Designing a Strategic Bipartite Matching Market

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1 Designing a Strategic Bipartite Matching Market Rahul Jain IBM T. J. Watson Research Center Hawthorne, NY rahul.jain@watson.ibm.com Abstract We consider a version of the Gale-Shapley matching problem and the Shapley-Shubik assignment game. It involves matching n buyers with m sellers. Each seller offers a distinct good and a buyer has a different valuation for various goods, and it only wants one. The matching may also involve an exchange of numeraire to be paid by the buyers, and to the sellers. The players are strategic. The key requirements are ex ante individual rationality, strong budget-balance and efficiency at equilibrium. The motivation is a advertiser-host matching system. We propose an auction-based matching mechanism. The mechanism is non-vcg (Vickrey-Clarke-Groves) type, ex ante individual rational and is able to achieve strong budget-balance. Further, we show that there exists a Nash equilibrium that is efficient. I. INTRODUCTION This paper is partly inspired by the seminal paper College admissions and the stability of marriage of Gale and Shapley [12], and partly motivated by the advertisement placement problem of the Google AdSense program. The formulation is as follows: Suppose there is a set B of m men to be matched with a set S of n women. Each man has a strict preference over each of the women. Similarly, each woman also has a strict preference over each of the men. Moreover, if a man and a woman are matched, the man pays a dowry in money to the woman (or her family), and no other side-payments are allowed. (This might be incompatible with cultural mores in some societies and compatible with others but men and women do have exchangeable roles in our formulation). The key question is whether a stable matching of men and women exists such that no man or woman has an incentive to break an engagement for another partner. If it does, then what kind of mechanisms can be used for matching and dowry determination that result in stable and, in a certain sense, optimal matchings. In [12], the formulation did not involve any dowry payments (or any other side-payments) and moreover, the players were regarded as straightforward or non-strategic. It was shown that a stable matching does indeed exist and moreover it is not unique. Furthermore, an iterative algorithm was proposed that achieved a stable matching. Moreover, it is known that while a stable matching exists for the marriage (without payments) problem, no stable matching exists for tripartite (man-woman-child) matching, nor for the many-toone matching problem [19]. A generalized model where money is also a continuous variable was considered in the (cooperative) assignment game of Shapley and Shubik [20]. Each B-agent i had a valuation v ij for an S-agent j, and each S-agent j had a reserve price c j, with σ ij = max{0, v ij c j } being the potential matching gain. The value of a coalition B S with B B and S S was considered to be the maximum matching gain within the coalition. It was established that the core of this cooperative game and the set of stable (in the sense of Gale-Shapley [12]) outcomes are the same. However, no explicit mechanism was given to achieve an outcome in the core. The Gale-Shapley marriage problem when the players are strategic and can misrepresent their true preferences was considered by Dubins and Freedman [8] and Roth [17]. In this non-cooperative game setting, it was established that no stable matching mechanism exists for which stating true preferences is a dominant strategy for every agent [17]. Further, it was established [8] that no coalition of agents may, by misrepresenting their true preferences arrange so that they all do better under a stable mechanism than when they all state their true preferences, unless they are able to make side-payments within the coalition. However, neither of these papers considered monetary payments as a possibility. But a realistic model for the problem at hand must consider agents to be strategic and must allow for monetary payments. Moreover, the mechanism solution must have the properties that it is ex post individual rational (i.e., it is rational for each player to participate in any such mechanism), it is strongly budget-balanced (i.e., the sum of payments made to all players is zero), and that it is efficient (i.e., the resulting matching maximizes the sum total of the matching gains over all feasible matchings). It would be desirable to have players reveal their true preferences but we already know from Roth s impossibility theorem that no such stable matching mechanism exists. The Shapley-Shubik assignment game was considered by Demange and Leonard independently [5], [16]. A pricing scheme motivated by VCG mechanisms was proposed for matching buyers to objects. It was proved that with such a pricing scheme, it is a dominant strategy for every buyer to be truthful. Further, in [7], Demange, Gale and Sotomayor proposed an iterative mechanism based on Hall s theorem [14] that finds transaction prices that are minimum quasicompetitive (see [7]). The resulting outcome turns out to be a competitive equilibrium. Thus, the mechanism is ex ante individual rational, efficient and strongly budget-balanced. However, the settings in all of the above mechanisms was matching of buyers to objects, and hence was a one-sided

