Core-Stable Rings in Auctions with Independent Private Values

We propose a semi-cooperative game theoretic approach to check whether a given coalition is stable in a Bayesian game with independent private values. The ex ante expected utilities of coalitions, at an incentive compatible (noncooperative) coalitional equilibrium, describe a (cooperative) partition form game. A coalition is core-stable if the core of a suitable characteristic function, derived from the partition form game, is not empty. As an application, we study collusion in auctions in which the bidders’ final utility possibly depends on the winner’s identity. We show that such direct externalities offer a possible explanation for cartels’ structures (not) observed in practice.


Introduction
Collusion in auctions is mostly studied as a mechanism design problem for a given ring (see, e.g., Graham and Marshall (1987), Mailath and Zemsky (1991) and McAfee and McMillan (1992) for early references and Marshall and Marx (2007) for a recent one). This framework imposes individual participation constraints to every member of the ring. In second price auctions with independent private values, Mailath and Zemsky (1991) further consider participation constraints for all subrings of any potential ring. In this particular framework, equilibria in weakly dominant strategies considerably limit the strategic externalities that coalitions might incur. Mailath and Zemsky (1991)'s analysis does not extend if equilibria in weakly dominant strategies do not exist, e.g., in the case of common values (see Barbar and Forges (2007)). In this paper, we keep the assumption of independent private values but except for that, allow for an arbitrary auction game. We ask whether a given coalition is stable, in the sense that no subgroup of players would like to leave it. Such collective participation constraints are traditionally captured by core-like solution concepts. However, two di¢ culties arise when trying to de…ne the core of an arbitrary auction game, or, more generally, a Bayesian game.
A …rst di¢ culty, which already appears under complete information, is that every coalition faces strategic externalities, so that it must make conjectures on the behavior of the players who are outside the coalition. To solve this di¢ culty, Aumann (1961) introduced the characteristic function, which measures the worth of a coalition in a strategic form game as the amount that it can guarantee whatever the complementary coalition does. However, the corresponding core, namely the core, can be criticized on the grounds that it involves incredible threats from the complementary coalition. As a remedy, Ray and Vohra (1997) and Ray (2007) construct a partition form game (as de…ned by Lucas and Thrall (1963)) in which, given a partition of the players, coalitions evaluate their worth at a Nash equilibrium of an auxiliary game between the coalitions. We extend Ray (2007)'s coalitional equilibrium to games with incomplete information and construct a partition form game from the noncooperative Bayesian game which models the auction. We then apply a notion of core for partition form games, the core with "cautious expectations" (see Hafalir (2007)). Under complete information, it is included in the core. In a coalitional equilibrium of a Bayesian game, it is understood that the strategy of every coalition is a function of its members'private information. The description of the previous paragraph hides a second di¢ culty, which is speci…c to incomplete information: every coalition faces incentive constraints. We establish (in proposition 1 and its corollary) that, in a class of Bayesian games which includes standard auctions (namely, games with independent private values and quasi-linear utilities), this di¢ culty can be ignored: every coalitional equilibrium can be made incentive compatible. More precisely, coalitional equilibria are "…rst best"solutions, in which every coalition plays a best reply to the strategies outside the coalition, as if information sharing was not an issue. We construct an incentive compatible revelation mechanism for the coalition, which involves exactly balanced monetary transfers among its members and achieves the "…rst best" reply of the coalition. The fact that the coalition maximizes its payo¤ given strategies in the complementary coalition is crucial to our construction, in particular, in the formulation of incentive constraints.
In order to associate a partition form (cooperative) game to a (noncooperative) Bayesian game, we assume that coalitions can commit to an incentive compatible mechanism at the ex ante stage, i.e., before their members get their private information. This assumption …rst requires that an ex ante stage can be identi…ed, which is true in many economic applications, like auctions, in which private information reduces to the value of some parameter, like a valuation or a cost. According to empirical data (see, e.g., Porter and Zona (1993), Pesendorfer (2000)), bidding rings often consist of well-identi…ed groups (e.g., "incumbents", as opposed to "newcomers") whose characteristics do not depend on particular information states. Such bidding rings typically form at an early stage. For instance, local suppliers may be aware that a procurement auction will take place and consider to collude before the precise project speci…cations are published. At the time they commit to a collusion mechanism, they do not …gure out their exact valuations, i.e., the costs incurred by the project.
