An Efficient Incentive-Compatible Combinatorial ... - Semantic Scholar

An Efficient Incentive-Compatible Combinatorial Market Mechanism Rahul Jain and Pravin Varaiya EECS Department University of California, Berkeley {rjain,varaiya}@eecs.berkeley.edu

Abstract Service providers lease bandwidth from owners of individual links to form desired routes. Bandwidth is leased in indivisible amounts, say multiples of 100 Mbps. We study the interaction between buyers and sellers of bandwidth. Within a conventional market, we showed in [8] that a competitive equilibrium exists if agents’ utilities are linear in bandwidth (and money) and they truthfully reveal them, and if the desired routes form a tree. However, strategic agents will not be truthful and the competitive equilibrium will not be realized. To ensure a good outcome among strategic agents, we propose a combinatorial double auction and examine the resulting Nash equilibria. We show that such an equilibrium exists and, more surprisingly, the resulting allocation is efficient. In reality, the players have incomplete information and are risk averse, so we consider the Bayesian-Nash equilibrium under the ex post individual rationality constraint. We show that the mechanism is asymptotically Bayesian incentive-compatible and hence asymptotically efficient. Surprisingly, without the ex post individual rationality constraint, the BayesianNash equilibrium strategy for the buyers is to bid more than their true value.

I. I NTRODUCTION We study the interaction among internet service providers who lease bandwidth from owners of individual links to form desired routes. Bandwidth is traded in indivisible amounts, say multiples of 100 Mbps. We study the interaction in two structured settings. The setting of a conventional market economy, in which the interaction results in a competitive equilibrium was considered in [8]. It was shown that such an equilibrium exists if agents’ utilities are linear in bandwidth (and money) and they truthfully reveal them, and the desired routes form a tree. The latter requirement is needed for the existence of an equilibrium in the presence of bandwidth indivisibility. Strategic agents, however, have an incentive not to be truthful. We propose a ‘combinatorial sellers’ bid double auction’ (c-SeBiDA) mechanism that achieves a socially desirable interaction among strategic agents. It is a double auction, because both buyers and sellers make bids. It is combinatorial because buyers make bids on several links that form a route, and they must receive the same bandwidth on all links. Sellers, however, offer to sell bandwidth on individual links. The mechanism takes all buy and sell bids, solves a mixed-integer program that matches bids to maximize the social surplus, and announces prices at which matched bids are settled. The settlement price for a link is the highest price asked by a matched seller (hence ‘sellers’ bid’ auction). Strategic behavior in the auction is modelled as a Nash equilibrium. It is shown that under full information a Nash equilibrium exists; it is not generally a competitive equilibrium. Nevertheless, the Nash equilibrium is efficient. Moreover, it is a dominant strategy for all buyers and for all sellers except the highest-ask price seller to be truthful. In an auction setting, players have incomplete information. We therefore, consider the BayesianNash equilibrium of the auction game. We show that when the players play only ex post individual Research supported by DARPA contract F 30602-01-2-0548 and the National Science Foundation grant ANI-0331659.

