential buyers and N potential sellers. Each buyer and seller obserses his/her private value v and cost c and knows that they need to incur a participation cost τ to enter the market. Both v and c are distributed on [0, 1]. However, not every trader will enter the market. Buyers and sellers use a cutoff strategy to decide whether to enter the market or not. Thus for a given potential market with N buyers and N sellers, the number of entering buyers and sellers are two random variables with binomial distribution. This implies that even we start with a symmetric potential market, the realized market configuration might be asymmetric in the sense that the numbers of entering buyers and sellers are different. Suppose q v denotes the minimum value buyer and q c denotes the highest cost seller that choose to participate in market trading. In the presence of a participation cost we must have q v > 0 and q c < 1 which implies the supports of entering buyers and sellers are not identical. Since the participation cost is sunk, all entering buyers and sellers will choose their strategy based on the number of participants and their beliefs about entrants value and cost. Therefore, except for the entry decision, the participation cost has no direct effect on traders strategic choice since it is irreversible. Suppose for a particular realization of N buyers values and N sellers costs, m buyers and n sellers enter the market. Each entering buyer submits a bid b and each entering seller makes an offer s. The market trading is modeled as a k-double auction which is defined in Satterthwaite and Williams (1989). That is, arrange all bids and offers in increasing order s (1) s (2) s (m) s (m+1) s (m+n) (1) where s (i) denotes the i th smallest bid/offer. Then the market price is a weighted average 3

4 of m th and (m +1) th smallest bid among all participating buyers and sellers: p = ks (m) +(1 k)s (m+1) (2) It is also worth noting that any k between 0 and 1 will clear the market with probability one, as shown in the following figure. It shows a realized market with three buyers bids (0.4, 0.52, 0.9) and three sellers offers (0.15, 0.375, 0.75). In this case, any price between 0.4 and 0.52 will clear the market. Offers Bids Supply Demand Quantity Figure 1: k - double auction The analysis of those k-double auctions with nonidentical supports is not a trivial extension of those literatures without participation cost since the equilibrium strategy is different in the sense that they might be kinked. Since the supports of entering buyers and sellers are only partially overlapped, there are some advantaged buyers who realize that there are no sellers in the market having cost above their value. Similarly, there exist some advantaged sellers whose cost is less than any value of entering buyers. Those traders might not be forced to tell the truth even when the market is large. Moreover, intuition 4

5 tells that when N goes to infinity, each buyers and seller will behave as a price taker. Suppose the market clearing price is p, then only buyers with value no less than p + τ and sellers with cost no larger than p τ will enter the market. This implies the support of entering buyers and sellers might be separated when the potential market size becomes large enough. I incorporate the considerations of those complications by studying two cases depending on whether the entering buyers and sellers support are overlapped or not. Depending on the information revealed, there are four efficiency concepts: Ex Ante efficiency, Interim I efficiency, Interim II efficiency and Ex Post Efficiency. The first and last one are common while two interim efficiency need some explanations. Interim I efficiency is defined as the expected efficiency when traders observe their private value and cost and before making their decisions on whether to enter the market or not. The Interim II efficiency is the expected efficiency when the numbers of entrants on each side are revealed to those traders. To prevent confusion, I also use Post Entry Efficiency to represent Interim II efficiency. My goal is to study the expected loss of post entry efficiency in the realized market when the size of potential market becomes large. The expected loss of efficiency is defined as the ratio of total loss of efficiency with respect to the total expected gains from trade. First, if the entering buyers and sellers support are overlapped, I show that the efficiency loss in the post entry market is at most O 1 N. In the paper, it is shown that 2 although there are buyers and sellers who tend not to tell the truth even when the market is large, they choose to do so because they are guaranteed to trade and will not cause efficiency loss. All those buyers and sellers who have positive probability of not trading will converge to truth telling as a result of competitive bidding. This asymptotically truth telling strategy contains the loss of efficiency in the post entry market. Second, for markets with non-overlapping support, there are two interesting questions: 5

6 how fast will the support be separated and whether the market is still asymptotically efficient? For the first question, if the distribution functions are well behaved 2, for any fixed participation cost τ, the entering traders will tend to have separated support at a very slow rate of order O ³ 1 N. This is measured by an upper bound of the expected payoff of the marginal traders when the support are overlapped. As for the second question, I show that there exist equilibria which realize all possible efficiency in the post entry market. My work directly stems from the pioneering paper by Myerson and Satterthwaite (1983), which analyzes the efficient bargaining mechanism between two agents. As I show later, the efficient mechanism defined in that paper is not robust to a small positive participation cost. The convergence result in this paper is directly related to a line of research starting with Gresik and Satterthwaite (1983) who study multilateral bargaining mechanisms. Satterthwaite and Williams (1989) considers a special case of bilateral trading mechanism: Buyer s bid double auction. Rustichini, Satterthwaite and Williams (RSW, 1994) obtains a convergence rate of order O 1 N for the case that the ratio of the number of buyers and 2 sellers is bounded. This paper advances those analysis by considering the rate of convergence to competitive market in terms of potential market sizes. The focus of the paper is on the efficiency of resulting double auction with asymmetric support caused by the participation cost, which is named as Interim II efficiency. This paper does not intend to study the efficiency loss in terms of participation cost, although it certainly could be a future research topic. My work is also related to another line of literature on auctions with participation cost: for example, McAfee and McMillian (1987a, 1987b), Stageman (1996). In those papers, the bidders entry decision can be explicitly derived using the closed form equilibrium strategies. 2 The distribution function has to satisfy a technical condition that it does not increase too fast such as convext distribution functions. This condition is satisfied in uniform and normal distribution. 6