2 market. The setting we consider in this paper involves both agents B (buyers) and agents S (sellers), and thus a two-sided market. It is well-known that a VCG-type mechanism will not achieve budget-balance, a key requirement in many such markets. We consider all players to be strategic, and moreover every transaction between a matching pair involves monetary payments. Moreover, we desire a mechanism that is ex ante individual rational, strongly budget-balanced and efficient. The mechanism that we propose and analyze is a one-shot mechanism with strategic players and has these properties. Specifically, we prove that there exists a Nash equilibrium at which the players are not necessarily truthful but the resulting assignment/matching is the efficient one. The Advertisement Placement Problem As mentioned at the very beginning, this problem is intimately related to a more contemporary problem, namely the Google AdSense problem (We call it a Google problem because of its lead in initiating such a program over other providers such as Yahoo and Quigo). Let us begin with a description of the Google Adwords program. Google, Yahoo, and many other search providers hold keyword auctions. Advertisers bid b i on keywords, and text ads of winners are placed in slots alongside search for specific keywords. For each slot and advertisement, a click through rate (CTR) is maintained which is the fraction of times an ad was clicked out of total number of times an ad was placed. Google maintains an aggregate (over all slots) CTR α i for each advertiser i while Yahoo maintains aggregate (over all advertisers) CTR γ k for each slot k. For each keyword, a next-price auction is held: In the Yahoo auction, bids are ranked in decreasing order and the ith highest bidder pays the (i+1)st highest bid per click on its ad (without loss of generality, relabel buyers in the decreasing order of their bids). In the Google auction, bids are ranked by α i b i (without loss of generality, relabel buyers in the decreasing order of α i b i ), and the ith highest ranked bidder pays α i+1 b i+1 /α i per click. The Google auction was analyzed in [11] where it was proved that certain symmetric Nash equilibria of the auction game are competitive equilibria of the Shapley-Shubik assignment game with a special utility structure. Further, it was shown that there is a symmetric Nash equilibria which achieves the maximum revenue at any Nash equilibrium. The Yahoo auction was analyzed in [9] where it was shown that truth-telling is not a dominant strategy in either the Yahoo or the Google auction. Further, they characterized the minimum revenue locally envy-free Nash equilibrium (see [9] for definition) in which the bidder assignments and payments are the same as in the dominant-strategy equilibrium of the VCG mechanism. In [15] and [13], [2], a more comprehensive equilibrium and revenue analysis of the two auction mechanisms was provided both in the complete and incomplete information setting. Similar results from analysis and simulations have also been obtained in [10]. In [1], a different mechanism, the laddered auction mechanism is proposed which is shown to be truthful. While [4] takes a different approach: Instead of choosing with payments to align incentives as in all the other proposals, it chooses the probability of an advertisers ad being placed. The payment equals the bid if the placed ad is clicked. Thus, the one-sided slot/position auctions as described above have been extensively analyzed. But Google, Yahoo and another company Quigo do run another program where the ads are placed not on their own websites but on third party websites. This pioneering program of Google is known as AdSense in which Google matches advertisers with websites. The mechanics are similar on one-side of this market: advertisers make bids on keywords. Based on relevance and the bids, Google places these ads on the websites of participating hosts. The advertisers pay every time their ad is clicked according to the Google payment rule, while the hosts get a share of the payment received from the advertiser. It is well-known that this mechanism is sub-optimal both from an assignment perspective as well as revenue. What is required is a double-sided market mechanism where both advertisers bid on keywords as well as the hosts. To the best of this author s knowledge, no efficient, individual rational, budget-balanced mechanism for this problem has been proposed. The strategic matching market mechanism that we propose in this paper solves this problem. II. BACKGROUND AND MODEL We will first state the formulation for the double-sided keyword slot auction. We will then reduce it to an assignment game formulation. Suppose there are n buyers (or advertisers) B, and m sellers (or ad-hosts) S and each seller has K slots to offer with q = min(n, m). If a buyer i s ad placed on slot k of host j is clicked he derives a value ṽ ijk. This could also be interpreted as buyer i s outside option for a similar slot on a similar host. Similarly, a seller j has a cost c ijk if an ad of buyer i is placed at the slot k. Again, this can be interpreted as seller j s outside option, i.e., what it can achieve outside the ad-exchange system. From the history of ad placements and clicks, a clickthrough-rate (CTR) is denoted by θ ijk. It is the empirical probability of a buyer i s ad being clicked by a random user if it is placed on host j s slot k. We will say it is completely separable if it is equal to some α i β j γ k, where α i is the empirical probability of buyer i s ad being clicked, β j is the empirical probability of an ad on seller j s site being clicked and γ k is the empirical probability of an ad at slot k on any host s site being clicked. In the model, we can also postulate certain weights w ijk that depend on more subjective criterion such as relevance and effectiveness of the ads at a particular host and a particular slot. We will assume that all this information about true valuations, costs and CTRs is common knowledge. This is the model in full generality.