rational strategies, symmetric Bayesian-Nash equilibrium strategies converges to truth-telling as the number of players becomes very large. Previous Work and Our Contribution Several papers study bandwidth allocation through a market mechanism [11], [13], [16], [17], [18], [19]. In the typical formulation, all available bandwidth is supplied inelastically and bandwidth consumers have a concave utility function. Welfare (consumer surplus) is maximized subject to the constraint that the bandwidth consumed on each link is less than the link capacity. At the optimal allocation, the Lagrange multipliers associated with each link capacity constraint can be interpreted as prices, and the optimal allocation together with these prices form a competitive equilibrium. For several reasons a competitive equilibrium is not a good prediction of bandwidth allocation. First, calculation of the welfare-maximizing allocation above requires centralized knowledge, although one can construct a decentralized calculation. In such a construction, an auctioneer proposes hypothetical prices to which buyers and sellers react, and based on this reaction, the auctioneer adjusts prices. Under fairly stringent conditions, this process of price adjustment converges to the competitive equilibrium prices. While no one seriously proposes implementing such a mechanism to allocate bandwidth in the internet, there are fruitful analogies between this iterative price process and some TCP protocols, see e.g. [16]. In this analogy, the delay experienced by a TCP flow serves as a surrogate price signal. Second, if bandwidth is traded in indivisible amounts as in actual markets [3], [15], a competitive equilibrium may not exist. However, it was shown in [8] that if utilities are linear in bandwidth and money, and if the routes that buyers desire form a tree, a competitive equilibrium does exist. The proof uses the fact that under the tree condition, the extreme points of the convex set of allocations are integer-valued. Third, and most importantly, the realization of the competitive equilibrium requires agents to truthfully report their utilities. While we showed in [7] that a competitive equilibrium exists, it is easily seen (example 4 in [8]) that strategic agents (who are aware of their ‘market power’) will not be truthful. Many proposals of auction mechanisms are designed to elicit socially desirable outcomes from the interaction of strategic agents, stemming from Vickrey’s fundamental contribution [27]. Attention has focused on single-item auctions [14]. In bandwidth trading, however, a buyer demands bandwidth in several links that constitute the desired route. A natural way to incorporate this demand is by combinatorial bids, in which a buyer’s bid is for a bundle of items and the auction mechanism must accept or reject the bid in its entirety. Combinatorial bids arise in many contexts, and so a growing body of research is devoted to combinatorial auctions [28]; the interplay between economic, game-theoretic and computational issues has sparked interest in algorithmic mechanism design [22]. Some iterative, ascending price combinatorial auctions achieve efficiencies close to the Vickrey auction. Some of them rely on approximation of the integer program [21] while others rely on linear programming [2]. There is little prior work on combinatorial double auctions. Yoon presents a modified Vickrey double auction [30] related to the k-double auction, while [4] considers truthful double auction mechanisms and obtains upper bounds on the profit of any such auction. But the setting for both papers is non-combinatorial in the sense that each bid is for an individual item. Ours is apparently one of the first proposals for a combinatorial double auctions mechanism. The rest of this paper is organized as follows. In Section II we present the combinatorial Seller’s Bid Double Auctions (c-SeBiDA) mechanism. In Section III we prove that under full information, the auction has a Nash equilibrium that is economically efficient, although it may not be a competitive equilibrium. In Section IV we show that when the players have incomplete information, a Bayesian-Nash equilibrium strategies for the mechanism with a single link under

the ex post individual rationality constraint converge to truth-telling as the number of players becomes sufficiently large. We end with some concluding remarks. II. BANDWIDTH A LLOCATION USING C OMBINATORIAL D OUBLE AUCTIONS We propose a combinatorial double auction mechanism for bandwidth trading. A buyer places buy bids for bandwidth along his route. A buyer’s bid is combinatorial or ‘all or nothing’: he must receive the same bandwidth on every link along the route or nothing at all. On the other hand, sellers make separate ask bids on individual links. The mechanism collects all announced bids, matches a subset of these to maximize the ‘surplus’ ((1), below) and declares a settlement price for bandwidth on each link at which the matched buy and ask bids—which we call the winning bids—are transacted. This constitutes the payment rule. As will be seen, each matched buyer’s buy bid is larger, and each matched seller’s ask bid is smaller, than the settlement price, so the outcome respects individual rationality. There is an asymmetry: buyers make multi-link combinatorial bids, but sellers make singlelink bids. This permits the mechanism to select uniform and anonymous settlement prices for each link. Agents’ bids may not be truthful. They know how the mechanism works and formulate their bids to maximize their individual returns. Since we examine Nash equilibria, we sometimes call an agent a player. A player can make multiple bids. The mechanism treats these as XOR bids, so at most one bid per player is a winning bid. Therefore the outcome is the same as if a matched player only makes (one) winning bid. Thus, in the formal description of the combinatorial Sellers’ Bid Double Auction (c-SeBiDA), each player places only one bid. c-SeBiDA is a ‘double auction’ because both buyers and sellers bid; it is a ‘sellers’ bid’ auction because the settlement price depends only on the matched sellers’ bids, as we will see. Formal mechanism Consider a network with links 1, · · · , L. Buyer i has (true) reservation value vi ∈ [0, V ] per trunk along route Ri , and submits a buy bid of bi per trunk and demands δi ∈ [0, D] trunks along the route. Seller j has (true) per trunk cost cj ∈ [0, C] and offers to sell up to σj ∈ [0, S] trunks in link lj at a unit price of aj . Denote Lj = {lj }. The mechanism receives all these bids, and matches some buy and sell bids. The matches are described by 0-1 variables xi , yj : xi is 1, if buyer i’s buy bid is accepted, 0 otherwise; similarly, yj is 1 if seller j’s ask bid is accepted, 0 otherwise. The mechanism determines the allocation (x∗ , y ∗ ) as the solution of the surplus maximization problem MIP: P P max xi bi − j yj aj (1) i x,y P P s.t. I ∈ Lj ) − i xi1(l I ∈ Ri ) ≥ 0, j yj1(l ∀l ∈ [1 : L], xi ∈ [0 : δi ], ∀i, yj ∈ [0, sj ]∀j. MIP is a mixed integer program: The buyer i’s bid is matched up to his maximum demand δi ; Similarly, seller j’s bid will also be matched up to his maximum supply σj . yj∗ will be integral since x∗i can only take integral values and the demand less than equal to supply constraint will be an equality. The settlement price is the highest ask bid among matched sellers, pˆl = max{aj : yj∗ > 0, l ∈ Lj }.