7 Most of literatures on Auctions with entry cost focuses on ex ante participation cost. That is, bidders pay the participation cost before they observe their private value 3. If bidders can choose whether to incur participation cost after they know their private value, Landsberger and Tsirelson (2001) show that any small participation cost will cause the number of bidders that enter the market to decrease exponentially. That is because there is only one unit of supply. The expected payoff of bidders with value below the upper bound of the support will decrease exponentially. It is in some sense equivalent to my result that the entering buyer and seller s support tend to separated except that the number of traders does not decrease drastically since it is not the case as in single sided auction that there is at most one unit of supply. The paper is organized as follows: Section 2 outlines the model. Section 3 studies the market with one buyer and one sellers and characterizes the equilibrium strategy to account for a positive participation cost. Section 4 consider the general N N market. Section 5 gives an example of a 2 2 double auction with linear strategies. Section 6 concludes the paper and discusses some conjectures. 2 The Model Consider a market with N buyers and N sellers interested in buying and selling one indivisible good. Each buyer demands only one unit of this good and each seller has only one unit for sale. Both buyers value v and sellers cost c are distributed on [0, 1]. Buyer s value v has absolutely continuous distribution function F (v) with positive density function f(v). Similarly, seller s cost c is distributed according to an absolutely continuous distribution function G(c) with positive density function g(c). Specifically, let δ =min{inf v f (v), inf c g (c)} > 0. 3 One exception is Monteiro, P. and F. Menezes (2000), which studies endogenous participation cost. 7

8 Suppose buyers and sellers observed their private value and cost before they decide whether to enter the market. Each buyer and seller needs to incur a small 4 positive participation cost τ to enter the trading market. We consider the subgame perfect equilibrium of this market. The timing of the game is as follows: First, all potential buyers and sellers observe their private value and cost. They use cutoff strategies to decide whether to enter the market or not. That is, a particular buyer with value v will enter the market if and only if v q v where q v [0, 1] Similarly, a particular seller with cost c will enter the market if and only if c q c where q c [0, 1] Second, all entering buyers and sellers submit their bids and offers simultaneously. To simplify the analysis, we assume the buyers and sellers are allowed to submit a vector of strategies contingent on the actual number of buyers and sellers in the market. This is equivalent to entering the market and observing the number of traders in the market before choosing their strategies. In each contingent market, the market clearing mechanism is the same as that of the k-double auction mechanism defined in Satterthwaite and Williams (1989). That is, suppose m buyers and n sellers enter the market since their values and costs satisfy the cutoff criterion, each buyer submits a bid and each seller submits an offer. Each trader s Bayesian Nash equilibrium strategy is a function of his/her own private value/cost. Let B (v) denote buyer v 0 s bid and S (c) denote seller c 0 s offer. All submitted 4 The participation cost is small in the sense that it is not large enough that no buyer or seller is willing to enter the market which is an uninteresting equilibrium. 8

9 bids and offers are arranged in increasing order s (1) s (2) s (m) s (m+1) s (m+n) where s (i) denotes the i th smallest bid/offer. Then the market price is a weighted average of m th and (m +1) th highest bid among all participating buyers and sellers: p = ks (m) +(1 k)s (m+1) (3) In our model, k is defined as k = 1 m<n 1 2 m = n 0 m>n With price defined as p = ks (m) +(1 k)s (m+1), a buyer will be able to obtain one unit if and only if his bid is greater than p and a seller will be able to sell her unit if and only if her offer is less than p. As shown in the introduction, it is innocuous to choose any k between 0 and 1. Specifically, when k =1, this is a Buyer s bid double auction, in which a buyer will only be able to purchase the good if his bid is strictly greater than p. Similarly, when k =0, it is a Seller s Offer Double Auction, in which a seller will only be able to purchase the good if her offer is strictly less than p. On the other hand, this choice of k significantly simplify the analysis. As I show later, whenever the numbers of participants are different, the market will become a Buyer s bid or Seller s Offer double auction. Each entering buyer v with bids greater than p will get one unit of good and thus realize his utility as v p τ. If his bid is less than p, then he does not get the good and his utility is τ since he has to incur the participation cost up-front. Similarly, each entering seller 9

