Consistent collusion-proofness and correlation in exchange economies

Journal of Mathematical Economics 38 (2002) 441–463

Consistent collusion-proofness and correlation in exchange economies Gaël Giraud a , Céline Rochon b,∗ b

a BETA, Université Louis Pasteur, Strasbourg, France THEMA, Université de Cergy-Pontoise, 33 bd du Port, 95011 Cergy-Pontoise, France

Received 3 July 2002; received in revised form 17 July 2002; accepted 2 September 2002

Abstract We present a feasible strategic market mechanism with finitely many agents whose Nash, semistrong Nash and coalition-proof Nash equilibria fully implement the Walrasian equilibria. We define a strategic equilibrium concept, called correlated semi-strong equilibrium, and show that the Walrasian equilibria can be implemented by these equilibria, and also by the coalition-proof correlated equilibria of our mechanism. We show that these two concepts, suitably modified with transfers, fully implement the Pareto optimal allocations. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: C72 D51 + implementation Keywords: Coalition-proofness; Correlation; Semi-strong equilibrium; Implementation

1. Introduction The objective of implementation theory, as stated in Jackson (1999), is to characterize the set of social choice correspondences that are obtainable as equilibrium outcomes when individuals interact through some game-form making strategic use of their knowledge. Specifically, given a social choice correspondence F which maps profiles of characteristics to allocations, does there exist a game-form G such that for any profile R of characteristics, the players’ outcome from the game is precisely F(R)? For a pure exchange economy with at least three players but a finite number of players, we present a simple strategic market mechanism, which is individually feasible (a player’s final allocation belongs to his consumption set), collectively feasible when no player goes ∗ Corresponding author. Tel.: +33-1-34-25-67-59; fax: +33-1-34-25-62-33. E-mail address: [email protected] (C. Rochon).

0304-4068/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 ( 0 2 ) 0 0 0 7 7 - 0

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bankrupt (markets clear), and which does not rely on any integer game. A strategy of a player consists in quoting a price and announcing a vector of net demands to every other player in the game. Using this strategic market mechanism, we show that the Walrasian equilibria can be fully implemented via the pure Nash and semi-strong Nash equilibria (Kaplan, 1992). We also obtain full implementation of Walrasian equilibria with coalition-proof Nash equilibria (Bernheim et al., 1987). Consequently, our mechanism is such that outcomewise, the Nash, semi-strong Nash and coalition-proof Nash equilibria are all equivalent. We then study implementation with correlation devices. There are at least two lines of justification for the use of such devices in our setting. First, from a purely game-theoretic viewpoint, many game theoretical attempts to provide foundations to strategic equilibrium concepts seem to designate the correlated equilibrium concept as the “right” solution concept of a non-cooperative game (by contrast with the mere pure Nash equilibrium concept) (see Aumann, 1987; Brandenberger and Dekel, 1987; Hart and Mas-Colell, 2000). Second, correlated equilibria are important when studying market economies, as they have been used to characterize sunspots (Azariadis, 1981; Maskin and Tirole, 1984; Forges, 1988; Peck and Shell, 1991; Forges and Peck, 1995). We introduce the concepts of correlated semi-strong and correlated strong equilibria and compare them to other notions through examples, namely the coalition-proof correlated equilibrium concept of Ray (1996). The correlated semi-strong and coalition-proof correlated equilibria of the strategic market mechanism that we present both implement the Walrasian equilibria. With transfers, these solution concepts fully implement the Pareto optimal allocations. The argument that drives the results stated above is essentially the following: given a Walrasian equilibrium or, more generally, a Pareto optimal allocation, if a coalition profitably deviates, it must do so by modifying its price quoting strategies. However, and this is the important observation, such a deviation is not internally consistent as a member of the coalition would then benefit from an arbitrage opportunity. The hypotheses on preferences required to get these results are minimal. There are several ways of defining the notion of coalition-proofness in the literature, especially when players are allowed to correlate their actions. In this paper, we adopt the concept of coalition-proof correlated equilibrium of Ray (1996), which rests on the coalition-proof Nash equilibrium notion of Bernheim et al. (1987). As defined in Bernheim et al. (1987), “an agreement is a coalition-proof Nash equilibrium (CPNE) if and only if it is Pareto efficient within the class of self-enforcing agreements; an agreement is self-enforcing if and only if no proper coalition of players, taking the actions of its complement as fixed, can agree to deviate in a way that makes all of its members better off.” The notion of Ray (1996) uses extensively that of a CPNE: “a coalition-proof correlated equilibrium of a game is a pair consisting of a correlation device and a CPNE of the game extended by the device.” In Ray (1996), deviations by coalitions take place ex ante, before the players learn their actions recommended by the correlation device, and a coalition that deviates cannot construct a new correlation device. Correlated semi-strong and correlated strong equilibria, as we define them in this paper, share these characteristics as well. In the spirit of Ray, they are the natural extensions of the semi-strong Nash (Kaplan, 1992) and strong Nash equilibria (Aumann, 1959). Recall that the strong Nash equilibrium concept is defined in terms

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of deviations by every possible coalition, and not only unilateral deviations as for Nash equilibria. A semi-strong equilibrium is defined in terms of deviations by coalitions that are Nash equilibria in the reduced game. Using these notions, we truly study implementation problems in a complete information setting. Two papers are related to this one. Hurwicz (1979) presents a strategic market mechanism which is collectively but not individually feasible, anonymous (the strategy sets of the players do not depend on their characteristics), smooth, and whose Nash equilibria implement the Walrasian equilibria. The author does not present any implementation results using coalition-proofness. The mechanism presented by Schmeidler (1980) is collectively but not individually feasible, discontinuous and anonymous. It allows the author to obtain implementation results in strong Nash equilibria. In both of these games, players are not allowed to go bankrupt, and both require that there be at least three players in the game. Dubey (1982) presents a strategic market mechanism that allows the players to exchange out of equilibrium, in contrast with Schmeidler (1980) where the no-trade is the generic outcome out of equilibrium. In Schmeidler (1980), announcing different prices prevents traders from trading, while traders generically do announce different prices. This is the key property that leads to his implementation result in strong Nash. As we show in an example, we do not get implementation in strong Nash precisely because we allow the players to trade even if they announce different prices. The no-trade equilibrium is always a Nash equilibrium of Dubey’s game and hence, his strategic market game is not competitive (a game being competitive when its Nash equilibria fully implement the Walrasian equilibria). Indeed, the no-trade is a Walrasian equilibrium only when initial endowments are Pareto optimal, which is generically not the case. In this paper, we assume that the mechanism designer ignores the preferences of the agents but knows their initial endowments. Indeed, Hurwicz et al. (1995) have shown that there exists no Pareto efficient strategic market mechanism with a finite number of players that does not depend on the initial endowments and which is individually and collectively feasible (any agent’s final allocation belongs to his consumption set and markets clear). It is also well known that if an economy admits a Walrasian equilibrium on the boundary of its feasible set, then the Walrasian correspondence is not Maskin monotonic (see Jackson, 1999 for details on the Maskin monotonicity) and thus not Nash implementable. We avoid this problem by requiring that all the players strictly prefer their initial endowments to any allocation on the boundary of their consumption set (Hong, 1995). Finally, we observe that the Nash equilibria of our mechanism fully implement the constrained Walrasian equilibria under weaker conditions on the initial endowments than those required to implement the Walrasian equilibria. Observe that the constrained Walrasian equilibria share the same efficiency properties as the Walrasian equilibria, and exist even in cases where Walrasian equilibria do not exist. Postlewaite and Wettstein (1989) present a strategic market mechanism that is feasible, continuous and implements in Nash equilibria the constrained Walrasian equilibria. However, they do not address the question of correlation and coalition-proofness. In Appendix A, we show by means of an example that the constrained Walrasian equilibria cannot be implemented in semi-strong Nash in Postlewaite and Wettstein’s game.