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The payments are determined by these prices. Matched buyers pay the sum of the link prices along their route; matched sellers receive a payment equal to the number of trunks sold times the price for the link. Unmatched buyers and sellers do not participate. This completes the mechanism description. P If i is a matched buyer (x∗i = 1), his bid bi ≥ l∈Ri pˆl ; for otherwise, the surplus (1) can be increased by eliminating this bid. Similarly, if j is a matched seller (yj∗ > 0), and l ∈ Lj , his bid aj ≤ pˆl , for otherwise the surplus can be increased by eliminating his bid. Thus the outcome of the auction respects individual rationality. It is easy to understand how the mechanism picks matched sellers. For each link j, a seller with lower ask bid will be matched before one with a higher bid. So sellers with bid aj < pˆl sell all their bandwidth (yj∗ = 1). At most one seller with ask bid aj = pˆl sells a fraction of his bandwidth (yj∗ < 1). On the other hand, because their bids are combinatorial, the matched buyers are selected only after solving MIP. III. NASH EQUILIBRIUM ANALYSIS : C -S E B I DA IS EFFICIENT Our focus is on price determination, so we will assume that the quantities in players’ bids (namely, {δi , σj }) are fixed. A strategy for seller i is a buy bid bi , a strategy for buyer j is an ask- bid aj . Let θ denote a collective strategy. Given θ, the mechanism determines the allocation (x∗ , y ∗ ) and the prices {ˆ pl }. So the payoff of buyer i and seller j are, respectively, X ubi (θ) = x∗i δi (vi − pˆl ), (3) usj (θ)

=

X

yj∗ σj (

l∈Ri

pˆl1(l I ∈ Lj ) − cj ).

(4)

l

The bids bi , aj may be different from the true valuations vi , cj , which however figure in the payoff calculations above. A collective strategy θ∗ is a Nash equilibrium if no player can increase his payoff by unilaterally changing his strategy. Single link, SeBiDA We study the one-link version SeBiDA, of c-SeBiDA. We construct a Nash equilibrium, and show it yields a unique and efficient allocation. The proof clarifies the more complex construction in the combinatorial case (Theorem 2). Each buyer bids for at most one unit of capacity, and each seller sells at most one unit of capacity on the single link (so δi , σj equal 1 in (3), (4)). There are M buyers and N sellers, whose true valuations and costs lie in [0, 1]. To avoid trivial cases of non-uniqueness, assume all buyers have different valuations and all sellers have different costs. The mechanism finds the allocation (x∗ , y ∗ ) that is a solution of the following integer program IP: P P max bi xi − j aj yj i x,y P P s.t. i xi ≤ j yj , xi , yj ∈ {0, 1}. As in (2) the settlement price is pˆ(b, a) = max{aj : yj∗ > 0}. It is trivial to find (x∗ , y ∗ ): We repeatedly match the highest unmatched buy bid with the lowest unmatched sell bid if the buy bid is greater than the sell bid.