10 c s utility will be either p c τ or τ depending on whether her bid is less or greater than p. Those buyers and sellers choosing not to enter the market will receive zero payoff. Given the possibility of multiple equilibria in the second stage k-double auctions, one equilibria will be randomly selected from the pool of such equilibria. 3 Bilateral Double Auction: kinked equilibrium strategy I first consider the classical one-on-one double auction. In this case, there is one buyer and one seller interested in entering the trading market. Each buyer and seller s strategy consists of two components: first, whether to enter the market; second, upon entering, the amount to bid or offer. In this case, whenever both buyer and seller enter the market, the market price will be determined by a 1 2- double auction. That is, the market price is the average of buyer s bid and seller s offer if buyer bids higher than the seller s offer. If there is no participation cost and buyer and seller s value and cost are distributed uniformly, both buyer and seller will enter the market and one pair of equilibrium strategies derived by Chatterjee and Samuelson (1983) 5 : 5 Algebraically, they are: B(v) = 2 3 v v 1 4 v v < 1 4 S(c) = 2 3 c c 3 4 c c >

11 λ 3 4 S(c) B(v) c,v Figure 2: 1 1 double auction without participation cost Figure 2 shows an equilibrium in one on one double auction without participating cost. λ denotes buyer s bid or seller s offer. B (v) denotes buyer s bidding strategy as a function of its value and S (c) denotes seller s offering strategy as a function of its cost. It is obvious from Figure 2 that those equilibrium strategies are strictly increasing. This property is critical in the implementation of the bilateral trading mechanism via the direct revelation approach since each bid corresponds to a unique type. Myerson and Satterthwaite (1983) also prove that this equilibrium is incentive efficient. However, there is a positive measure of buyers and sellers who submit bids/offers with zero probability of winning, which accounts for 25% of the total measure of traders. Had there been a participation cost, they would not have entered the market. Now suppose there is a small participation cost. We need to consider four cases 6 depending on the number of buyer and seller present in the market, among which the only meaningful case is the one when both the buyer and the seller enter the market. 6 The cases are as follows: no buyer and seller are present; only one buyer but no seller or only one seller but no buyer enter the market; one buyer and one seller enter the market. 11

12 Lemma 1 There is no strictly increasing pure strategy equilibrium in the bilateral double auction with a small positive participation cost τ. Proof. Suppose not. Let q v and q c denote the minimum value buyer and maximum value seller that enter the market. Since buyer with value v =0will not have positive gain from and trade, we must have q v > 0 Similarly q c < 1 Let b(v) denotes buyer s equilibrium strategy and we know b(v) is increasing for v q v. Similarly let s(c) denote seller s equilibrium strategy and s(c) is also increasing for c q c. Consider the buyer q v first. Let U(v) denote buyer s expected gain from trade. Since buyer q v is the lowest buyer entering the market, we must have U(q v )=τ>0 (4) That is, there exists a positive measure of seller whose bids are smaller than b(q v ). However, in equilibrium, the seller with cost 0 knows the minimum value buyer that enters the market and his strategy b(v). Thus this seller will bid as large as the minimum bid from the buyer. Since the participation cost is paid before they enter the market, thus once the buyer enter the market, the participation cost is sunk. Therefore, even if the buyer with minimum value meets a seller bidding at his level (or minus ), he still would like to accept the bid. Since both buyer and seller s strategy must be monotone increasing, the set of sellers bidding less than or equal to b(q v ) has measure zero. In this case, the minimum value buyer s expected gain from trade will be zero. This contradict the condition (4). Hence this cannot be an 12

13 equilibrium. Similar argument for seller with cost q c. The above Lemma shows that for the existence of the equilibrium, there must be a positive measure of seller who bids at the lowest value buyer s bid and a positive measure of buyers who bids at the highest cost seller s bid. Graphically, this requires the buyer s strategy has a flat on the top and seller s strategy has a flat on the bottom. Suppose entering buyer s value has support on [q v, 1] and entering seller s cost has support on [0,q c ], then the following theorem derives an equilibrium with increasing strategies based on an equilibrium with no participation cost. Claim 1 Assume [b (v),s(c)] is a pair of strictly increasing strategy for the 1 1 double auction without participation cost and v and c satisfy b (v) =v and s (c) =c. Then for a bilateral double auction with participation cost τ, q v v q c c and the following modified strategies constitute a Nash equilibrium B (v) = S (c) = s (q c ) 1 v b 1 (s (q c )) b (v) q v v b 1 (s (q c )) b (q v ) c s 1 (b (q v )) s (c) 0 c s 1 (b (q v )) where in this case q c and q v satisfies U (q v ) = τ H (q c ) = τ 13