3 In subsequent discussion in this paper, we will consider a simplified model though the analysis should extend to the above general model with some difficulty. We will assume that each seller offers only one slot, and moreover a seller s cost is independent of the buyer whose ad it hosts, i.e., c ijk = c j, ṽ ijk = ṽ ij and θ ijk = θ ij. We will assume that each player has a quasi-linear payoff function. Thus, a seller j s expected payoff is u s j = p c j (with c j = c j ) if he is paid p each time he hosts an ad. Similarly, a buyer i s expected payoff u b i = θ ij ṽ ij p = v ij p if his ad is placed at host j, where v ij = θ ij ṽ ij. Thus, we can simplify our model and just work with expected value per ad placement derived by the buyers. Those buyers whose ad is not placed anywhere and those sellers who do not host any ad will have payoff zero. Denote the set of feasible assignments by X := { x ij 1 i, x ij 1, j, x ij 0, i, j}, j i i.e., the sum probability of assignment of a buyer i to any seller j is at most 1, and the sum probability of assignment of a seller j to any buyer is at most 1. Thus, x ij = 1 will mean that i and j are matched. We define a social welfare function as S(x) = (v ij c j )x ij. ij Our objective is an assignment x arg max S(x) s.t. x X. (1) We will call such an assignment efficient. The players are strategic and act to maximize their own payoff. Thus, we would like to design a mechanism consisting of an assignment rule that matches buyers B with sellers S and a payment rule that specifies payments to be made by or to, matched players. We will say that the mechanism is stable if u b i 0 and u s j 0, and ub i + us j (v ij c j ) +, where (z) + denotes max(0, z). This notion was introduced by Shapley-Shubik [20]. We will see that this is closely related to our objective of efficient, i.e., a matching that maximizes the social welfare function. We define a Pareto-optimal assignment to be one such that no agent is better off (in terms of his payoff) at another assignment without any other agent being worse off (in terms of his payoff), i.e., an assignment that cannot be changed to improve at least one agent s payoff without decreasing any other agent s payoff. If the payment rule of the assignment mechanism is such that it is strongly budget-balanced (formally defined in the next section), then i ub i + j us j = ij (v ij c j )x ij with x X. Thus, with budget-balance, an assignment that is efficient is also Pareto-optimal. Moreover, the set of Paretooptimal outcomes is in the core [20] (see [3] for a definition). Thus, we have argued that Theorem 1: An efficient assignment x for the assignment game above is compatible with a stable payoff. Thus, we can focus on designing an assignment mechanism that yields an efficient assignment at equilibrium since stability is then guaranteed in a budget-balanced mechanism. III. A BUDGET-BALANCED EFFICIENT ASSIGNMENT MECHANISM We now propose an assignment mechanism for strategic players. First, bids are elicited from both the buyers and sellers indicative of their true valuations and costs. Thus, a buyer i bids b ij for seller j s slot and seller j places a minimum reservation price bid a j. Let ij (b ij a j )x ij denote the total matching surplus. ASSIGNMENT RULE. We determine an assignment that maximizes the total matching-surplus x arg max{ ij (b ij a j )x ij : x X }. (2) Payment by buyer i will be denoted µ i and payment to seller j will be denoted λ j. We will call a payment function pairwise strongly budgetbalanced if µ i = λ j if x ij = 1, i.e, if buyer i is matched with seller j, and their absolute payments are equal. We will say that the mechanism is ex ante individual rational if for any outcome assignment x ij, u b i(b i, b (i ) ; a) = j v ij x ij (b, a) µ i (b, a) 0, where b i := (b i1,, b in ) and b (i ) denotes the bids of all the other buyers, and u s j(a j, a j ; b) = i c j x ij (b, a) λ j (b, a) 0, where a j denotes the bids of all the other sellers, i.e, every buyer and seller has a non-negative payoff at every feasible outcome. Thus, the strategy space of buyer i is Γ i = j [0, v ij ] and of seller is Ξ j = [0, c j ]. We will define a Nash equilibrium to be the bid profile (b, a ) such that i j, b i arg max u b i(b i, b (i ) ; a ) i, (3) a j arg max u s j(a j, a j; b ) j. (4) We shall say that a Nash equilibrium (b, a ) is efficient if the corresponding allocation x is