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Before presenting the details of the model, it may be useful to add a few words about Shapley–Shubik games. In a completely different context, Shapley and Shubik (1977) present a strategic market game which is individually and collectively feasible, and such that all allocations induced by the Nash equilibria of the game are individually rational, in the sense that for any player, the utility derived from his final allocation is larger than that obtained from his initial endowment. This is so as a player can choose to play the no-trade. However, the no-trade equilibrium is always a Nash equilibrium of the game and hence, their strategic market game is not competitive. More generally, generically, no Nash equilibrium is Pareto efficient in this kind of game (Dubey and Rogawski, 1990). However, when the number of players tends to infinity, the allocations induced by the non-trivial Nash equilibria of this game converge to the Walrasian equilibria (Dubey and Shubik, 1978). The main departure between that type of literature and our approach is that we want to get efficiency even for finite economies. The paper is organized as follows. In Section 2, we present the economy, the game-form and the solution concepts. In Section 3, we present a full implementation result using Nash and semi-strong Nash equilibria. In Section 4, we present implementation results involving correlation and some examples to relate to one another the different solution concepts introduced. In Section 5, we get full implementation of Pareto optimal allocations via correlated semi-strong equilibria and coalition-proof correlated equilibria with transfers.

2. The model 2.1. The economy Notations: |S| denotes the cardinality of the set S; intS and ∂S respectively denote the interior and the boundary of S; BL (x, ) := {y ∈ RL : y − x ≤ }. Consider a pure exchange economy E = (Xi , ui , ωi )i∈I with N agents i ∈ I = {1, . . . , N} and L commodities h ∈ {1, . . . , L}. For each agent i, let Xi ⊂ RL be his consumption set. : Xi → R and ωi ∈ Xi denotes his initial endowment. The utility function of agent i is ui
ˆ i := {x ∈ RL The aggregate endowment is ω¯ := i ωi . Let X ¯ + : x ≤ ω}. Assumption 1. • N ≥ 3, and for all i ∈ I, Xi = RL +. ˆ i. • For all i ∈ I, ui is quasi-concave on X ˆ i such ˆ • Local non-satiation: For all x ∈ Xi , for all > 0, there exists y ∈ BL (x, ) ∩ X that ui (y) > ui (x). ˆ i , ui (ωi ) > ui (x).1 • Boundary condition: For all x ∈ ∂Xi ∩ X Definition 1. • (p∗ , (xi∗ )i ) is a Walrasian equilibrium if and only if p∗ = 0 and 1

This condition implies that for all i ∈ I, ωi >> 0.

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◦ for I, xi∗ ∈ argmaxx ui (x) under the constraint p∗ x ≤ p∗ ωi ,
all∗ i ∈
◦ i xi = i ωi . • (p∗ , (xi∗ )i ) is a constrained Walrasian equilibrium if and only if p∗ = 0 and
◦ for I, xi∗ ∈ argmaxx ui (x) under the constraints p∗ x ≤ p∗ ωi and x ≤ i ωi ,
all∗ i ∈
◦ i xi = i ωi . We denote by WE(E) the set of Walrasian equilibria of the economy E and CWE(E) the set of constrained Walrasian equilibria. Note that CWE(E) induces Pareto optimal allocations and WE(E) ⊂ CWE(E). 2.2. The game-form A game Γ is defined by Γ = (Ai , φi )i=1,...,N where for all i ∈ I, Ai denotes the set of  pure actions of player i, while φi : A = h∈I Ah → RL + denotes the strategic outcome function of player i. Ai will be defined as the set of prices proposed by player i, pi ∈ RL + , and the vector of net demands addressed by player i to player j = i, zi = (zij )j=i ∈ (RL )N−1 : L N−1 }. Ai = {(pi , zi ) ∈ RL + × (R )

The strategic budget constraint of player i is  j pj (zij − zi ) ≤ 0.

(1)

j=i

Note that this constraint does not depend on the price quoted by player i. Let ψi : A → RL be the auxiliary function defined by 
j i j=i (zj − zi ), if (1) is satisfied, ψi (a) = −ωi , otherwise. We can now define the strategic outcome function of player i, φi : A → RL +: φi (a) = proji ◦ projF ((ωk + ψk (a))k=1,...,N ), where



F = (xi )i ∈



Xi :

i∈I

 i

xi ≤



 ωi

i

is the feasible set, which is convex, compact and non-empty; projF : (RL )N → F is the orthogonal projection over F and proji : (RL )N → RL is the canonical projection on the ith coordinate.

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The payoff function of player i with respect to the actions, gi : A → R, is defined by gi (a) = ui ◦ φi (a). Remark. The mechanism is feasible, in the sense that any strategy profile induces an outcome such that demand does not exceed supply (given by the initial endowments) and the agents receive a bundle which belongs to their consumption set. Moreover, the mechanism is not balanced out of equilibrium if some player goes bankrupt. In this way, part of the initial resources are wasted. Finally, it is not individually rational out of equilibrium since if someone is kept in the red, he necessarily gets zero. 2.3. Solution concepts We now recall a first definition of a coalition-proof Nash equilibrium (Bernheim et al., 1987).2 Definition 2. • If N = 1, a∗ ∈ A is a coalition-proof Nash equilibrium (CPNE) if and only if a∗ ∈ argmaxa∈A gi (a). • For N > 1, suppose that the notion of CPNE has been defined for less than N players. Then ◦ a¯ ∈ A is self-enforcing if for all coalition S ⊂ I, a¯ S = (¯ai )i∈S is a CPNE in the game a¯ 3 Γ a¯ ,S with outcome  function for player i ∈ S defined by φi (aS ) := φi (aS , a¯ −S ) for all aS ∈ AS = i∈S Ai . • a∗ ∈ A is a CPNE if it is self-enforcing and Pareto-undominated among the self-enforcing strategy vectors, namely, for all a¯ ∈ A self-enforcing, ∃ io ∈ I such that gio (¯a) ≤ gio (a∗ ).

(2)

We now present a definition of a CPNE equivalent to that just defined. Definition 3. Let a¯ ∈ A and S ⊆ I, S = ∅. • An internally consistent improvement (ICI) of S upon a¯ is defined by induction on the cardinality of S: ◦ If S = {i}, ai ∈ Ai is an ICI  of {i} upon a¯ if and only if gi (ai , a¯ −i ) > gi (¯a). ◦ If |S| > 1, then aS ∈ AS = i∈S Ai is an ICI of S upon a¯ if and only if for all i ∈ S, gi (aS , a¯ −S ) > gi (¯a), •

a∗

(3)

and there exists no coalition S  ⊆ S, S  = ∅ which has an ICI upon (aS , a¯ −S ). ∈ A is a CPNE if and only if there exists no S ⊆ I, S = ∅ which has an ICI upon a∗ .

2 X ⊆ Y (respectively X ⊂ Y ) means that X is included in and possibly equal to (respectively strictly included in) Y . 3 −S = I \ S, a −S = (ai )i∈−S .