14 where U ( ) and H ( ) are the buyer and seller s equilibrium expected utility respectively. Proof. The proof is routine by checking the first order conditions. It is omitted but available upon request. Chatterjee and Samuelson (Theorem 3, 1983) obtains a similar theorem in terms of asymmetric support only. The following figure shows a typical equilibrium when both buyer s value and seller s cost are distributed uniformly. λ S(c) B(v) c * q v q c v * c,v Figure 3: 1 1 double auction with a participation cost Theorem 1 shows that the equilibrium strategy in a bilateral double auction with a small participation cost are kinked. However, it is not the case that in multilateral double auction all strategies have to be kinked. In Section 5 I derive an example with strictly increasing strategies for a 2 2 double auction admitting a small participation cost. 14

15 4 N N Double Auction We come to the general case. Suppose there are N (N 2) potential buyers and N potential sellers interested in entering the market. Unlike one-on-one double auction, it is not possible to produce an equilibrium with closed form strategies in multilateral double auction. Further, the existence of equilibrium with increasing strategies is itself a challenging problem which has recently studied in several papers like Jackson and Swinkels(2001) and Fudenberg et.al. (2003). Especially, the second stage of our model can be viewed as a combination of three types of double auctions: Buyer s bid double auction (k =0), Seller s Offer double auction (k =1) and 1 2 double auction k = 1 2. For the first two types, Williams (1991) obtains an existence result. While for the general double auctions, Jackson and Swinkels (2001) prove the existence of symmetric equilibrium. They start with mixed strategies and purify them later. To directly tackel the equilibrium with pure strategies, Fudenberg, Mobius and Szeidl (2003) provide the existence conditions of large double auctions. We restrict ourselves to the study of equilibrium with continuous increasing strategies for two reasons: first, this is the most interesting equilibrium especially for its smoothness; Second, it is very difficult to track down step function equilibriums. Even in one-onone double auction, Leininger, Linhart and Radner (1989) shows the computation is too formidable to produce any meaningful results. The following lemma shows the cutoff strategy indeed constitute a Nash equilibrium. Lemma 2 If second stage is modeled as a k-double auction, then the following entry strategy is a Nash equilibrium: Let U ( ) and H ( ) denote the buyer and seller s expected payoff respectively. Buyers will enter the market if and only if v q v and sellers will enter the market if and only if 15

16 c q c, where U (q v ) = τ H (q c ) = τ Proof. Let us consider buyer v. Suppose all other buyers and sellers are using the above entry strategy. If v q v, he will enter the market since his expected payoff is at least τ while he will receive zero payoff if he chooses not to enter. If v<q v, he will enter the market if and only if his expected payoff is at least τ. But this is impossible since otherwise q v can mimick buyer v 0 s strategy and achieve higher payoff. Similarly arguments apply to the seller q c. If the participation cost τ>0, we must have q v > 0 q c < 1 since buyers with value 0 and sellers with cost 1 will not be able to earn any positive payoff in this market. Given those observations, there are two immediate implications once we introduce a positive participation cost. First, the entering buyer and entering seller s supports are asymmetric. Sellers with cost less than q v understand that there is no entering buyer having value less than their cost and buyers with value greater than q c realize that there is no entering seller having cost greater than their value. This asymmetry might provide some market power to those advantageous traders. For example, if the lowest value buyer tell the truth by bidding q v, then the sellers with cost less than q v need not tell the truth if they are guaranteed of 16

17 trading by offering q v. Second, even we start with a symmetric market where the number of potential buyers is equal to that of potential seller, we might still end up with an asymmetric market since not every potential trader will enter the market. Both the lower bound of entering buyer s support (q v ) and the upper bound of entering seller s support (q c ) are function s of N. Specifically, suppose m buyers having value no less than q v and n sellers having cost no greater than q c and they choose to enter the market. Lemma 3 If m<n,then truth telling is a dominant strategy for the entering buyers; If m>n,then truth telling is a dominant strategy for the entering sellers; Proof. If m<n,then k =1. That is, the market price is determined by the m th lowest bid/offer. A buyer will obtain one unit of goods if and only if his bid is strictly greater than the price p; thus whenever the buyer succeeds in trading, he cannot affect the price. On the other hand, truth telling could maximize his probability of winning and it is therefore a dominant strategy for entering buyers. Similarly, if m > n, then k =0. Any seller will succeed in selling her unit of goods if and only her offer is strictly less than (m +1) th offer/bid which she can not affect. Hence, she is willing to tell the truth too. Lemma 4 Suppose (B (v),s(c)) is a pair of continuously increasing equilibrium strategies. If m<n,then in the resulting seller s bid double auction S (q c )=q c. If m>n,then in the resulting buyer s bid double auction B (q v )=q v. 17