efficient. We will define an ε-nash equilibrium to be a bid profile (b, a ) such that each player is satisfied in picking a bid that yields a payoff within ε of the optimal in (3) and (4) above. We now specify the payment rule for the assignment mechanism but first, some notation to ease readability: Denote the true matching surplus for a buyer i and seller j as σ ij = (v ij c j ), and without loss of generality, we can relabel buyers and sellers, and assume that σ 11 σ 22 σ qq. Given the bids b ij and a j, we shall denote the revealed matching surplus between buyer i and seller j as: ρ ij = (b ij a j ). We shall also denote η ij = (v ij a j ) and ζ ij = (b ij c j ). We shall denote ˆ1 to be the buyer with the highest

4 matching surplus ρ ij, and ˇ1 to be the corresponding matched seller. Thus, ρˆ1ˇ1 ρˆ2ˇ2 ρˆqˇq. (5) Suppose there are k matches in x. Then Lemma 1: Given the bids of all the other players, for i k, the buyer î must bid above bîǐ = aǐ + (max j>i ρ ĵ,ǐ max j>i ρ î,ǰ )+, (6) to match with seller ǐ. Similarly, given the bids of all the other players, the seller ǐ for i k, can bid up to āǐ = bîǐ (max j>i ρ î,ǰ )+. (7) and still match with buyer î. (Here, denotes a max of joined quantities). Proof: Let us first consider buyer ˆ1. We know from equation (5) above that his matching with the seller ˇ1 generates the largest revealed matching surplus. Thus, given the bids of all the other players, if buyer ˆ1 wants to be matched with seller ˇ1, he must bid high enough to beat any competing bids from other buyers. Further, his bids for any other sellers must also have a lower revealed matching surplus. Thus, the requirement on buyer ˆ1 s bid is that the revealed surplus ρˆ1,ˇ1 := (bˆ1ˇ1 aˇ1) max j>1 ρ+, and max ĵ,ˇ1 j>1 ρ+ˆ1,ǰ where the term in the first inequality corresponds to the revealed matching surplus of competing bids from other buyers for seller ˇ1 and the term in the second inequality corresponds to buyer ˆ1 s own competing bids for other sellers. The first inequality above follows because buyer ˆ1 has higher matching surplus with seller ˇ1 than any other buyer. For subsequent buyers î, i > 1, since here we are primarily concerned with the question of the minimum bid for a buyer î to match with seller ǐ, it need not worry about beating the bids of buyers ĵ, j < i nor worry about choosing competitive bids for sellers ǰ, j < i. From this, we easily get equation (6) by using forward induction. The argument to derive equation (7) is similar. Consider seller ˇ1 first. Again, we know from equation (5) that his matching with buyer ˆ1 generates the largest matching surplus. Thus, given the bids of all the other players, to match with buyer ˆ1, it must bid to beat any competing bids from other sellers for buyer ˆ1. Thus, the requirement on seller ˇ1 s bid is that ρˆ1,ˇ1 := (bˆ1ˇ1 aˇ1) max j>1 ρ+ˆ1,ǰ. For subsequent sellers ǐ, i > 1, we note that since they want to match with buyer î, i > 1 they need not bid to beat the bids of sellers ǰ, j < i. From this, we easily get equation (7) by using forward induction. Thus, given the bids of all the other players, seller i can at most bid ā i to match with buyer i. Similarly, given bids of all the other players, a buyer i must bid at least b ii to match with seller i. PAYMENT RULE. In general, we can take the transaction price between buyer î and seller ǐ to be p i = γāǐ + (1 γ)bîǐ, i.e., if x = 1, then µ îǐ î = λǐ = p i. For the remainder of this paper though we will take γ = 0 so that the prices only depend on the buyers lower bounds. We will make the following assumptions. (A1) For i k: v ii max k v li v ij max k l>j v lj, j i (A2) For i k: v ii max v li v ij c j, j > k (A3) For i > k: v il max j>l v jl, l k (A4) For i > k: v il c l, l > k Theorem 2: If the buyer valuations and seller costs satisfy the above assumptions (A1)-(A4), then an efficient Nash equilibrium exists. Further, the mechanism is ex ante individual rational and pairwise strongly budget-balanced. Proof: We will formulate the problem of existence of an efficient Nash equilibrium as a fixed point problem. We will show that it has a solution under the assumptions made. Let k denote the number of matches in the efficient assignment x from the optimization (1), with players truthful bids. We consider buyers and sellers relabeled so that σ 11 σ 22 σ kk 0 σ k+1,k+1 σ qq. Note that from the efficient matching conditions, we have i k : b ii a