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Remark. A strict CPNE is defined as a CPNE with requirements (2) and (3) replaced respectively by • there exists no a¯ ∈ A self-enforcing such that gi (¯a) ≥ gi (a∗ ), for all i ∈ I, with strict inequality for some io ∈ I; • for all i ∈ S, gi (aS , a¯ −S ) ≥ gi (¯a) and for some io ∈ S, the strict inequality holds. Definition 4. Let a¯ ∈ A and S ⊆ I, S = ∅. • Let Γ a¯ ,S be the game induced by S with outcome function for all a ∈ AS and for all i ∈ S defined by φia¯ (a) := φi (a, a¯ −S ). Then NE(Γ a¯ ,S ) = {a∗ ∈ AS : a∗ is a pure Nash equilibrium of Γ a¯ ,S }. • A strategy a∗ ∈ A is a semi-strong Nash equilibrium (ssNE) if for every coalition ∅ = ∗ ∗ ) S ⊆ I and every aS ∈ NE(Γ a ,S ), there exists i ∈ S such that gi (a∗ ) ≥ gi (aS , a−S (Kaplan, 1992). • A strategy a∗ ∈ A is a strong Nash equilibrium (sNE) if and only if for all coalition S ⊆ I ∗ ) and for all strategy aS ∈ AS , there exists an agent i ∈ S such that gi (a∗ ) ≥ gi (aS , a−S (Aumann, 1959). Remark. Neither the CPNE nor the sNE involve any binding agreements. They are both, as well as the intermediate notion of semi-strong Nash equilibrium, strictly non-cooperative notions. By analogy, the cooperative concept corresponding to sNE is the Core, the one corresponding to semi-strong Nash equilibrium is the Bargaining set, and the one corresponding to CPNE is the consistent Bargaining set.

3. Full Nash and semi-strong Nash implementation Proposition 1. Under Assumption 1, the game Γ with complete information is such that • for all a∗ ∈ NE(Γ), p∗i = p¯ for all i and (p, ¯ (φi (a∗ ))i∈I ) ∈ WE(E); ∗ ∗ • for all (p , (xi )i∈I ) ∈ WE(E), there exists a∗ ∈ NE(Γ) such that for all i, p∗i = p∗ and for all i ∈ I, φi (a∗ ) = xi∗ . Proof. We first show that every Nash equilibrium induces a Walrasian equilibrium (“NE(Γ) ⊆ WE(E)”). ∗ Let a∗ = (ai∗ )i ∈ NE(Γ), where ai∗ = (p∗i , (z∗i j )j=i ); then a is individually rational: for all i ∈ I, gi (a∗ ) ≥ ui (ωi ). Indeed, suppose there exists io ∈ I such that gio (a∗ ) < uio (ωio ). Then by playing the strategy a¯ io = (p¯ io , (¯zijo )j=io ) defined by p¯ io = p∗io and for all j = io z¯ ijo =



h∈{i / o ,j}

∗j

∗j

(zh − z∗h j ) + zio ,

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∗ ,a player io can get φio (a−i ¯ io ) = ωio . By definition and assumption, o ∗ ∗ , a¯ io ) = uio (φio (a−i , a¯ io )) = uio (ωio ) > gio (a∗ ), gio (a−i o o

which contradicts the definition of a Nash equilibrium. Consequently, each player satisfies his strategic budget constraint. For if io violates his strategic budget constraint, then φio (a∗ ) = 0 as ψio (a∗ ) = −ωio and Xi = RL + for all i, in which case the boundary condition is not met. ˆ i ). Indeed, suppose φio (a∗ ) ∈ ∂(X ˆ io ) for some io ∈ I. Then Now, for all i, φi (a∗ ) ∈ int(X ∗ φi (a ) ∈ ∂(Xi ) for some i ∈ I possibly different from io . But then the boundary condition is violated.  Now, we show that for all i ∈ I, p∗i = p¯ ∈ RL ++ . Suppose that there exists j, j such that ∗h ∗h  pj > pj ≥ 0 for some commodity h ∈ {1, . . . , L}. Some player i ∈ / {j, j } is such that ∗ ˆ φi (a ) ∈ int(Xi ), as we have seen. By the local non-satiation hypothesis, for every > 0 ˆ i such that yi − φi (a∗ ) ≤ and ui (yi ) > ui (φi (a∗ )). there exists yi ∈ X Consider the following strategy for player i: ai = (p˜ i , (˜zij )j=i ) where • p˜ i = p∗i , / {i, j, j  }, • z˜ ik = z∗i k , for all k ∈ ∗h ∗h ∗ ∗ ∗ • z˜ ij = z∗i j  + [pj  (yi − φi (a ))/(pj − pj  )]I{h} + yi − φi (a ), ∗h ∗h ∗ ∗ • z˜ ij = z∗i j − [pj  (yi − φi (a ))/(pj − pj  )]I{h} ,

where I{h} = (0, . . . , 0, 1, 0, . . . , 0) ∈ RL with non-zero hth coordinate. This strategy allows player i to obtain yi , a contradiction. Now, since every player i ∈ I satisfies his strategic budget constraint, then   φi (a∗ ) = (ψi (a∗ ) + ωi ). i

i

Let us verify that for all i ∈ I, φi (a∗ ) ∈ argmax ui (x), under the constraint p¯ x ≤ p¯ ωi . Suppose that there exists io ∈ I for which there exists yio ∈ Xio = RL ¯ yio ≤ p¯ ωio and uio (yio ) > uio (φio (a∗ )). By quasi-concavity and + with p local non-satiation of uio , every element z ∈]φio (a∗ ), yio ] = {λφio (a∗ ) + (1 − λ)yio : λ ∈ [0, 1]}, ˆ io since φio (a∗ ) ∈ is such that uio (z) > uio (φio (a∗ )). For z close enough to φio (a∗ ), then z ∈ X ˆ io ). Moreover, p¯ z ≤ p¯ ωi . Hence, using the strategy a¯ io = (p˜ io , (˜zio )j=io ) which int(X j consists in playing the following price and quantity setting actions when he observes p: ¯ • p˜ io = p∗io = p, ¯

o ∗ • z˜ ijo = z∗i j + z − φio (a ) for some j,

o / {io , j}, • z˜ iko = z∗i k for all k ∈

allows player io to get z as a final outcome. Indeed, (φk (a∗ ))k∈I belongs to the interior of the ∗ ,a feasible set F ; for z close enough to φio (a∗ ), we get that (ψk (a−i ¯ io )+ωk )k∈I still belongs o

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∗ ,a to F . If no player goes bankrupt in spite of the deviation of io , then φio (a−i ¯ io ) = z. If o player j = io goes bankrupt, then he gets zero and player io still gets z. We now show that WE(E) ⊆ NE(Γ). Let (p∗ , (xi∗ )i ) ∈ WE(E). Consider the strategy profile a∗ = (ai∗ )i defined for all i ∈ I by ∗i ∗ ∗ ∗ ∗ ∗ ai = (p∗i , (z∗i j )j=i ) where pi = p and zj = (1/N)(xi −ωi ). We then see that a ∈ NE(Γ) ∗ ∗ and for all i ∈ I, φi (a ) = xi . Indeed, player i has no incentive to deviate in prices as his own price does not affect his final outcome (recall that the budget constraint of agent i does not depend on his own price). In addition, player i has no incentive not to satisfy his budget constraint since he then receives zero. The Walrasian equilibrium being individually rational, he is better off satisfying his constraint. 䊐