18 Proof. Suppose m<n. First, we know the seller with cost q c will have zero expected utility since her probability of trade is zero given the insufficient demand in an equilibrium of increasing strategies. Suppose S (q c ) >q c then since the equilibrium strategy S (c) is continuous over c, thus there exists a positive measure of sellers with cost c<q c but S (c) >q c. Thus the seller q c could achieve positive probability of trade by lowering her offer below S (q c ) by ε such that S (q c ) ε>q c By doing this, the seller q c could achieve positive expected utility even in the case of m < n since it is a positive probability event that the market price is above S (q c ) ε. This contradicts that S ( ) is an equilibrium strategy. Thus we must have S (q c )=q c. The case when m>ncan be proved similarly. Since participation cost is sunk, once buyers and sellers enter the market, their equilibrium strategies will be defined as a m nk-double auction with buyers value distributed on [q v, 1] with distribution function F (v) and sellers cost distributed on [0,q c ] with distribution function G (c), where F (v) = F (v) F (q v) 1 F (q v ) G (c) = G (c) G (q c ) Assumption 1 1 F (q v )=G (q c ) This assumption says in any equilibrium, the expected number of traders must be equal 18

19 on two sides. In later section we will show this assumption must be true for large N almost surely. We introduce this assumption here to make our analysis more tractable. Generally speaking, obtaining a closed form expression over buyers and sellers utility is averydifficult task since it involves multiple integral of unknown functions 7, although in Section 5 I characterize a linear strategic equilibrium for 2 2 case with a small participation cost. This difficulty prevents us from directly characterizing the sellers and buyers s entry decision. On the other hand, intuition tells that if all buyers and sellers are price taker and the market price is p τ where the participation cost is τ. Then only buyers with value at least as large as p τ + τ will enter the market and make the purchase. Similarly, only sellers with cost no larger than p τ τ will enter the market and sell her goods. This shows that the support of entering buyers and sellers might become separated. To circumvent the difficulty of direct calculation of expected utilities, we need to discuss two cases depending on whether the supports are overlapped or not. 4.1 Markets with overlapping supports Our goal is to study the rate of convergence to efficiency in this market. Although we start with a symmetric potential market with N buyers and N sellers,thepostentrynumbers of buyers and sellers probably are not the same. However, Lemma 3 shows those traders on the short side of the market will always tell the truth since they could never affect the market price. In this case, we only need to consider the seller s strategy when m<nand buyer s strategy when m > n.both strategies are discussed when m = n. Since there are N 2 different cases, we introduce a two-step strategy: first we fix short side of the market and study the expected rate of convergence in this market. After that, 7 Although RSW(1994) provides methods to trace down the equilibrium strategies, calculating expected payoff based on those numerically calculated equilibrium is another formidale task. 19

20 we derive the expected rate of convergence over the whole market. Let p denote the Walrasian price satisfying 1 F (p )=G (p ). Given Assumption 1, we know q v <p <q c Case 1: m<n λ S(c) B(v) c * q v q c c,v Figure 4: Equilibrium Strategies when m<n From Lemma 3, when m < n, in equilibrium all buyers will tell the truth. Now let us consider seller s equilbirium strategy. Suppose there exist a continuous piecewise differentiable equilibrium strategy S (c) for the seller. Then for any seller with cost c [0,q c ), given all other sellers are using strategy S (c) and all other buyers are using strategy B (v) =v, if S (c) is differentiable at c, thesellerc has the following first order condition: (λ c) (n 1) X n,m (λ) g (c) S 0 (c) + my n,m (λ) f (v) Z n,m (λ) =0 (5) X n,m (λ) the probability that bid λ lies between s (m 1) and s (m) in a sample of m buyers using strategy B and n 2 sellers using S; 20

21 Y n,m (λ) the probability that bid λ lies between s (m 1) and s (m) in a sample of m 1 buyers using strategy B and n 1 sellers using S; Z n,m (λ) the probability that bid λ lies between s (m 1) and s (m) in a sample of m buyers using strategy B and n 1 sellers using S; To understand equation (5), let us pick one seller. if seller c s offer λ lies between s (m 1) and s (m) among other m buyers and n 2 sellers, by raising her offer by 4λ, she might pass the selected seller with probability (n 1) X n,m (λ) g (c) S 0 (c) and lose the payoff λ c. On the other hand, pick one buyer, if seller c 0 s offer λ lies between s (m 1) and s (m) among other m 1 buyers and n 1 sellers, by raising her offer by 4λ, she might pass the selected buyer with probability my n,m (λ) f (v) and lose the payoff λ c. Hence her expected h i loss of payoff by raising her offer is (λ c) (n 1) X n,m (λ) g (c) S 0 (c) + my n,m (λ) f (v). On the other hand, by raising her offer by 4λ, she might not pass anybody s offer/bid with probability Z n,m (λ). In this case, she raises her payoff by 4λ by raising the market clearing price. However, this strategic path is only valid on the range of bids/offers that both sellers and buyers will use. That is to say, λ (q v, 1]. Although no buyers will go below q v, it is possible that some sellers will go below q v since sellers understand they outnumber buyers and competition force them to bid below q v. Let c bethecostthats (c )=q v. That is, seller s strategy S ( ) is kinked at c. This strategy is represented by Figure 3. The following theorem obtains a bound for seller s misrepresentation based on above analysis. Theorem 1 Let the number of entering buyers is given as m<n. Then if the number of sellers entering the market is n, then S (c) c is O 1 n. Note that from seller s first order condition, we must have S (c) c< Z n,m (λ) (n 1) X n,m (λ) g (c) S0 (c) 21