Theorem 1: (i) A Nash equilibrium (b∗ , a∗ ) exists for the SeBiDA game. (ii) Except for the matched seller with the highest bid on each link, it is a dominant strategy for each player to bid truthfully. The highest matched seller bids min{v, c}, in which c is the true reservation cost of the unmatched seller with lowest bid and v is the reservation value of the matched buyer with the lowest bid. (iii) The equilibrium allocation is efficient. Proof: Set a0 = c0 = 0, b0 = v0 = 1. Order the players so v1 ≥ · · · ≥ vM and c1 ≤ · · · ≤ cN . Let k = max{i : ci ≤ vi }. We will show that the set of strategies, ∀i, bi = vi ; ∀j 6= k, aj = cj ; ak = min{ck+1 , vk }, is a Nash equilibrium. The first k buyers and sellers are matched and the settlement price is pˆ = ak . Consider a matched buyer i ≤ k. This buyer has no incentive to bid lower, since by doing so he may be able to lower the price but then he will also become unmatched; since he is already matched, he certainly will not bid higher. Consider an unmatched buyer i > k. He has no incentive to bid lower, as he will remain unmatched. He can become matched by bidding above ak , but then, if he does get matched, his payoff will be negative. Consider an unmatched seller j > k. He has no incentive to bid higher, as he will remain unmatched. He can get matched by bidding lower than ak , but since his cost is aj > ak his payoff will be negative. Consider a matched seller j < k. By bidding lower, this seller will not improve his payoff. If he bids higher to increase the settlement price, this will happen only if he bids above ak , but then he will become unmatched. Lastly, consider the ‘marginal’ matched seller, k. He will not bid lower, as that will decrease his payoff. If he bids more than ak , his bid will exceed either bk or ak+1 , and in either case he will become unmatched. This proves (i), (ii). This Nash equilibrium yields the allocation (x∗ , y ∗ ) which matches buyers with highest valuation and sellers with least cost. Hence it is efficient. We now prove uniqueness. Suppose (˜ x, y˜) is another Nash equilibrium. Suppose that the two Nash equilibria differ in allocation to a buyer who goes from being matched in the first allocation (x∗ , y ∗ ), to being unmatched in the second allocation (˜ x, y˜). Then, either there is another buyer who goes from being unmatched to matched, or there is a seller who also goes from being matched to unmatched. Thus, we can see that as we go from (x∗ , y ∗ ) to (˜ x, y˜) one of four cases must occur: (i) An unmatched buyer i1 is made matched and a matched buyer i2 is made unmatched; (ii) An unmatched seller j1 is made matched and a matched seller j2 is made unmatched; (iii) An unmatched buyer i and unmatched seller j are made matched; (iv) A matched buyer i and seller j are made unmatched. Case (i) We must have vi1 < vi2 and the new bids must satisfy ˜bi2 < ˜bi1 . But then either i2 ’s payoff is negative or i1 can also bid just above i2 ’s bid. In either case (˜ x, y˜) cannot be a Nash equilibrium. Case (ii) An argument similar to that for case (i) shows that (˜ x, y˜) cannot be a Nash equilibrium. Case (iii) Since both are unmatched in the first allocation, it must be that vi < cj . Since both are matched in the second allocation, it must be that bi > aj , so that one of them must have a negative payoff. Again, (˜ x, y˜) cannot be a Nash equilibrium. Case (iv) An argument similar to that for case (iii) shows that (˜ x, y˜) cannot be a Nash equilibrium. Above, we constructed a Nash equilibrium for the game described by (1)-(4) in the case of a single link. The result can be extended to multiple links.

Theorem 2: (i) A Nash Equilibrium (b∗ , a∗ ) exists in the c-SeBiDA game. (ii) Except for the matched seller with the highest bid on each link, it is a dominant strategy for each player to bid truthfully. (iii) The Nash equilibrium outcome (x∗ , y ∗ ) is always efficient. The proof is omitted due to space constraints and can be found in [8]. Although it yields an efficient allocation, the Nash equilibrium is not a competitive equilibrium as the following example shows. Example 1: Consider a single link with two trunks, one buyer who demands one trunk, with valuation b = 1, one seller with cost a = 0. The Nash equilibrium settlement price is pˆ = 1, but the competitive equilibrium price is 0, because there is excess capacity.