Remark. The reader who dislikes the boundary condition for its strong implications on the endowments should note that the above result is true for constrained Walrasian equilibria, once the boundary condition in Assumption 1 is replaced with the assumption that for all i ∈ I, ωi > 0 and preferences are strictly monotone. Indeed, any CWE(E) can be obtained as a NE(Γ), by the same argument as above since it makes no use of the boundary condition. Now, to show that any Nash equilibrium induces a constrained Walrasian equilibrium, the following modifications of the proof are in order: • No player goes bankrupt, since the no-trade is preferred to 0. • Every player fixes the same price. Otherwise, there is an arbitrage opportunity for some player i which is not saturated by some allocation which is feasible (if he were saturated, then he would possess all of the endowments, hence some other player would not hold anything; but we are at a Nash equilibrium, and consequently every player is individually rational). The rest of the proof follows verbatim. We shall use again constrained Walrasian equilibria in the proof of Proposition 5. Proposition 2. Under Assumption 1, the semi-strong equilibria of Γ fully implement every Walrasian equilibrium of E. Sketch of the proof. Take (p∗ , (xi∗ )i ) ∈ WE(E) and consider b∗ = (bi∗ )i defined for all ∗i ∗ ∗ ∗ i ∈ I by bi∗ = (p∗i , (z∗i j )j=i ), where pi = p and zj = (1/N)(xi − ωi ). By the Pareto ∗ optimality of (xi )i and feasibility, the grand coalition has no improvement b upon b∗ . Now, assume S ⊂ I has an improvement b upon b∗ . Then there must exist i ∈ S such that when playing bi instead of bi∗ , i quotes a different price vector p∗i = p∗ . Hence, some player ∗ io ∈ S \ {i} faces an arbitrage opportunity in Γ b ,S if b is played. Indeed, |S| ≥ 2 since b∗ is a Nash equilibrium. Now by using some strategy b˜ io , io could improve his payoff. Hence, ∗ b∈ / NE(Γ b ,S ) and therefore is not an improvement. The converse argument follows the proof of Proposition 1. Remarks. • In the proof of Proposition 3 , we show in Appendix A that WE(E) ⊆ CPNE(Γ). As CPNE(Γ) ⊆ NE(Γ) = WE(E), then WE(E) = CPNE(Γ). Hence, our strategic

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market game is such that outcomewise, the Nash, semi-strong Nash and coalition-proof Nash equilibria are all equivalent. In general, the following inclusions hold, ssNE(Γ) ⊆ CPNE(Γ) ⊆ NE(Γ), but may be strict. • One may wonder why we cannot implement WE in strong Nash equilibria while this is possible in Schmeidler’s (1980) game. The following example highlights what is going on. Let L = {x, y, z} and N = 3. Let u1 (x, y, z) = 2x + 2y + z, u2 (x, y, z) = x + 2y + 2z, u3 (x, y, z) = 2x + y + 2z, ω1 = (1, 1, 4),

ω2 = (1, 1, 1) = ω3 .

The unique Walrasian equilibrium of this economy is given by p = (1, 1, 1), x1 = (3, 3, 0),

x2 = (0, 0, 3) = x3 .

The corresponding strategies played by the players are: p¯ 1 = p¯ 2 = p¯ 3 = (1, 1, 1), j

z¯ 1 = ( 23 , 23 , − 43 ),

j = 1,

j

j = 2,

j

j = 3.

z¯ 2 = (− 13 , − 13 , 23 ), z¯ 3 = (− 13 , − 13 , 23 ),

Now, let us consider a deviation by players 2 and 3 consisting in playing p2 = p3 = (0, 0, 0) and the following quantity setting strategy: z∗1 = ( 23 , 23 , − 43 ), 2 5 2 z∗1 2 = ( 3 , 3 , 3 ),

z∗3 2 = (−1, 1, 0),

5 2 2 z∗1 3 = ( 3 , 3 , 3 ),

z∗2 3 = (0, 0, 0).

Following this deviation, the final allocations are x1 = (0, 0, 0),

x2 = (0, 3, 3),

x3 = (3, 0, 3).

We can consider a further deviation by player 3 alone z˜ 13 = z∗1 3 ,

z˜ 23 = (0, 3, 3).

Following this new deviation, the final allocations are x˜ 1 = (0, 0, 0),

x˜ 2 = (0, 0, 0),

It is a profitable deviation.

x˜ 3 = (3, 3, 6).

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For coalition {2, 3} to be able to deviate profitably, she needs to modify her prices. By contrast, in Schmeidler’s (1980) game,4 if coalition {2, 3} modifies her prices, she is no longer allowed to trade with {1}. As WE is a subset of the Core, {2, 3} cannot improve her situation. In our setup, nothing prevents coalition {2, 3} from trading once she has changed her price quoting strategy. Hence, the impossibility to implement in strong Nash equilibria with our game. More generally, the reason why Schmeidler (1980) succeeds in implementing the Walrasian correspondence by means of strong Nash equilibria is that generically in the product strategy space, there is no trade in Schmeidler’s game. Indeed, traders announcing different prices cannot trade, which would be the case for a generic choice of price quoting strategies. On the other hand, given some pure Nash equilibrium strategy profile, the induced allocation is a Walrasian equilibrium, hence belongs to the Core. Since, on the other hand, a deviating coalition must modify its price quoting strategies for the deviation to be profitable, such a coalition, when deviating, will no more be able to trade with the non-deviating players. As a consequence of the Coreproperty of the original equilibrium allocation, the coalitional deviation cannot be profitable, whence the implementation result in strong Nash in Schmeidler’s paper. In our game, players can trade even if they announce different prices, and consequently it is possible for a coalition to profitably deviate from a given Walrasian equilibrium. However, what we have just shown in Proposition 2, is that such a deviation cannot be consistent. Thus, the non-implementability of WE by means of strong Nash equilibria in our game is the price to pay for allowing players to trade even out of equilibrium.

4. Correlation In this section, we extend the analysis presented so far by introducing correlation devices. From a purely game-theoretic viewpoint, many game theoretical attempts to provide foundations to strategic equilibrium concepts seem to designate the correlated equilibrium concept as the “right” solution concept of a non-cooperative game (by contrast with the pure Nash equilibrium concept). Aumann (1987) provides a Bayesian interpretation of correlated equilibria, Brandenberger and Dekel (1987) provide a justification of this concept in terms of rationalizability while Hart and Mas-Colell (2000) give a foundation of correlated equilibria in terms of learning mechanisms with limited rationality. On the other hand, correlated equilibria are important when studying market economies, as they have been used to characterize sunspots in various institutional situations (Azariadis, 1981; Maskin and Tirole, 1984; Forges, 1988; Peck and Shell, 1991; Forges and Peck, 1995). We now recall the notion of a coalition-proof correlated equilibrium introduced by Ray (1996) and give a definition of correlated semi-strong and correlated strong equilibria. We present some examples to relate to one another the various solution concepts and give an implementation result using the strategic market mechanism presented above. 4

See Appendix A for the construction of Schmeidler’s mechanism.