22 By following the similar strategy to the proof of Theorem 3.1 of Rustichini, Satterthwaite and Williams (1994), we can show the convergence result as stated in the theorem. There is one complexity to deal with: the posibility of kinked strategy. As shown in Figure 4, since the number of sellers is more than that of buyers, sellers have to compete for buyer s demand. This pressure will induce some sellers bidding below the lowest bid of buyers. However, once the offer goes below the lowest buyer s bid, the sellers are essentially competing with themselves. This change will cause kink at the equilibrium strategies. Let c denote the kinked point which is defined as S (c )=B(q v ). The proof discusses those two cases separately and show that they have the same convergence rate. To see why, note that given m entering buyers telling truth in equilibrium, the sellers are essentially competiting over themselves when choose the overbid according to their cost. It is this behavior cause them to tell the truth when number of fellow sellers gets large. Convergence to efficiency In the case of m<n,all buyers tell the truth and there exists η>0 such that S (c) c η n. We know missed trades only happen when buyer s value is greater than seller s cost but seller s offer is higher than buyer s bid. Given the bounds of seller s overbidding, we know those loss of efficiency due to seller s strategic behavior is also bounded by η n too. Given the continuity of the density function, we can bound the number of mistrades and this will give us a convergence rate of O 1 n of the total loss of efficiency. On the other hand, the total expected efficiency is at least of order O (m). Combining this two results, we will have the following Lemma: Lemma 5 For given n m with n>mthe post entry loss of efficiency is O 1 nm 22

23 Proof. The loss of efficiency is defined as the ratio of total loss of efficiency proportional to the total expected efficiency. The denominator is clearly O (m) since any combination of a buyer and a seller will produces a positive expected gain of trade. Hence we only need to prove the total loss of efficiency is O 1 n. Let t (m) be the cost/value of seller or buyer who post the m th smallest offer/bid. We will consider the loss of efficiency conditional on t (m). Note that in the case that c > 0, if t (m) <c, then s (m) S (c )=q v. In this case, there is no loss of efficiency since all buyers having bids above s (m) and will obtain the good sucessfully. Suppose t (m) >c, then by Lemma 1 S (c) c< η n for c [0, q c] (6) And we know B (v) v =0 From Theorem 1, t (m) η n s (m) t (m) + η n A buyer with value v and a seller with cost c inefficiently fail to trade only if B (v) < s (m) (7) and S (c) s (m) (8) and v > c (9) Statement (7) implies v s (m), (8) implies c s (m) η n, and together with (9) imply s (m) η n <c<v<s (m) (10) 23

24 The value v c of a missed trade is thus no more than η n. Hence we only need to bound the expected number of missed trades given t (m).from the above result, those missed trades must have cost and value satisfying t (m) 2 η n c<v t (m) + η n The expected number of missed trades conditional upon t (m) is bounded by the minimum of two expected numbers: one is the expected number of sellers with cost below t (m) but no smaller than t (m) 2 η n.the other is the expected number of buyers with cost above t (m) but no larger than t (m) + η n. Note that the conditional density of sellers having cost below t (m) is g( ) 1 G(t (m)) and the conditional density of buyers having value above f( ) t (m) is 1 F(t (m) ). Because t (m) (q v,q c ), and f and g are continuous, those two densities are bounded above by some number ζ(f,g) independent of m. Hence, conditional upon t (m), expected number of miss trades is bounded by min{n 2η n ζ (F, G),m2η n ζ (F, G)} 2ηζ (F, G) Case 2: m = n If the number of entering buyers equal to the number of entering sellers, then we know the market clearing price will be set by a 1 2 double auction. In this case, both buyers and sellers have incentive to misrepresent their values or costs. First, the probability of occurence of this case goes to zero as the number of potential entrants gets large. Lemma 6 The probability of equal number of traders is O 1 N. 24

25 Proof. We know Pr (m = n) = = NX µ µ N (1 F (q v )) m (F (q v )) N m N G (q c ) m (1 G (q c )) N m m m NX µ µ N (1 F (q v )) m (F (q v )) N m 1 N +1 G (q c ) m+1 (1 G (q c )) N m m (N +1)G (q c ) m m=1 m=1 1 (N +1)ω where ω = G (p ) <G(q c ) since is the unique Walrasian price p is less than q c. Since m = n, both buyers and sellers will have incentive to overbid or underbid. However, for the highest cost seller q c, making an offer of 1 is equivalent of quitting the market without trades. Similarly, the lowest buyer q v will never make a bid equal to 0. On the one hand, seller q c understand that if the equilibrium strategy is strictly increasing, she will not be able to earn any positive payoff even she tells the truth since her probability of trading is zero. However, if the equilibrium strategy is kinked as shown in Figure 4 below, then there exists a value v such that B (v )=S(q c ). It is possible for seller q c to earn positive payoff since it is a positive probability event that all buyers value are greater than v. 25