The ex ante formation of rings is assumed explicitly in Bajari (2001), Marshall et al. (1994) and Waehrer (1999). In these papers, ring mechanisms are investigated within an a priori given partition of the bidders. The partition itself does not depend on the bidders' private information, which re ‡ects the ex ante formation of the rings. To the best of our knowledge, interim formation of rings has only been investigated to a limited extent, e.g., in Graham and Marshall (1987), McAfee and McMillan (1992), Caillaud and Jehiel (1998) and Marshall and Marx (2007). These papers focus on an ex ante given bidding ring, the grand coalition for instance, and formulate interim participation constraints for the individual members of the ring. More precisely, every member of the ring can decide to leave the ring once he knows his private information. The precise form of the participation constraints depends on the reaction of ring members when one of them leaves the ring. Mailath and Zemsky (1991) start with the latter model but restrict themselves on ex ante expected payo¤s when studying the stability of rings. Being interested in the participation constraints of coalitions, rather than individuals, we assume, to keep the analysis tractable, that coalitions can commit to an incentive compatible mechanism ex ante, as in Forges and Minelli (2001) and Forges, Mertens and Vohra (2002). These papers introduce the notion of ex ante incentive compatible core in exchange economies with di¤erential information. In this setup, there are no externalities, i.e., only the second di¢ culty above arises. The basic solution concept in the present paper is an ex ante incentive compatible core for Bayesian games. 1 Coming back to auctions, we construct a partition form game, which re ‡ects the ex ante commitments of bidding rings. A coalition is core-stable if all its subcoalitions agree to participate in its collusion mechanism. In this de…nition, we focus on a single ring and assume, as in Marshall and Marx (2007), that the bidders outside the ring do not collude. We …rst apply the solution concept to standard auctions, without direct externalities. In the case of second price auctions, possibly with asymmetric players, our partition form game reduces to a characteristic function and we prove in proposition 2 that all rings are core-stable. In particular, strategic externalities have limited e¤ects on collusion. Mailath and Zemsky (1991) already obtain this result. They directly focus on the equilibrium in weakly dominant strategies so that they can deal with every coalition separately, without taking account of possible externalities. They thus face a single mechanism design problem for every coalition and derive a characteristic function.
In …rst price auctions, we derive a genuine partition form game. As is well-known, asymmetric bidders are di¢ cult to handle in this case (see 1 Forges, Minelli and Vohra (2002) discuss interim collective participation constraints in the absence of externalities. The latter assumption takes the form of "orthogonal coalitions" in Myerson (1984)'s study of interim binding agreements in Bayesian games and in Myerson (2007)'s de…nition of an interim incentive compatible core. A. Kalai and E. Kalai (2009) propose a cooperative-competitive solution to two-person Bayesian games. They consider interim participation but with only two players, incentives only matter for the grand coalition, which does not face externalities. Krishna (2002)). Lebrun (1991Lebrun ( , 1999, Marshall et al. (1994), Waehrer (1999) and Bajari (2001) introduce bidding rings in …rst price auctions as prototypes of asymmetric bidders. However, in these papers, rings operate as single entities, which automatically share their information, without relying on any (incentive compatible) mechanism. It follows from our proposition 1 that this simplifying assumption is fully justi…ed if bidding rings can make inside transfers. Thanks to results of Lebrun (1999) and Waehrer (1999), we establish that the grand coalition is always core-stable in a …rst price auction (proposition 3). In the absence of general, analytical solutions for …rst price auctions with asymmetric bidders, we only check that all coalitions are corestable in two speci…c examples, borrowed from McAfee and McMillan (1992) and Marshall et al. (1994).