IV. S E B I DA IS A SYMPTOTICALLY BAYESIAN I NCENTIVE C OMPATIBLE We now consider the incomplete information case and analyze the SeBiDA market mechanism in the limit of large number of players. We will only consider n ≥ 2, where n is the number of buyers as well as the number of sellers. By adding dummy agents, the results can be extended to the case when the market is not symmetric. Suppose c ∼ U1 and v ∼ U2 are such that the corresponding pdfs u1 and u2 have full support [0, 1]. Let α : [0, 1] → [0, 1] denote the strategy of the sellers and β : [0, 1] → [0, 1] denote the strategy of the buyers. We will consider symmetric equilibria, i.e., equilibria where all buyers ˜ use the same strategy β and all sellers use the same strategy α. Let α ˜ (c) = c and β(v) =v denote the truth-telling strategies. Under strategies α and β, we will denote the distribution of ask-bids a and buy-bids b as F and G respectively. We will denote [1 − F (x)] by F¯ (x). Under ˜ F = U1 and G = U2 . We will consider only those bid strategies which satisfy the ex α ˜ and β, post individual rationality constraint, i.e., α(c) ≥ c and β(v) ≤ v. Denote X = {α : α(c) ≥ c} and Y = {β : β(v) ≤ v}. We will only consider single unit bids and assume that a symmetric Bayesian-Nash equilibrium exists. Theorem 3: Consider the SeBiDA auction game with (α, β) ∈ X × Y, i.e., both buyers and sellers have ex post individual rationality constraint. Let (αn , βn ) be a symmetric Bayesian Nash ˜ equilibrium with n buyers and n sellers. Then, (i) βn (v) = β(v) = v ∀n, and (ii) (αn , βn ) → ˜ as n → ∞, i.e., SeBiDA is asymptotically Bayesian incentive compatible. (˜ α, β) We will prove this in the form of two lemmas below. Lemma 1: Consider the SeBiDA auction game with n buyers and n sellers. Suppose the sellers use bid strategy α with f (a), the pdf of its ask-bid under strategy α. Then, the best-response strategy of the buyers βn satisfies βn (v) ≥ v for all n. Proof: Fix a buyer j with valuation v. Suppose sellers use a fixed bidding strategy α and denote the buyers best-response bidding strategy by βn . Consider the game denoted G −j , where all players except buyer j participate and bid truthfully. Denote the number of matched buyers and sellers by K = sup{k : a(k) ≤ b(k) }, which is a random variable. Here a(k) denotes the order statistics increasing with k over the ask-bids of the participating sellers and b(k) the order statistics decreasing with k over the buy-bids of the participating buyers. Denote X = a(K) , Y = a(K+1) and U = b(K) . Then, when buyer j participates and bids b = β(v), it gets a payoff ( v − X, if X < U < b and U < Y ; πj0 (b) = (5) v − Y, if X < Y < b and Y < U.

Its expected value denoted by π ¯j0 satisfies the differential equation ¶ ¶ µ n−1 µ X d¯ πj0 n − 1 ¯k n−1 k n−1−k ¯ = F (b)F (b)nf (b) G (b)Gn−1−k (b) (v − b) (6) db k k k=0 ¶ µ ¶ Z bX n−1 µ n−1 n − 2 ¯ k−1 k−1 n−k ¯ + F (x)nf (x)F (b) G (b)Gn−1−k (b)(n − 1)g(b) (v − x) dx, k − 1 k − 1 0 k=1 with the boundary condition π ¯j0 (0) = 0. The first term above arises from the change in payoff when U > Y > b > X, and b + ∆b > Y . Similarly, the second term is the 0change in payoff d¯ π when Y > U > b > X and b + ∆b > U . It is clear from (6) that for b ≤ v, dbj > 0. Given that the sellers play strategy α, the best-response strategy of the buyers, βn is such that b = βn (v) d¯ π0 and dbj = 0. From this it is clear that b = βn (v) ≥ v, ∀n.