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Definition 5. • A correlation device d consists  of finite sets of messages, {M1 , . . . , MN }, and a probability distribution P over M = i∈I Mi . The device selects private messages for each player according to P. • The extended game Γd is the game where the correlation device d sends private messages to the players and then the game Γ is played. • A pure strategy σi for player i in the game Γd is a mapping σi : Mi → Ai . • The payoff to player i when the pure strategy vector σ = (σi )i is played is Ui (σ) =



P(m)ui ◦ φi ((σi (mi ))i ).

m∈M

Definition 6. • A coalition-proof correlated equilibrium (CPCE) of the game Γ is a pair (d, σ) such that σ is a CPNE of the extended game Γd . • A correlated semi-strong equilibrium (CssE) of the game Γ is a pair (d, σ) with d a correlation device and σ a ssNE of the game Γd . • A correlated strong equilibrium (CsE) of the game Γ is a pair (d, σ) with d a correlation device and σ a sNE of the game Γd . Remarks. • In Aumann’s (1987) Bayesian interpretation of correlated equilibria, uncertainty bears on strategies. Here, typically, the space of strategies of a player is not finite. Hence, to impose |Mi | < ∞ for all i is a real restriction, that could however be suppressed without modifying any results of this paper. One can in fact define in all generality (Forges, 1986) a correlation device d as a probability space (C, C, P) together with sub σ-fields (Ci )i∈I of C. The extension Γd of the game Γ by the device d is the game where nature first selects c ∈ C according to P, each player i ∈ I is then informed of the events in Ci which contain c, and finally the game Γ is played. In the game Γd , a pure strategy σi for player i is a Ci -measurable mapping σi : C → (Ai , B(Ai )). The payoff to player i when σ = (σi )i∈I is played is EP (ui (φi (σ))). The probability measure πi induced on Ai by σi and P is defined as follows: for all B ∈ B(Ai ), πi (B) = P((σi )−1 (B)). • As pointed out in Ray (1996), coalition-proofness depends on the structure of the correlation device. The revelation principle, which normally allows one to restrict attention to canonical representations, namely the equilibria consisting of a canonical device and the obedient strategy, does not hold for CPCE. • Example 2 in Ray (1996) can be used to show that the revelation principle does not hold for CssE. Recall simply that in a two-player game, CPCE and CssE are equivalent notions. • Deviations by coalitions take place ex ante, before the players learn the actions recommended by the correlation device, and a coalition which deviates cannot construct

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a new correlation device. Without this restriction, the game is no longer with complete information, but with incomplete information (see Giraud and Rochon, 2001). • Ray (1996) has shown that every pure CPNE is a direct CPCE. It is also true that any pure ssNE is a direct CssE. This property will be used in a crucial way in the next proposition. • For N = 2, ssNE and CPNE are equivalent notions, as are CssE and CPCE. Ray (1996) proves the existence of such equilibria. However, existence is not guaranteed for N ≥ 3, as he shows in a counterexample. • By its very definition, the notion of correlation used here involves devices used by the players only. No central planner is making use of these devices. For the sake of completeness, the next example reminds that CPNE, ssNE and sNE are distinct notions, as are their correlated counterparts. Example 1. Consider the three-player game in Fig. 1, where player A chooses rows, player B chooses columns and player C chooses matrices (Fig. 1). The two Nash equilibria of this game are N1 = (A2 , B2 , C1 ) and N2 = (A1 , B1 , C2 ). Note that N2 is not a sNE (as it is Pareto dominated by N1 ), nor a ssNE. Holding the action played by C fixed to C1 , (A2 , B2 ) is the unique Nash equilibrium (note that to play A2 is a dominant strategy for player A). (A1 , B1 ) is a profitable deviation from (A2 , B2 ) which is not a Nash equilibrium; hence (A2 , B2 , C1 ) is a ssNE but not a sNE.

Fig. 1. ssNE = sNE.

The game presented in introduction in Bernheim et al. (1987), which inspired the game in Fig. 1, gives an example of a CPNE which is not a ssNE and hence, of a CPCE which is not a CssE. It is identical to the game in Fig. 1, but for the entry (A2 , B1 , C1 ) which reads (−5, −5, 0). Example 2. We now observe that for the no-trade Walrasian equilibrium associated to the following economy, the equivalence ssNE = CPCE = CssE holds. We therefore observe that in some cases, correlation does not modify the strategic equilibrium outcomes. Moreover, in this example, the unique ssNE is not a sNE. Let I = {1, 2, 3, 4}, L = 2 and Xi = R2+ . For all i ∈ I, let ui (x, y) = xα y1−α , with α = 1/2. For each player, the marginal rate of substitution between x and y is α y px = y. 1−αx p Let ωi = (1/2, 1/2) for all i. Then there exists a unique Walrasian equilibrium, the no-trade equilibrium, with p∗ = (px , py ) = (1/2, 1/2) and xi = ωi for all i ∈ I.

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Note that the CPCE corresponding to the no-trade Walrasian equilibrium is efficient. Indeed, if it were not Pareto optimal, then the grand coalition could play the no-trade, whatever the message m ∈ M delivered by the device. This would represent a profitable deviation, which is self-enforcing. By Proposition 2, the no-trade equilibrium is a semi-strong equilibrium, hence a correlated semi-strong equilibrium (trivially). We now exhibit a coalition with a profitable deviation that is not a Nash equilibrium to show that the no-trade equilibrium is a ssNE but not a sNE. Assume ph1 = p∗h = ph3 is observed by player 2 and ph2 = p∗h = ph4 is observed by player 1, for h ∈ {x, y}. By playing the following strategy ai , player i ∈ S = {1, 2} would get ω/2 ¯ > ωi : ai = (p˜ i , (˜zij )j=i ) where • p˜ i = p∗i , • z˜ ik = 0 for all k ∈ / {i, j, j  }, • z˜ ij = [pj yˆ i /(phj − phj )]I{h} + ω/2 ¯ − ωi , • z˜ ij = −[pj yˆ i /(phj − phj )I{h} ],

¯ and where I{h} = (0, . . . , 0, 1, 0, . . . , 0) ∈ RL with non-zero hth coordinate, yˆ i = ω/2 j  = j + 2. Proposition 3. Assume Assumption 1 holds. Every Walrasian equilibrium can be implemented as a correlated semi-strong equilibrium (CssE). Proof. Let (p∗ , (xi∗ )i∈I ) be a Walrasian equilibrium. Consider the trivial correlation device d ∗ with |Mi | = 1 for all i ∈ I, and let σ¯ = (σ¯ i )i be such that for all i ∈ I, σ¯ i (mi ) = ∗i ∗ ∗ ∗ ¯ is a (p∗i , (z∗i j )j=i ) where pi = p and zj = (1/N)(xi − ωi ) for all j. We show that σ ssNE, and hence a CssE: there is no coalition S ⊂ I which has a profitable deviation σ¯ S upon σ¯ which is a Nash equilibrium. Indeed, in order to improve his payoff, player j ∈ S must change his price quoting strategy. Suppose there exists j ∈ S such that σ¯ S,j (mj ) = ∗j (pj , (zk )k=j ) with pj = p∗ . Assume without loss of generality phj < p∗h for some h ∈ {1, . . . , L}. Then there exists j  ∈ I \ S and i ∈ S \ {j} such that pj = p∗ and i faces an arbitrage opportunity. Note that S is a strict subset of I as (xi∗ )i is Pareto optimal and {j} is a strict subset of S as a Walrasian equilibrium is a Nash equilibrium. As (p∗ , (xi∗ )i∈I ) is a Walrasian equilibrium, the boundary condition implies that xi∗ ∈ ˆ i for all i. intX ˆ i such that By the local non-satiation hypothesis, for every > 0 there exists yi ∈ X ∗ ∗ yi − xi  ≤ and ui (yi ) > ui (xi ). Consider the following strategy for player i: σi (mi ) = (p˜ i , (˜zij )j=i ) where • p˜ i is chosen according to σ¯ S , • z˜ ik = z∗i / {i, j, j  }, k , for all k ∈ i ∗i • z˜ j = zj + [pj (yi − xi )/(phj − phj )]I{h} + yi − xi , h h • z˜ ij = z∗i j  − [pj (yi − xi )/(pj  − pj )]I{h} ,

where I{h} = (0, . . . , 0, 1, 0, . . . , 0) ∈ RL with non-zero hth coordinate, (z∗i l )l=i is the quantity setting allocation of player i according to σ¯ (respectively σ¯ S ) if i ∈ I\S (respectively