We …nally consider the e¤ect of direct externalities on collusion. We …rst assume, as in Jehiel and Moldovanu (1996)'s model, that a bidder su¤ers more if a competitor wins the auction than if the object is not sold at all ("negative externalities"). We check to which extent the grand coalition is (not) core-stable in this case. We then propose a three person …rst price auction game in which a two bidder ring is not stable. If direct externalities can possibly be positive, we show that the grand coalition is not core-stable and that there exist "small"(i.e., non-singleton) rings which are core-stable.
All these examples con…rm that direct externalities make cooperative behavior di¢ cult, which was already suggested in Jehiel and Moldovanu (1996), but we give a more precise content to that phenomenon. Indeed, Jehiel and Moldovanu (1996) only show that, under reasonable assumptions, no agreement between (some of) the buyers and/or the seller can be stable. They thus depart from collusion of the bidders in the original auction game.
The paper is organized as follows. Section 2 is devoted to the model and solution concept. In subsection 2.1, we de…ne coalitional equilibria in games with incomplete information. In subsection 2.2, we address the issue of incentives. Proposition 1 and its corollary state that every coalitional equilibrium become incentive compatible once appropriate balanced transfers are made in every coalition. In subsection 2.3, we propose a notion of core-stability for a bidding ring, which does not necessarily gather all the bidders. In section 3, we apply core-stability to auctions. As a benchmark, we consider standard auctions (second price in subsection 3.1 and …rst price in subsection 3.2). In subsection 3.3, we turn to auctions with direct externalities.

Model and solution concept 2.1 From Bayesian games to cooperative games
Let us …x a Bayesian game with independent, private values N; fT i ; q i ; A i ; u i g i2N , namely a set of players N and for every player i, i 2 N , Let P be a coalition structure, namely a partition of N . From and P , we construct an auxiliary Bayesian game (P ) P; fT S ; q S ; A S ; U S g S2P , in which the players are the coalitions S, S 2 P , and A strategy 2 of S in (P ) is a mapping S : T S ! A S . Such a de…nition makes sense if the members of coalition S fully share their information in T S before jointly deciding on an action pro…le in A S . We justify such strategies in the next subsection by showing that they can be derived from coalitions' mechanisms, which allow for appropriate transfers between the coalitions' members. Thanks to these mechanisms, utilities become transferable and incentive compatibility conditions are satis…ed (see proposition 1).
As in Ray and Vohra (1997) and Ray (2007), we de…ne a coalitional equilibrium relative to P as a Nash equilibrium ( S ) S2P of (P ). We assume that for every P , there exists a coalitional equilibrium relative to P and in case of multiple equilibria, we …x a mapping associating a coalitional equilibrium (P ) with every P . 3 We denote as v (S; P ) the expected utility of S at (P ), for every S 2 P , namely where r.v.'s are denoted with a e and (P )( e t) = (P ) K ( e t K ) K2P . (1) de…nes a partition form game, which is constructed from and , with (P ) as an intermediary step.
# for every and P . (2) v(N ) is thus the …rst best Pareto optimal payo¤ of the grand coalition. Given any coalitional equilibrium mapping and any partition P of N , (P ) is a feasible strategy for N (i.e., (P ) 2 A T ). Hence, v is "grand coalition superadditive", or, according to an equivalent terminology, N is e¢ cient in

Coalitions'mechanisms
Let us …x a coalition S. A mechanism S for S is a pair of mappings S = ( S ; m S ): 3 Ray and Vohra (1997) give su¢ cient conditions for the existence of a coalitional equilibrium but their result is not useful in our applications to auctions. However, many speci…c results are available in this context (see section 3). Ray (2007) argues that the partition form game only makes sense if a unique coalitional equilibrium can be associated with every partition (possibly up to transfers). We rather take the view that in case of multiple equilibria, some "standard of behavior" allows us to select among them. Again, this seems appropriate in the context of auctions.