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This implies that under the ex post individual rationality constraint, the buyer always uses the ˜ strategy βn = β. Lemma 2: Consider the SeBiDA auction game with n buyers and n sellers and suppose the buyers bid truthfully, i.e., βn = β˜ and let αn be the sellers’ best-response strategy. Then, ˜ → (˜ ˜ as n → ∞. (αn , β) α, β) Proof: Fix a seller i with cost c. Consider the auction game, denoted G−i , in which seller i does not participate and all participating buyers bid truthfully. As before, denote the number of matched buyers and sellers by K = sup{k : a(k) ≤ b(k) }, U = b(K) , the bid of the lowest matched buyer, W = b(K+1) , the bid of the highest unmatched buyer, X = a(K) , the bid of the highest matched seller, Y = a(K+1) , the bid of the lowest unmatched seller, and Z = a(K−1) , the bid of the next highest matched seller. Consider the payoff of the i-th seller when it participates as well. It’s payoff when it bids a = α(c) is given by   x − c, if a < Z < X < W, or      Z < a < X < W;     a − c, if Z < X < a < W, or      Z < a < W < X, or  (8) πi (a) = Z < W < a < X, or    W < Z < a < X;      z − c, if a < Z < W < X, or     a < W < Z < X, or     W < a < Z < X. Its expected value denoted by π ¯i satisfies the differential equation d¯ πi (a) = [P n (Aa ) + P n (Ba ) + P n (Ca )] − [ng(a)P n (Da ) + (n − 1)f (a)P n (Ea )](a − c), (9) da with the boundary condition π ¯i (1) = 0 where Aa denotes the event that there are n − 1 sellers and n buyers and X < a < W . As a is increased, the payoff to the seller also increases by ∆a since seller i is the price-determining seller. Similarly, Ba denotes the event that there are n − 1 sellers and n buyers and Z < a < W < X and seller i is the price-determining seller. In the same way, Ca denotes the event that there are n − 1 sellers, n buyers and max(Z, W ) < a < X

and seller i is the price-determining seller. Da denotes the event that there are n − 1 sellers and n − 1 buyers, X < a (with the n-th buyer bidding a) and W ∈ [a, a + ∆a] so that the seller i becomes unmatched as it increases its bid. Similarly, Ea is the event that there are n − 2 sellers, n buyers, W < a (with the (n − 1)-th seller bidding a) and X ∈ [a, a + ∆a]. And so as it increases its bid, it becomes unmatched. The following can then be obtained easily: ¶ µ ¶ n−1 µ X n−1 n n k n−1−k ¯ k+1 (a)Gn−(k+1) (a) ¯ P (Aa ) = F (a)F (a) G k k + 1 k=0 ¶ µ ¶ n−1 µ X n−1 n n k−1 n−k ¯ ¯ k+1 (a)Gn−(k+1) (a) P (Ba ) = F (a)F (a) G k − 1 k + 1 k=1 ¶ µ ¶ n−1 µ X n ¯k n−1 n k−1 n−k ¯ P (Ca ) = F (a)F (a) G (a)Gn−k (a) k k − 1 k=1 ¶ ¶ µ n−1 µ X n − 1 ¯k n−1 n k n−1−k ¯ P (Da ) = F (a)F (a) G (a)Gn−1−k (a) k k k=0 ¶ µ ¶ n−1 µ X n−2 n ¯k n k−1 n−1−k ¯ P (Ea ) = F (a)F (a) G (a)Gn−k (a). (10) k − 1 k k=1 Let a = αn (c) be the best-response strategy of the sellers. Then, πi any a < c, d¯ > 0. Thus, da a = αn (c) ≥ c, ∀n.

d¯ πi da

= 0 at a = αn (c). For (11)

Then, for a = αn (c), if a = c, the sellers’ best-response strategy is to bid truthfully. If a 6= c, setting (9) equal to zero and rearranging, we get [P n (Aa ) + P n (Ba ) + P n (Ca )] − ng(a)P n (Da )(a − c) f (a) = ≥ 0, (n − 1)P n (Ea )(a − c) from which we obtain [P n (Aa ) + P n (Ba ) + P n (Ca )] (αn (c) − c) ≤ ng(a)P n (Da ) P Pn−1 ¡n−1¢2 zk Pn−1 ¡n−1¢2 kzk n−1 ¡n−1¢2 z k ¯ ¯ 1  k=0 k (k+1) ¯ k=1 k=1 k (n−k) GF k (n−k)2 ≤ + Pn−1 ¡n−1¢2 Pn−1 ¡n−1¢2 k G + Pn−1 ¡n−1¢2 k g(a) F z z zk k=0 k=0 k=0 k k k

(12)  ¯ F , F

¯ G(a) ¯ ¯ F¯ (a) and F¯ (a) in the numerator are where z = FF¯ (a) . Observe that the terms G(a), G(a) (a)G(a) upper-bounded by one, and the term F (a) in the denominator is lower-bounded by F (c). It can now be shown that each of the terms converges to zero for all z > 0 as n → 0. Thus, ˜ → (˜ ˜ (αn , β) α, β).