i max (b li a i ) + max (b il a l ) +, i, j > k : b ij a j 0. (8) Since, the payments are determined by lower bounds on buy-bids for matching, buyer i would like to match with seller i only if i k : v ii b ii v il b ll, l k, l i (9) v ij a j, j > k, i > k : v il b ll, l k (10) a l, l > k. These are the Buyer Incentive Constraints (BIC). The first set of inequalities states that the matched buyer i obtains maximum payoff if it matches with seller i. The second set of inequalities states that an unmatched buyer i will obtain a negative payoff if tries to match with any seller given the current bids. The Seller Incentive Constraints (SIC) are i k : b ii c i a i + max j k,j i (b ji a i ) + c i (11) b ji c i, j > k, i > k : 0 b ji c i, j. (12) The first set of inequalities states that the matched seller i obtains maximum payoff if it is matches with buyer i. The second set of inequalities states that an unmatched seller i will obtain a negative payoff if it tries to match with any buyer given the current bids.

5 Consider a matched seller i k. Then, given the bids of all the other players, if seller i matches with buyer i, his payoff is maximized if a i = b ii max (b il a l ) +. This is because the payment to seller i depends on a i, so it bids the maximum it can while still remaining matched with buyer i. If a matched buyer i( k) matches with seller i, then his payoff is maximized if and b il, for l > i such that b il = v il, l < i, b ii a i + max (v li a i ) +, max (b il a l ) + max (v li a i ) +. The above is true because b il, l > i affects the payment to be made only if we have the reverse inequality above. We can now obtain the fixed point equations (along with the incentive constraints) that an efficient Nash equilibrium should satisfy. Consider the following: i > k : b il = v il, l, (13) i > k : a j = c j, j > k, (14) i k : a i = b ii max il a i ) + (15) i k : b ii a i + max li a i ) + (16) max il a l ) + max li a i ) +. Together, they imply b ii a i = max (b il a l ) + = max (v li a i ) +, which yields the following as a solution: v ij j < i i k : b ij = max v li c i, j = i 0 j > i (17) i k : a i = b ii, (18) i > k : b il = v il, l (19) i > k : a i = c i. (20) We claim that buyer i is matched with seller i before the pair j, for j > i, and the pair k are the last ones matched. To see this note that from the matching conditions (8), ρ ii ρ ij, j > i. Thus, buyer 1 matched with seller 1, buyer 2 matches with seller 2, and so on until k since ρ ij = σ ij 0 for i, j > k. Thus, î = i and ǰ = j, and buyers i > k and sellers j > k do not match. Thus, the constructed bids yield an efficient allocation that satisfies (1). Further, we claim that the payment between matched buyer i and seller i is p i = b ii = b ii. Note that buyer i s bid on seller i, b ii is the minimum it needs to bid to match with seller i. Thus, p i as defined above is the transaction payment. We will now argue that solution (17)-(20) is a best response of each player. First consider any matched buyer i( k). Note that by assumptions (A1) and (A2), (17) satisfies buyer incentive constraints (9), and it maximizes buyer i s payoff if it matches with seller i. Thus, it is a best response of buyer i. Now, consider any unmatched buyer i(> k). By (17) and matching condition (8), buyer incentive constraints (10) are satisfied. Thus, (19) is a best response of unmatched buyer i. Consider a matched seller i( k). By (17), seller incentive constraints (11) are clearly satisfied. Further, (18) maximizes seller i s payoff if it matches with buyer (i). Thus, it is a best response of matched seller i. Now, consider an unmatched seller i(> k). From (19), (20) and (8), seller incentive constraint (12) is satisfied for unmatched buyers. From the efficient matching condition (8) for matched buyers, the seller incentive constraint (12) is also satisfied for matched buyers. Thus, (20) is a best response of unmatched sellers. This proves that (17)-(20) is a Nash equilibrium. Since it satisfies (8), it is efficient. Note that the mechanism is strongly budget-balanced by design µ i = λ j if x ij = 1. Further, it is ex ante individual rational since for a matched buyer i, v ij µ i b ij b ii 0, and for a matched seller j, λ j c j ā j c j 0. Above, we designed an assignment mechanism that employed two-parameter bids for buyers. An interesting question is whether it is possible to design an assignment mechanism