S is S's decision scheme and m S is a balanced transfer scheme. As usual, the interpretation is that members of S are invited to report their types to a planner who then chooses a pro…le of actions and transfers as a function of these reports only 4 . According to Marshall and Marx (2007)'s terminology, we use "bid submission mechanisms", in which the bidders' delegate their decision power to a planner (as opposed to "bid coordination mechanisms", in which the planner just recommends bids to the players).
We assume that utilities over mechanisms are quasi-linear. More precisely, the utility of S for player i 2 S, given his type t i , reported types r S = (r j ) j2S , a "strategy" N nS : T N nS ! A N nS for the players outside S (e.g., N nS = ( K ) K2P;K6 =S , for some partition P of N ) and types t N nS for the players outside S is As this expression explicitly shows, every member i of S incurs an externality from the strategic choices of the players in N n S but, thanks to the private value assumption, does not face any direct informational externality. We de…ne the incentive compatibility (I.C.) of the mechanism S given a mapping N nS : T N nS ! A N nS . More precisely, S is I.C. given N nS i¤ for every i 2 S, every type t i and reported type r i , This de…nition makes sense because, in any coalitional equilibrium, coalition S must take account of the behavior of the players in N nS in elaborating its own strategy. In the case of complete information, S just looks for a best reply to N n S's action pro…le. In the case of incomplete information with private values, S looks for an I.C. best reply to N n S's strategy N nS , without entering the details of N nS (whether the players lie or not, how they possibly gather into subcoalitions, etc.). The next proposition justi…es the coalitions'strategies in the auxiliary Bayesian game; in particular, we show that explicit I.C. conditions are not necessary. The construction, which goes back to Arrow (1979) and d' Aspremont andGérard-Varet (1979, 1982), has been widely used in economic frameworks which do not involve externalities (see, e.g., ). Proposition 1 Let S N ; let N nS : T N nS ! A N nS be an arbitrary strategy of N n S and let S be a best response of S to N nS in (fS; N n Sg). There exists a transfer scheme m S such that Proof: Let us …x S, N nS and S as in the statement. For every i 2 S; t i 2 where the inequality is due to (4) w.r.t. the type vector (t i ; r Sni ). Hence, the mechanism ( S ; b m S ) is I.C. given N nS , but not yet balanced.
]. By taking expectations in (5) we conclude that given N nS and X i2S m i S (r S ) = 0 for every r S 2 T S .
As a direct consequence of this proposition, we get the following Corollary Every coalitional equilibrium can be made incentive compatible: let P be a partition of N and be a coalitional equilibrium relative to P ; for every S 2 P , there exists a transfer scheme m S such that ( S ; m S ) is I.C. given We say that R is core-stable (w.r.t. ) i¤ the (standard) core of w R , C(w R ), is not empty. The interpretation is the following: The coalitional equilibrium mapping is given.
The ring R considers to form; the players outside R are supposed to act individually. R proposes to every i 2 R a share x i of the total expected payo¤ w R (R) = v (R; n R; fkg k2N nR o ), to be achieved by means of an I.C. mechanism R = ( R ; m R ).
Every subcoalition S of R considers non-participation; if S does not participate, the players outside R remain singletons, the players in R n S partition themselves as they wish. Hence S can guarantee the total expected payo¤ w R (S) to its members.
If the participation constraint of every S R is satis…ed, R forms; every player observes his type; R implements R .

Basic properties
Every singleton fkg, k 2 N , is core-stable.
Recalling (2), for every , w N (N ) = v(N ); by (3) and (6), w N is grand coalition superadditive (N is e¢ cient in w N ). This property does not necessarily hold for w R , R N (see example 2 in section 3.3).
C(w R ) corresponds to cautious expectations of the subcoalitions of R.