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S), and xi is the final allocation received by i when σ¯ is played by I \ S and σ¯ S is played by ˆ i , and thus the improvement by coalition S is not a Nash equilibrium. S. Player i gets yi ∈ X Hence, σ¯ is a ssNE. 䊐 Remark. We show in Appendix A (Proposition 3 ) that every Walrasian equilibrium can be implemented as a CPCE.

5. Transfers In Proposition 3, we have shown that WE(E) ⊆ CssE(Γ), and in Proposition 3 in Appendix A, we show WE(E) ⊆ CPCE(Γ). In order to get full implementation, we need to allow for transfers and substitute the whole set of Pareto optimal allocations to the set of Walrasian equilibria. Indeed, without transfers we cannot show that a CPCE is efficient. Before presenting an example to support this idea, we briefly sketch the underlying intuitive argument: assume a given CPCE is not efficient. Then there exists a Pareto optimal allocation which Pareto dominates the allocation induced by this CPCE. The grand coalition should be able to set a deviation to play the Pareto optimal allocation. However, any Nash equilibrium is a Walrasian equilibrium (as shown in Proposition 1) and any consistent deviation from a CPCE must be a Nash equilibrium. Hence, the grand coalition can deviate and play the Pareto optimal allocation only if she finds a way to induce this allocation as a Nash equilibrium and hence as a Walrasian equilibrium. There is a priori no reason to believe that this Pareto optimal allocation is a Walrasian equilibrium of the economy without transfers. The only way to let the grand coalition deviate while playing in Nash is to use transfers, which would allow to see any Pareto optimal allocation as a Walrasian equilibrium. Example 3. Consider a two goods–two agents economy with three Walrasian equilibria, A, B, C, as in Fig. 2 (an example of such an economy can be found in Mas-Colell et al., 1995, p. 521). The grand coalition can play a with probability 1/2, 1/2. Such a strategy is a CPCE, but it is not efficient (it is in the coalition’s own interest not to follow the recommendation of the device).

Fig. 2. Efficiency of a CPCE vs. transfers.

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We now define the notions of coalition-proof correlated equilibrium and correlated semi-strong equilibrium with transfers, and obtain full implementation of all Pareto optimal allocations with these solution concepts applied to our strategic market mechanism. For all i ∈ I, let Mi be the finite space of messages of player i, with generic element mi . Definition 7. • A correlated contract, (d, y), is a correlation device d = (M
1 , . . . , MN , P) together with an N-tuple of transfers y = (yi )i∈I ∈ RN which satisfies i∈I yi = 0. y • Γy is the modified game obtained from Γ by replacing ψi (·) for all i ∈ I with ψi (·) defined by 

j j i i y j=i (zj − zi ), if j=i pj (zj − zi ) ≤ yi , ψi (a) = −ωi , otherwise. • The corresponding outcome function is defined by y

y

φi (a) = proji ◦ projF ((ωk + ψk (a))k=1,...,N ). • The extended game Γd,y is the game where the correlation device d sends private messages to the players, and then the game Γy is played. • A pure strategy σi for player i in the game Γd,y is a mapping σi : Mi → Ai . • The payoff to player i when the pure strategy vector σ = (σi )i is played is  y Ui (σ) = P(m)ui ◦ φi ((σi (mi ))i ). m∈M

N • Let R : (Ω, A, λ) → (RL + ) be any random allocation, where (Ω, A, λ) is an arbitrary
probability
space. R is said to be Pareto optimal if it is ex post feasible, namely, R (ω) = i i∈I i∈I ωi a.s., and there does not exist any other feasible random allocation N ν  λ R : (Ω , A , ν) → (RL + ) such that E (ui (R (ω))) ≥ E (ui (R(ω))) for all i, with strict inequality for some i ∈ I.

Remark. We call a Pareto optimal random allocation ex post feasible in order to distinguish our feasibility notion from the one introduced by Prescott and Townsend (1984) where feasibility is only required in expectation. Definition 8. • A collusion-proof correlated equilibrium (cpCE) of the game Γ is a triple (d, y, σ) such that σ is a CPNE of the extended game Γd,y , i.e. (d, σ) is a CPCE of the game Γy . • A correlated semi-strong equilibrium with transfers (cssE) of the game Γ is a triple (d, y, σ) such that σ is a ssNE of the extended game Γd,y , i.e. (d, σ) is a CssE of the game Γy . Assumption 2. • N ≥ 3, and for all i ∈ I, Xi = RL +.

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ˆ i. • For all i ∈ I, ui is quasi-concave on X ¯ then for all i ∈ I, there • Constrained local non-satiation: For all i ∈ I, if 0 ≤ xi∗ < ω, exists 0 ≤ xi ≤ ω¯ such that ui (xi ) > ui (xi∗ ). Proposition 4. Assume Assumption 2 holds and preferences are strongly monotone and continuous. Every cssE induces a deterministic Pareto optimal outcome. Conversely, every Pareto optimal allocation can be obtained as a cssE. Proof. Consider a Pareto optimal allocation (xi∗ )i∈I . By the constrained second welfare theorem (see Appendix A), as preferences are strongly monotone and continuous, is
there ∗ = ∗ = 0 and an assignment of wealth levels w∗ = (w∗ ) a vector of prices p with w i∈I i i i
p∗
∈ I, xi∗ ∈ argmaxx ui (x) under the constraints p∗ x ≤ w∗i and i ωi such that
for ∗all i
x ≤ i ωi , and i∈I xi = i∈I ωi . Consider the contract (d ∗ , y∗ ) where y∗ = (yi∗ )i with yi∗ = w∗i − p∗ ωi , and d ∗ is trivial, namely |Mi | = 1, and the strategy of player i is defined ∗i ∗ by σ¯ i (mi ) = (p∗ , (z∗i ¯ is a cssE using j )j=i ) where zj = 1/N(xi − ωi ). We can show that σ the proof of Proposition 3. Indeed, Proposition 3 shows that σ¯ is a ssNE of the game Γd ∗ ,y∗ . ¯ is a CssE of Γy∗ . Hence, (d ∗ , σ) Observe that by Jensen’s inequality, every Pareto optimal random allocation must be in fact deterministic. Now, let (d, y, σ) be a cssE. If the random allocation (xi )i∈I induced by (d, y, σ) is not Pareto optimal, there exists a deterministic Pareto optimal allocation which Pareto dominates it. (Indeed, if (xi )i∈I is not deterministic, it is Pareto dominated by (E(xi ))i∈I . If this last deterministic allocation is itself not Pareto optimal, then there exists a Pareto optimal deterministic allocation (˜xi )i∈I in E which dominates (E(xi ))i∈I , hence dominates (xi )i∈I .) Using the constrained second welfare theorem, the grand coalition can choose to play a constrained Walrasian equilibrium with transfers which gives an improvement upon (d, y, σ). 䊐 Remarks. • All the transfers used in the above argument are chosen by the grand coalition, which therefore must know all the characteristics of the economy. This admittedly very strong assumption could be dropped. • We show in Appendix A (Proposition 5 ) that every cpCE induces a deterministic Pareto optimal outcome and every Pareto optimal allocation can be obtained as a cpCE. 6. Conclusion In this paper, we studied implementation problems in a complete information pure exchange economy, making use of a feasible strategic market mechanism that does not rely on the use of any integer game. The solution concepts used are the following: Nash, semi-strong Nash, coalition-proof Nash, coalition-proof correlated (with and without transfers), and correlated semi-strong equilibria (with and without transfers). In a complementary paper (Giraud and Rochon, 2001), we study mechanism design problems in an incomplete information setting.