26 s(c) b = s(q) b(v) s=b(q v ) c * q v q c v * c,v Figure 5: Kinked Strategy when m = n The following Claim shows that this partially flat strategy is possible and necessary in terms of generating some positive payoff for the marginal buyers and sellers. Claim 2 In an m m (m >2)double auction with buyer s value v [q v, 1] and seller s cost c [0,q c ]. The following strategies are the unique form of continuous increasing strategies that admit a participation cost: B (v) = S (c) = b b (v) s s (c) v v v v q v c c q c c c 26

27 where b (v) and s (c) are defined through RSW method which are strictly increasing and b = b (v )=s (q c ) s = s (c )=b (q v ) Proof. The proof is routine by checking first order conditions. It is a generalized version of the proof of Claim 1. Intuitively, if m>2, any equilibrium strategy should be strictly increasing except at the lowest offer and highes bid. The flat parts at those two ends are essential for the highest cost seller and lowest value buyer to obatain positive payoff. Remark 1 Note that the equilibrium strategy defined in the above claim contains the strictly increasing equilibrium strategies which is the case when c = 0 and v = 1 if we do not require positive payoff for the lowest value buyer and highest cost seller. The next Lemma consider relevant sellers and buyers strategy. Those relevant traders are the ones whose bids/offers do not fall in the flat part of the equilibrium strategies. Those traders strategies are strictly increasing. We call them relevant traders since they are the ones that causes inefficiency since any seller bidding the lowest offer and any buyer making the highest bid will be able to trade for sure. For those relevant traders, they are facing the same problem: misrepresentation might bring them higher payoff if they succeed in trading and affect the trading price in their favor, it also will cost them positive payoff if they fail to trade due to misrepresentation. Similar pressure exists for those high cost sellers and low value buyers since they are less likely to trade. Following the similar strategy of the proof of Theorem 1, we can show the following Lemma which bound the misrepresentation of those traders. 27

28 Lemma 7 If the post entry market is an m m 1 2 double auction, then we must have S (c) c η 2m for all c (c,q c ) v B (v) η 2m for all v (q v,v ) Proof. If the equilibrium strategy chosen is strictly increasing over the support of entering buyers and sellers, then the proof is the same as part 1 of the proof of Theorem 1. On the other hand, if equilibrium strategy is kinked at c and v by invoking the similar argument of the part 2 of the proof of Theorem 1, we will have the result. The number 2 in the denominator comes from the assumption that when m = n, the market mechanism is a 1 2 double auction. Convergence to Efficiency Realizing that only those buyers and sellers whose bids/offers within the nonflat part of the equilibrium strategy will cause inefficiency and their strategy is bounded by Lemma 8, we can easily bound the lost of efficiciency in this case. Lemma 8 In an m m double auction, the post entry efficiency loss is at most O 1 m We relegate this proof to the Appendix since it is essentially the same as that of Lemma Now let s come back to the question asked in Section 1: will all buyers and seller converge to truth telling when the market gets large in the presence of a participation cost? The answer is probably not in the case of equal number of traders. Let us call the highest cost seller and lowest value buyer marginal buyers. When they know their marginal position in the sense no seller having cost higher than her and no buyer having value lower than him are in the market, their bargaining power are undermined which force them to restraint their misrepresentation. On the other hand, they really want flat strategy on the 28

29 other side since that is only possibility that will bring them positive payoff. Inconstrast of the literatures without participation cost, there must be some high value buyers and low cost sellers not telling the truth even the market is large. Fortunately, those trader s behavior will not cause loss of efficiency The Expected loss of efficiency With the result of three cases depending on the number of entrants, I can easily derive the main theorem of this paper which shows the loss of efficiency is converging to zero at a quardratic rate. Theorem 2 The loss of efficiency is O 1 N 2 in all markets with overlapping supports. Proof. This is proved by bounding the expected loss of efficiencies. Let L m,n denote the relative loss of efficiency when m buyers and n sellers enter the market. By the above results, we can assume that there is a χ>0 L m,n χ (n +1)(m +1) Thus let α =1 F (q v ) and β = G (q c ). Then µ N α n (1 α) N n n E (L m,n m n) X n m χ (n +1)(m +1) where ξ = < X m ξ (N +1) 2 χ (1 F (p ))G(p ). µ N m 29 χ (n +1)(m +1) α m (1 α) N m X n m µ N β n (1 β) N n n