In particular, C(w N ) contains the usual variants of the core of the partition form game v (see Hafalir (2007)). For instance, the core with singleton expectations, or s core, of v , denoted as C s (v ), is de…ned as the standard core C(f s ) of the characteristic function Similarly, the core with merging expectations (see Maskin (2003)), or If is a game with complete information, let v (S) = max In particular, v (N ) = v(N ). The core of is de…ned as C(v ) (see Aumann (1961)). It is easily checked that, for every and every S N , . 5 The extension of the de…nition of the core to incomplete information may be delicate in the presence of incentive constraints. In particular, our previous construction of transfers, which made any coalitional equilibrium incentive compatible (see proposition 1), cannot be used for the maxmin, since the latter solution concept requires that coalition S considers any possible strategy of coalition N n S. 6 However, in the framework of standard auctions, the di¢ culties disappear. Indeed, every coalition S guarantees itself a total expected payo¤ of 0, whatever the mechanism adopted by N n S, by having all its members bidding 0 independently of their types, a strategy that is clearly I.C. for S. Furthermore, S cannot guarantee more than 0, since the members of N n S can all bid the maximal possible amount, which is I.C. for N n S. Hence, the core is well-de…ned and not empty in standard auctions. But the usual objection against maxmin strategies applies: why should S fear costly overbidding from N n S? 5 Hafalir (2007) focuses on abstract partition form games, which are not necessarily generated by a strategic form game. Hence he does not distinguish the core with cautious expectations from the core. In our framework, at least under complete information, Aumann (1961)'s original de…nition of the core can be used. 6 Our construction applies to the minmax, i.e., to the characteristic function, in the sense that we can dispense with I.C. constraints in the best replies of the coaltion under consideration.

Applications
In this section, we apply our solution concept to auctions with independent private values. In the …rst two subsections, we consider standard auctions, that is, without direct externalities. We check the core-stability of coalitions in several speci…c auction models which have been proposed in the literature. In subsections 3.1 and 3.2, we illustrate that, in absence of direct externalities, coalitions are core-stable. In subsection 3.3, we allow for direct negative externalities. We show that the grand coalition can be made core-stable in this case. However, the s core and the m core of the underlying partition form game can be empty (example 1) and small coalitions may not be corestable (example 2). Finally, if externalities are possibly positive, the core may be empty (example 3).

Standard second price auctions
Let player i's type e t i be a continuous random variable over t i ; t i , 0 t i t i , to be interpreted as his valuation for a single object. A i = [0; M ] is the set of possible bids, where M max i2N t i . Let a = (a k ) k2N be an n tuple of bids. A second price auction is de…ned by the following utility functions where (a) = j fk 2 N : a k = max j2N a j g j.
As is well-known, this game has an equilibrium in weakly dominant strategies. More generally, let P be a partition of N . The auxiliary Bayesian game (P ) has a coalitional equilibrium in weakly dominant strategies described by k S (t S ) = t k for some k 2 S such that t k = max j2S t j and i S (t S ) = 0 for i 2 S, i 6 = k, for every S 2 P and t S = (t j ) j2S . It is easily checked that for every P and S 2 P , where f + = max ff; 0g. The previous expression shows that, at the equilibrium in weakly dominant strategies, the external e¤ects disappear, so that v reduces to a plain characteristic function. In particular, for every S R N and every 2 P(R n S), v (S; n S; ; fkg k2N nR o ) = '(S) and a ring R is core-stable i¤ C('j R ) is not empty, where 'j R (S) = '(S) for every S R.
Proof: Mailath and Zemsky (1991) establish that ' is balanced. Barbar and Forges (2007) further show that ' is supermodular (convex). If the bidders are symmetric, namely if the types e t i , i = 1; :::; n, are i.i.d., an easy direct argument shows that giving the same amount '(N ) jN j to every member of N de…nes a payo¤ n tuple in C('): …rst, I denoting the indicator function, Further, it is easily checked that where F is the distribution function of any e t i . It follows then from (10) that '(S) '(N ) jSj jN j .