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Some questions remain open. In particular, • Moreno and Wooders (1996) have defined a coalition-proof correlated equilibrium concept intermediate to that of Ray (1996) and Einy and Peleg (1995). In their model, deviations occur before the mechanism recommends actions but coalitions are allowed to construct new mechanisms. We leave for further research the task of extending our result to that concept. • the strategic outcome function used here is discontinuous, as in Dubey (1982) and Schmeidler (1980), although for a reason different from theirs. The discontinuity of our game is linked to the possibility of default. It would be possible to construct a continuous strategic outcome function by defining a continuous penalty rule. Whether it is possible or not to get a full-blown, feasible and smooth Nash competitive mechanism is still an open problem. • the two-player case is largely unexplored in this literature and is covered by none of our results. Appendix A. A.1. Schmeidler’s (1980) mechanism For the sake of completeness, we recall the definition of Schmeidler’s mechanism: each L player’s strategy set is given by: Si := {(pi , zi ) ∈ RL + × R : pi · zi = 0} (hence default  never occurs, even out of equilibrium). Given some strategy profile s = (si ) ∈ S := i Si , compute the set Ti (s) = {h ∈ I : ph = pi }, and denote by ni (s) ≥ 1 its cardinal. The strategic outcome function is given by:  ϕi (s) := zi − 1/ni (s) zh + ω i . h∈Ti (s)

If, say, player 1 is the unique one to quote the price vector p1 , then her final allocation is ω1 , whatever her quantity setting strategy. A.2. Implementation in coalition-proof correlated equilibria Proposition 3 . Assume Assumption 1 holds. Every Walrasian equilibrium can be implemented as a CPCE. Proof. Let (p∗ , (xi∗ )i∈I ) be a Walrasian equilibrium. Consider the trivial correlation device d ∗ with |Mi | = 1 for all i ∈ I, and let σ¯ = (σ¯ i )i be such that for all i ∈ I, σ¯ i (mi ) = ∗i ∗ ∗ ∗ (p∗i , (z∗i ¯ is a j )j=i ) where pi = p and zj = (1/N)(xi − ωi ) for all j. We show that σ ¯ Indeed, CPNE, and hence a CPCE: there is no coalition S ⊂ I which has an ICI σ¯ S upon σ. in order to improve his payoff, player j ∈ S must change his price quoting strategy. Suppose ∗j there exists j ∈ S such that σ¯ S,j (mj ) = (pj , (zk )k=j ) with pj = p∗ . Assume without loss of generality phj < p∗h for some h ∈ {1, . . . , L}. Then there exists j  ∈ I \ S and i ∈ S \ {j} such that pj = p∗ and i faces an arbitrage opportunity. Note that S is a strict subset of I as

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(xi∗ )i is Pareto optimal and {j} is a strict subset of S as a Walrasian equilibrium is a Nash equilibrium. As (p∗ , (xi∗ )i∈I ) is a Walrasian equilibrium, the boundary condition implies that xi∗ ∈ ˆ i for all i. intX ˆ i such that By the local non-satiation hypothesis, for every > 0 there exists yi ∈ X ∗ ∗ yi − xi  ≤ and ui (yi ) > ui (xi ). Consider the following strategy for player i: σi (mi ) = (p˜ i , (˜zij )j=i ) where • p˜ i is chosen according to σ¯ S , • z˜ ik = z∗i / {i, j, j  }, k , for all k ∈ i ∗i • z˜ j = zj + [pj (yi − xi )/(phj − phj )]I{h} + yi − xi , h h • z˜ ij = z∗i j  − [pj (yi − xi )/(pj  − pj )]I{h} ,

where I{h} = (0, . . . , 0, 1, 0, . . . , 0) ∈ RL with non-zero hth coordinate, (z∗i l )l=i is the quantity setting allocation of player i according to σ¯ (respectively σ¯ S ) if i ∈ I\S (respectively S), and xi is the final allocation received by i when σ¯ is played by I \S and σ¯ S is played by S. ˆ i , and thus the improvement by coalition S is not internally consistent. Player i gets yi ∈ X Hence, σ¯ I is a CPNE. 䊐 Proposition 5 . Assume Assumption 2 holds and preferences are strongly monotone and continuous. Every cpCE induces a deterministic Pareto optimal outcome. Conversely, every Pareto optimal allocation can be obtained as a cpCE. Proof. Consider a Pareto optimal allocation (xi∗ )i∈I . By the constrained second welfare theorem, as preferences are strongly monotone and continuous, is a
vector of prices p∗ =
there ∗ ∗ ∗ ∗ 0 and an assignment of wealth levels w = (wi )i∈I with i wi = p
i ωi such
that for all i
∈ I, xi∗ ∈ argmaxx ui (x) under the constraints p∗ x ≤ w∗i and x ≤ i ωi , and i∈I xi∗ = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i∈I ωi . Consider the contract (d , y ) where y = (yi )i with yi = wi − p ωi and d is ∗i ∗ trivial, namely |Mi | = 1, and the strategy of player i is defined by σ¯ i (mi ) = (p , (zj )j=i ) ∗ where z∗i ¯ is a cpCE using the proof of Proposition j = (1/N)(xi − ωi ). We can show that σ  ¯ is a CPCE of Γy∗ . 3 . Indeed, it shows that σ¯ is a CPNE of the game Γd ∗ ,y∗ . Hence, (d ∗ , σ) Observe that by Jensen’s inequality, every Pareto optimal random allocation must be in fact deterministic. Now, let (d, y, σ) be a cpCE. If the random allocation (xi )i∈I induced by (d, y, σ) is not Pareto optimal, there exists a deterministic Pareto optimal allocation which Pareto dominates it. (Indeed, if (xi )i∈I is not deterministic, it is Pareto dominated by (E(xi ))i∈I . If this last deterministic allocation is itself not Pareto optimal, then there exists a Pareto optimal deterministic allocation (˜xi )i∈I in E which dominates (E(xi ))i∈I , hence dominates (xi )i∈I .) Using the constrained second welfare theorem, the grand coalition can choose to play a constrained Walrasian equilibrium with transfers which gives an ICI upon (d, y, σ). 䊐 A.3. Constrained second welfare theorem We here extend to constrained Walrasian equilibria the well-known second welfare theorem, following the proof presented in Mas-Colell et al. (1995).