with same or similar properties if buyers only make one-parameter bids. Our conjecture is that it is impossible. IV. CONCLUSIONS AND FURTHER WORK We posed the problem of bipartite matching with strategic players. Motivated by the Google AdSense advertisement problem, strong budget-balance was a key requirement, and stability and efficiency of the assignment was our objective. We then proposed a mechanism that can used to solve such a problem. Determining the matching is easy, as it can be posed as a linear program. However, this doesn t guarantee that the incentives of the players will be aligned by any payment rule that will result in the efficient assignment as an equilibrium outcome. Thus, we designed a payment rule that despite strategic behavior is able to achieve the efficient assignment. Moreover, we are able to achieve strong budget-balance. Specifically, we showed that the designed assignment mechanism is ex ante individual rational, strongly budget-balanced and has an efficient Nash equilibrium. We considered a simplified version of the general problem posed. As part of further work, we would like to extend this to each seller offering multiple slots. Moreover, we only showed that an efficient Nash equilibrium exists. We would like to investigate whether there are other Nash equilibria and if those are efficient. Further, we assume the complete information model, but in a real setting, players might only

6 have incomplete information. It would also be of interest to investigate if the mechanism is robust to collusion. While the problem was motivated primarily by the advertisement placement problem, it has implications (after further extensions) for another related problem from networks: QoS assignment problem. Suppose there are multiple network service providers each offering a different quality of service (QoS), and there are buyers who would like to buy network services, each of whom may have different utility for service at a particular QoS, how do we design an assignment mechanism that is efficient in some sense. A problem along these lines was posed in [21] but the VCG mechanism used was one-sided only. REFERENCES [1] G. Agarwal, A. Goel and R. Motwani, Truthful auctions for pricing search keywords, ACM Electronic Commerce Conference, [2] G. Agarwal, J. Feldman and S. Muthukrishnan, Bidding to the top: VCG and Equilibria of Position-based Auctions, unpublished, [3] K. Arrow and F. Hahn, General Competitive Analysis, North-Holland, [4] J. Chen, D. Liu and A. Whinston, Designing share structure in auctions of divisible goods, ACM Electronic Commerce, [5] G. Demange, Strategyproofness in the assignment market game, Laboratoire d Econometrie de l Ecole Polytechnique Paris, Mimeo, [6] G. Demange and D. Gale, The strategy structure of two-sided matching markets, Econometrica, 53:873-78, 1985 [7] G. Demange, D. Gale and M. Sotomayor, Multi-item auctions, J. of Political Economy, 94: , [8] L. Dubins and D. Freedman, Machiavelli and the Gale-Shapley algorithm, American Mathematical Monthly, 88: , [9] B. Edelman, M. Ostrovsky and M. Schwarz, Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords, Mimeo, Fall [10] J. Feng, H. Bhargava and D. Pennock, Implementing sponsored search in web search engines: Computation evaluation of alternative mechanisms, INFORMS J. of Computing, 19(1), [11] H. Varian, Position Auctions, Mimeo, December [12] D. Gale and L. Shapley, College admissions and the stability of marriage, American Mathematical Monthly, 69:9-15, [13] D. Garg, Y. Narahari and S.S. Reddy, An optimal mechanism for sponsored search auction, unpublished, [14] P. Hall, On representatives of subsets, J. of the London Mathematical Society, 44:26-30, [15] S. Lahaie, An analysis of alternative slot auction designs for sponsored search, ACM Elecronic Commerce Conference, [16] H. Leonard, Elicitation of honest preferences for the assignment of individuals to positions, J. of Political Economy, 44: , [17] A. Roth, The economics of matching: stability and incentives, Math. of Operation Research, 7: , [18] A. Roth, Misrepresentation and stability in the marriage problem, J. of Econ. Theory, 34: , [19] A. Roth and M. Sotomayer, Two-sided matching, Cambridge University Press, [20] L. Shapley and and M. Shubik, The assignment game I: the core, International J. of Game Theory, 1: , [21] J. Shu and P. Varaiya, Pricing Network Services, Proc. INFOCOM, 2002.