Standard …rst price auctions
In this subsection, we assume that the n initial bidders are symmetric, namely that the valuations e t i , i = 1; :::; n are i.i.d. Let a = (a k ) k2N be an n tuple of bids. A …rst price auction is de…ned by the following utility functions where (a) is de…ned as for the second price auction. Obviously, given a nontrivial partition P of N , the players of the auxiliary Bayesian game (P ) are not symmetric. By Lebrun (1999), (P ) has a unique equilibrium, for every partition P . In other words, there exists a unique coalitional equilibrium mapping . However, no general analytical solution is available. Waehrer (1999, proposition 2) shows that for every partition P and every coalitions R, S 2 P such that jRj jSj v (S; P ) jSj v (R; P ) jRj (11) In words, at a …rst price auction, the per capita expected payo¤ of a cartel's member is greater in small cartels 7 . This result enables us to deduce the following Proposition 3 In a standard …rst price auction with symmetric bidders, the grand coalition is core-stable.
Proof: We will show that the vector payo¤ allocating the amount v(N ) jN j to every member of N is in the s core of the underlying partition game v . Let S N and P = From (11), we deduce that for every j 2 N n S, v (fjg ; P ) v (S; P ) jSj while, from the grand coalition superadditivity of v (recall (3)), The latter two inequalities yield (12).
The previous reasoning can be applied to establish the stability of a bidding ring R N if v is superadditive on R. Such a property indeed holds in examples proposed by McAfee and McMillan (1992) and Marshall et al. (1994). McAfee and McMillan (1992) assume that e t i 2 f0; 1g, i = 1; :::; n. They show (in inequality (13)) that, for every coalition S N and or, equivalently, recalling our notation f s (see (7)) One can also check that (11)  jSj is increasing with the size of S (i.e., (13) holds) so that, in their example too, all rings are core-stable.

First price auction with complete information and direct externalities
Following Jehiel and Moldovanu (1996), we consider …rst price auctions with complete information, in which every bidder incurs an externality if a competitor acquires the object. The basic game reduces to N; fA i ; u i g i2N , where A i = f0; ; 2 ; :::g is the set of possible bids, given a smallest money unit > 0. The utility functions are described by an n n matrix E = [e ij ]; for every i, e ii t i is agent i's utility for the object and for every i 6 = j, e ij is the externality incurred by agent j if agent i gets the object. If all bids are 0, the seller keeps the object; agent i's utility is normalized to 0 in this case. Let a = (a k ) k2N ; the utility of player i is To complete this description, we assume that if several players make the highest bid, they all get the object with the same probability.
Since is a game with complete information, the characteristic function v is de…ned by (9).

Core-stability of the grand coalition under negative externalities
Except in example 3, we assume negative externalities, i.e., e ij 0 for every i 6 = j. In this case, given any strategy pro…le (a i ) i2S of S, N n S can in ‡ict a negative payo¤ on S by bidding over max i2S a i ; hence v (S) 0 for S N ; since v(N ) 0, the core C(v ) is not empty. A similar argument shows that, for a coalitional equilibrium mapping proposed in Jehiel and Moldovanu (1996) 8 , w N (N ) = v(N ) 0, while for every S N , w N (S) 0. 8 Jehiel and Moldovanu (1996) prove that, under appropriate genericity conditions, the following strategies (b j ) j2N form an equilibrium in : if t i min j e ji 0 for every i = 1; :::; n, then b i = 0 for every i. Otherwise, let (i; k) be a pair of bidders i 6 = k such that t i e ki is maximal over all t j e lj , j 6 = l (that is, bidder i is willing to pay the highest price for the object, given his valuation and the externalities he might su¤er); take b i = t i e ki , b k = t i e ki 2 and b j < b k , j 6 = i; k. At this equilibrium, which typically involves weakly dominated strategies, bidder i's payo¤ is e ki + 0 and all other bidders j 6 = i get e ij 0 (see Biran (2009), Appendix A, for a full characterization of equilibria).
Hence, for that particular choice of , the grand coalition N is core-stable, namely C(w N ) 6 = ;. Jehiel and Moldovanu (1996) establish the emptiness of the -core of a quite di¤erent market game, in which all agreements between the bidders and the seller are possible. Here, we stick to the original format of the …rst price auction, so that we only allow for collusion between the potential buyers.