Proof of Theorem 1: By Lemma 1 when the sellers use strategy αn , the buyers under the ex post individual ˜ By Lemma 2, when the buyers bid truthfully, sellers’ bestrationality constraint use strategy β. ˜ is a Bayesian-Nash equilibrium with n players on each side of response is αn . Thus, (αn , β) ˜ → (˜ ˜ as n → ∞, which is the the market. Further, Lemma 2 shows that (αn , βn ) = (αn , β) α, β) conclusion we want to establish.

Thus, under the ex post IR constraint, SeBiDA is asymptotically Bayesian incentive compatible. Unlike in the complete information case when the mechanism is not incentive compatible, yet the outcome is efficient, in the incomplete information case, the mechanism is only asymptotically efficient. Remarks The mechanism proposed in this paper is related to the buyer’s bid double auction (BBDA) mechanism [25], [26]. While the spirit of the two mechanisms is the same (maximizing the efficiency of trading), the prices and the payments are different. In SeBiDA, the prices are determined by the bids of the sellers only. This makes the market asymmetric: In the complete information case, all buyers have no incentive to bid non-truthfully but at least one seller does. In BBDA, the determined price could be either a buyer’s bid or a seller’s bid. While the claim in Theorem 2.1 of [25] is not correct, in [26] it is simply assumed that the buyers bid truthfully which need not be true. In fact, we found that for SeBiDA, even though under complete information it is a dominant strategy for buyers to bid truthfully, this is not the case for incomplete information. We would like to acknowledge that the proof techniques used in this paper owe a lot to those developed by [25], [26]. The rate of convergence of SeBiDA can be obtained from the analysis in the proof of Lemma 2. Strangely, Nash equilibrium analysis was ignored in [25]. And lastly, the ex post individual rationality constraint seems restrictive at first glance. However, in human subject experiments conducted using this mechanism, it was observed that all subjects in fact always used strategies that were ex post individual rational. Thus, it may be concluded that the predictive power of the result does not diminish in real-world settings with the assumption made. V. C ONCLUSION The paper presents two results. The first result concerned the existence of a Nash equilibrium for a combinatorial, sellers’ bid, double auction (c-SeBiDA). In c-SeBiDA, settlement prices are determined by sellers’ bids. In the case of a single item, a similar mechanism, is proposed in [25], [26]. We showed that the allocation of c-SeBiDA efficient. Moreover, truth-telling is a Nash strategy for all players except the highest matched seller on each link. The second result concerned Bayesian-Nash equilibrium of the mechanism under incomplete information. We showed that under the ex post individual rationality constraint, symmetric Bayesian-Nash equilibrium strategies converge to truth-telling. Thus, the mechanism is asymptotically Bayesian incentive compatible, and hence asymptotically efficient. In the rest of this section we situate our contribution in the literature that relates Nash and competitive equilibria. The basic idea is that as the economy gets large (in our context the number of buyers and sellers and link capacities all go to infinity), Nash equilibrium strategies should converge to competitive equilibrium strategies, because ‘market power’ diminishes. Thus in [25], [26] it is shown that Bayesian-Nash equilibrium strategies converge to truthful bidding as the market size goes to infinity. The relationship is first investigated in [23]. In a later paper [6], it is shown that under certain regularity conditions, a sufficiently replicated economy has an allocation which is incentive-compatible, individually-rational and ex-post ²efficient. Similarly [9] shows that the demand functions that an agent might consider based on strategic considerations converge to the competitive demand functions. Further, [10] shows that under certain conditions on beliefs of individual agents, not only do the strategic behaviors of individual agents converge to the competitive behavior but the Nash equilibrium allocations also converge to the competitive equilibrium allocation. The formulation in [29] is a buyer’s bid double auction with single type of item that maximizes surplus. It is shown that with BayesianNash strategies, the mechanism is asymptotically incentive efficient in the sense that a player’s

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