30 Similarly E (L m,n m>n) < ξ (N +1) 2 Thus the expected efficiency loss is E (L m,n ) < ξ (N +1) 2 The essential idea of double auction market with overlapping support is that when the market gets large, competitive pressure force those relevant buyers and sellers to tell the truth, especially for those marginal buyers and sellers whose bargaining power are weakened by the nonpresence of those zero probability traders. Comparison with Vickerey Double Auction Yoon (2001) defines a modified Vickerey double auction that force everybody to tell the truth which is similar to the two price double auction defined by McAfee (1992). One problem of this two price mechanism is that seller gets more and buyers pay less which result in a loss for the market operator. Hence participation fee is collected to cover market operator s loss. Yoon derived a convergence rate of 1 N while our simple market mechanism gives an rate of 1 N 2 everybody to tell the truth. without requiring 4.2 Markets with Non-overlapping supports Market efficiency when the supports are separated In the last section we study the interim efficiency when the supports of entering buyers and sellers are not separated. It turns out that those traders pressure of losing trade induce them to truth telling. When the support of entering buyers and sellers become non- 30

31 overlapped, for those traders who enter the market, each buyer can trade with any seller and each seller can trade with any buyer. The k-double auction is becoming essentially a pure bargaining game. In terms of the Interim II efficiency, however, there exists equilibria that there is no loss of efficiency if the supports are not overlapped. Lemma 9 If m = n and q v >q c, then there exists infinitely many equilibrium strategies as follows B (v) = λ S (c) = λ where λ [q c,q v ]. We call those strategies as fixed price strategies. Proof. For a specific buyer v that enter the market, when m = n, biding less than λ will render loss of trade and bidding anything above λ will increase his probability of trading. Hence bidding λ is a best response when other buyers and sellers are using this fixed price strategy. Similarly we can prove the result in the sellers side. Theorem 3 If in the post entry market, the entering buyers and sellers supports are not overlapped, assume when m = n, buyers and sellers will employ fixed price strategies defined in Lemma 9, then there is no loss of efficiency in the post entry market. Proof. Note that when m = n, all possible gain of trade are realized. Hence it suffices to consider the cases of m<nand m>n. If m<n,lemma 3 tells us that no buyers will bid below q v. On the other hand, Lemma 4 tells us that no sellers will bid above q c. Hence all buyers bid are above all seller s offer. Since the seller s equilibrium strategy is increasing, the largest possible gain of trade are realized and thus there is no efficiency loss. 31

32 Similarly, we can have the same result when m>n. There might be other equilibria in the case of m = n, but the fixed price equilibrium maximizes the expected trading probability of each trader since each trader is certain to trade in this case. Hence it is always selected The Occurrence of Non-overlapping supports In this section, I show that given any positive participation cost, the participation ratio is decreasing in the potential market size until the supports become non-overlapped. Now let s consider some conditions that must be satisfied in any equilibrium with non trivial marginal buyers and sellers. In any equilibrium, for the lowest value buyer to earn a positive gain from the market, there are at least two conditions to satisfy: 1. There exist a positive measure of sellers who would trade with the lowest value buyer 2. It is a positive probability event that the trading price will be no greater than the bid of lowest value buyer. In the proof of Lemma 1, the increasing strategy assumption violate condition 1. If the number of buyers is greater than that of sellers almost surely, than any market clearing price will be greater than the bid of lowest value buyer since buyers are competing for sellers short supply. Hence, we have the following Lemma. Lemma 10 For a double auction with a small positive participation cost, no equilibrium exists if the number of buyers entering the market is strictly less or greater than that of sellers almost surely. Proof. Suppose the number of entering buyers is strictly greater than that of sellers 32

33 almost surely, then for any fixed participation cost, buyer q v will have to tell the truth almost surely. This means he will have zero expected payoff almost surely and therefore he will not enter the market. Thenextlemmageneralizestheabovelemmainthesensethatforlargemarket,buyer and sellers will enter the market if and only if the expected demand is equal to the expected supply. Intuition tells that buyers and sellers will only want to enter the market if they can guarantee themselves a positive expected gain if they have to pay a participation cost. However, as the number of market participants gets large, the market clearing price will converge to Walrasian price and this convergence is relatively fast. Hence for those buyers with value less than the Walrasian price and sellers with cost greater than Walrasian price, their primary gain will come from the potential market asymmetry which allow them to maintain some positive expected gain. Nevertheless, if q v q c, and individual rationality will require b(q v ) q v and s(q c ) q c, hence buyer q v and seller q c won tbeabletotradeif the market price is close to Walrasian price. On the other hand, as the number of market participants gets large, the possibility of getting enough seller with cost less than q v and enough buyer with value greater than q c is decreasing fast. Hence, if the buyers and sellers have to pay a small but positive amount of participation fee, as the market gets large, those marginal buyer and sellers expected gain from market asymmetry will be less than the participation fee ultimately unless they can guarantee themselves a trading probability of one, in which case we must have q v >q c. The following two Lemmas are needed for the proof of Theorem 4. Lemma 11 shows that the expected number of trader on each side must equal when N gets large while Lemma 12 provides a probability bound for the difference in the number of entering buyers and sellers. Their proofs are relegated to the Appendix. 33