At the above coalitional equilibrium mapping , all conceivable cores (e.g., the s core and the m core, see subsection 2.3) are nonempty. The example below illustrates that this property does not necessarily hold for coalitional equilibrium mappings which lead to possibly positive payo¤s. 9 Example 1: n = 4; the matrix of valuations/externalities is  (14) and (15) imply that the characteristic function f s is not grand coalition superadditive, hence that the s core C s (v ) is empty in that example. Let us take t 1 = 4. We now have v(N ) = 1 . Let us assume that bidder 3 competes with the cartel f1; 2; 4g. The matrix is where the …rst row corresponds to the utilities in case the cartel obtains the object. The strategies a 1 = 0, a 2 = 1, a 3 = 1 , a 4 = 0 form an equilibrium. Hence v (f3g ; ff3g ; f1; 2; 4gg) = 0 (16) and similarly for bidder 4. Let us assume again that the …rst two bidders collude, but facing the opposite ring f3; 4g. The relevant matrix is now The strategies a 1 = , a 2 = a 3 = a 4 = 0 are in equilibrium, so that v (f1; 2g ; ff1; 2g ; f3; 4gg) = 2 (17) (16), the analog of (16) for bidder 4 and (17) imply that the characteristic function f m is not grand coalition superadditive, hence that the m core C m (v ) is empty in that example.
Core-stability of a "small" coalition under negative externalities Jehiel and Moldovanu (1996) relate the case of two European …rms who did not cooperate in a procurement auction opposing them to an Asian competitor. They suggest that negative externalities might explain the failure of the natural partners'association but, as explained above, the emptiness of the core that they consider only shows that no stable agreement can be found between the three potential buyers and the seller. In this particular example, cooperation between the European …rms and the Asian one looked unlikely, but the stability of the European coalition could be considered. We illustrate below that, in the presence of externalities, a two …rm cartel may not be stable. If a …rst price auction takes place between the 3 agents, in every equilibrium, agent 1 wins and agent 3 is the second highest bidder; in undominated strategies, 10 p 12; at the lowest price p = 10, the utilities are ( 5; 4; 3). Provided that p < 11, bidders 1 and 2 get a total utility > 10. If they form a joint venture, in every equilibrium, agent 2 represents R = f1; 2g at the auction and wins; in undominated strategies, p = 12: the price raises when agent 1 and agent 2 do not compete. The total utility of f1; 2g is 10, which is less than the sum of agents 1 and 2's individual payo¤s (in our previous notation, w R (f1g) = 5, w R (f2g) = 4, w R (f1; 2g) = 10). The interpretation is the following: if agents 1 and 2 get together, they cannot expect more than 10; if agent 3 plays a dominated strategy, they will even get less. If agent 1 breaks the agreement, he does not expect that agents 2 and 3 (like a European …rm and the Asian …rm above) will collude, but considers a noncooperative equilibrium between the three competitors. At an equilibrium leading to the lowest price, he can expect 5. Similarly, agent 2 can expect 4.

Core-stability of the grand coalition under possibly positive externalities
In example 2, the grand coalition is core-stable. If externalities are negative, the grand coalition can decide not to participate in the auction so as to guarantee 0 to its members, a strategy that is not feasible for small coalitions. More generally, the next proposition, proved in the appendix, states that, if n 3, the grand coalition is core-stable w.r.t. every coalitional equilibrium mapping, even if externalities can be positive. Recall that f + = max ff; 0g.

Proposition 4
In every 3-player …rst price auction with direct externalities such that t i > e + ji for every i; j 6 = i, the grand coalition is core-stable w.r.t. every coalitional mapping .
We conclude this section by illustrating that, if su¢ ciently many players face possibly positive externalities, the grand coalition may not be stable. In the next example, with …ve players, the core, C(v ), is empty.
Example 3: n = 5; every agent i has two neighbors (i 1 mod 5, i+1 mod 5); t i = 3, e ji = 2 if agent j is a neighbor of agent i, e ji = 2 otherwise.
One computes that v(N ) = 3 . By symmetry, if C(v ) 6 = ;, the payo¤ vector in which every agent gets 3 5 must be in C(v ). Let us consider a coalition of the form S = fi; i + 1; i + 3g where + is mod 5, i.e., S contains agent i, a neighbor of agent i and a non-neighbor of agent i. S guarantees max f3 ; 2g if agent i bids and the other members of S bid 0; hence, v (S) 2 3 3 5 , contradicting C(v ) 6 = ;. 10 Hence the grand coalition is not stable in this example. It can be shown that the same holds for all coalitions of 4 players but that all coalitions of 2 or 3 players are stable, for any coalitional equilibrium mapping in undominated strategies.

Concluding remarks
In this paper, we study collusion in auctions, possibly with direct externalities, by associating a cooperative game to the initial Bayesian game modelling the auction. Such a simple "semi-cooperative"approach, which constructs a direct "bridge" between the initial noncooperative game and a cooperative one, abstracts from the details of the strategic negotiation between coalitions (see Ray (2007) and A. Kalai and E. Kalai (2009) for recent discussions of this issue). The partition form game v (S; P ) constructed in this paper is tractable and enables us to use a well-founded solution concept as the core. The auxiliary game Bayesian game behind v (S; P ) was considered in Marshall et al. (1994) and Waehrer (1999) without any justi…cation for the absence of incentive compatibility constraints. The corollary of proposition 1 provides such a justi…cation.
We give a precise content to the idea that "direct externalities make collusion harder". According to the available results, without direct externalities, bidding rings are stable. Examples based on Jehiel and Moldovanu (1996), i.e., with complete information, show that this property no longer holds in the presence of direct externalities. A natural setup to pursue the analysis is the second price auction with externalities proposed by Caillaud and Jehiel (1998), in which the valuations of the initial bidders are independently and identically distributed. They show that if interim individual participation constraints are imposed, the grand coalition may fail from being ex post e¢ cient but do not address the question of its ex ante stability. 10 Equivalently: S = ffi; i + 1; i + 3g ; i = 1; :::; 5g is balanced (with weights S = 1 3 ) and X

S2S
S v (S) 10 In this paper, we focused on independent private values. This assumption, which is standard in the auction framework, plays a crucial role in dispensing with explicit incentive compatibility constraints in the de…nition of coalitional equilibria (i.e., in proposition 1). In more general models, we expect that "second best" coalitional equilibria will need to be considered. Our de…nition of incentive compatibility can be extended to cover general utility functions but the study of coalitional equilibria with I.C. binding constraints is potentially complex.
A Appendix: proof of proposition 4 Let us …x an arbitrary coalitional mapping , namely, for every partition P of N = f1; 2; 3g, a Nash equilibrium (P ) of the auction game in which the players are the coalitions in P . We will show that the core with singleton expectations C s (v ) is not empty, i.e., that C(f s ) 6 = ;, where the characteristic function f s is de…ned by (7).
Let us assume w.l.o.g. that player 1 is e¢ cient in N , namely that t 1 + e 12 + e 13 max ft 2 + e 21 + e 23 ; t 3 + e 31 + e 32 g Then f s (N ) = [t 1 + e 12 + e 13 ] + We will consider the modi…ed characteristic function g de…ned by g (N ) = t 1 + e 12 + e 13 g (S) = f s (S) for every S N and show that C(g ) 6 = ;. Let us set x i = g (fig), i = 1; 2; 3. x i is player i's payo¤ at the equilibrium (ff1g ; f2g ; f3gg) induced by in the 3-person original auction game. Since t i > 0 for every i, the seller cannot keep the object at . If player i gets the object at a positive price p at , x i = t i p < t i ; if player j 6 = i wins the object at , x i = e ji < t i by assumption. Hence x i < t i . Furthermore, x 2 + x 3 e 12 + e 13 . Indeed, if player 1 wins the object at , x 2 + x 3 = e 12 + e 13 . If, say, player 2 wins the object at , the price p must exceed t 1 e 21 , otherwise player 1 would deviate from : x 2 + x 3 = t 2 p + e 23 t 2 t 1 + e 21 + e 23 e 12 + e 13 , where the last inequality follows from (18).