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Under strongly monotonic preferences, the following local non-satiation condition holds: Constrained local non-satiation condition: For all i ∈ I, if xi∗ < ω, ¯ then for all i ∈ I, there exists xi ≤ ω¯ such that ui (xi ) > ui (xi∗ ). In words, the above condition states that if an agent does not possess all of the endowments, then he is not saturated on the feasible set. Definition. An allocation x∗ and a price vector p = 0 constitute a constrained price quasi-equilibrium with transfers if there is an assignment of wealth levels (wi , . . . , wN )
with i wi = p ω¯ such that (1) for i, if xi ≤ ω¯ and ui (xi ) > ui (xi∗ ), then p xi ≥ wi ,
every ∗ (2) ¯ i xi = ω. Remark. A constrained price equilibrium with transfers is defined analogously, by replacing condition (1) above with (1 ) for every i, xi∗ ∈ argmaxx ui (x) under the constraints p x ≤ wi and x ≤ ω. ¯ Proposition 5. Assume preferences are quasi-concave, continuous, strongly monotone and ωi > 0 for all i. Then for every Pareto optimal allocation x∗ , there is a price vector p = 0 such that (x∗ , p) is a constrained price equilibrium with transfers. Proof. Let Vi = {xi ∈ RL ¯ + : xi ≤ ω

and

ui (xi ) > ui (xi∗ )},

and V =

 i

• • • • •

 Vi =



 xi : xi ∈ Vi for all i .

i

Every set Vi is convex. The sets V and {ω} ¯ are convex. V ∩ {ω} ¯ = ∅. There is p = 0 and a number r such that pz ≥ r for
all z ∈ V and pω¯ ≤ r. For every i, if xi ≤ ω¯ and ui (xi ) ≥ ui (xi∗ ), then p i xi ≥ r. ◦ Assume xi ≤ ω¯ and ui (xi ) ≥ ui (xi∗ ) for all i. For all i such that xi < ω, ¯ there exists xˆ i ≤ ω¯ such that ui (ˆxi ) > ui (xi ) (by local non-satiation). If there exists i
o such that
xio = ω, ¯ then take xˆ io = xio . Every xˆ i ∈ Vi and thus i xˆ i ∈ V and p i xˆ i ≥ r. Taking the limit as xˆ i → xi gives the result.
• p i xi∗ = p ω¯
= r.

◦ We have p i xi∗ ≥ r by the previous step. As i xi∗ = ω, ¯ then p i xi∗ ≤ r, and hence the required inequality. • For every i, if xi ≤ ω¯ and ui (xi ) > ui (xi∗ ), then p xi ≥ p xi∗ . ◦ Assume xi ≤ ω¯ and ui (xi ) > ui (xi∗ ); then

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 p xi +

 k=i





xk∗  ≥ r = p xi∗ +

 k=i

461

 ¯ xk∗  = p ω,

and hence the result. • The wealth levels wi = p xi∗ for i ∈ I support (x∗ , p) as a constrained price quasi-equilibrium with transfers.  • If there is a consumption vector xi ∈ RL + such that p xi < wi (a cheaper consumption ∗ for (p, wi )), then xi ≤ ω¯ and ui (xi ) > ui (xi ) imply p xi > wi . ◦ By contradiction: Suppose that there is an xi with xi ≤ ω¯ and ui (xi ) > ui (xi∗ ) such that p xi = wi . By the cheaper consumption assumption, there exists xi ∈ RL + such that p xi < wi . Then for all α ∈ [0, 1), we have p (αxi + (1 − α)xi ) < wi . For α close to 1, the continuity of the preferences implies that ui (αxi + (1 − α)xi ) > ui (xi∗ ), a contradiction. 䊐 A.4. Postlewaite and Wettstein (1989) versus coalitions In this section, we show by means of an example that the semi-strong Nash equilibria of the mechanism presented in Postlewaite and Wettstein (1989) do not implement the constrained Walrasian allocations. A strategy of player i in Postlewaite and Wettstein’s mechanism consists in announcing an initial endowment profile, vi , a net trade, zi , a price vector, pi , and two positive numbers (ri , qi ) determining penalties associated with differences in individual announcements and weights assigned to the agents’ demands. In their mechanism, a weighted average of the prices announced by the agents is computed. It is defined in such a way that if all agents other than agent i announce the same price, then i cannot affect the price as the weight corresponding to his announcement is zero. The allocation that i receives, αi , is determined by projecting his announced net trade onto the price hyperplane determined by the weighted average price. A rationing takes place if the allocation is not feasible and moreover, incentives to announce truthfully the endowments are provided. Consider an exchange economy with three agents and three goods, x, y, z. Let px = py = pz = 1, u1 (x, y, z) = x + 10−5 y + 10−10 z, u2 (x, y, z) = z + 10−5 y + 10−10 x, u3 (x, y, z) = y + 10−5 x + 10−10 z, and ω1 = ( 18 , 21 , 18 ),

ω2 = ( 18 , 21 , 18 ),

ω3 = ( 21 , 41 , 21 ).

Solving max u1 (x, y, z),

subject to x + y + z ≤

3 4

462

G. Giraud, C. Rochon / Journal of Mathematical Economics 38 (2002) 441–463

and x ≤ 3/4, y ≤ 5/4, z ≤ 3/4 yields the solution (x1∗ , y1∗ , z∗1 ) = (3/4, 0, 0). Similarly, solving max u2 (x, y, z),

subject to x + y + z ≤ 43 ,

and x ≤ 3/4, y ≤ 5/4, z ≤ 3/4 yields (x2∗ , y2∗ , z∗2 ) = (0, 0, 3/4). Finally, solving max u3 (x, y, z),

subject to x + y + z ≤ 45 ,

and x ≤ 3/4, y ≤ 5/4, z ≤ 3/4 yields (x3∗ , y3∗ , z∗3 ) = (0, 5/4, 0). Consider a deviation in prices by the coalition composed of players 1 and 2 as follows: p1 = p2 = ( 21 , 2, 21 ), while player 3 still plays p3 = (1, 1, 1). We now construct the average price vector, p, ¯ as in Postlewaite and Wettstein (1989), using p1 , p2 and p3 , as follows: Let   pj − pk 2 and a = ai . ai = i

j,k=i

Define



bi =

ai a, 1 3,

if a > 0, if a = 0.


With the above, the average price vector p¯ is defined by p¯ := i bi pi . In our example, we find a1 = a2 = 3/2 and a3 = 0, hence a = 3, and b1 = b2 = 1/2 and b3 = 0, yielding p¯ = (1/2, 2, 1/2). At the price p, ¯ solving max u1 (x, y, z),

subject to

1 2 x + 2y

+ 21 z ≤ 98 ,

and x ≤ 3/4, y ≤ 5/4, z ≤ 3/4 yields the solution (x1∗ , y1∗ , z∗1 ) = (3/4, 3/8, 0). Similarly, solving max u2 (x, y, z),

subject to

1 2 x + 2y

+ 21 z ≤ 98 ,

and x ≤ 3/4, y ≤ 5/4, z ≤ 3/4 yields (x2∗ , y2∗ , z∗2 ) = (0, 3/8, 3/4). The deviation in prices by players 1 and 2 combined with a strategy which consists in declaring a large ri , with r1 = r2 (to ensure the desired final allocation) and truthfully revealing endowments is a Nash equilibrium and hence the constrained Walrasian equilibrium cannot be implemented in semi-strong Nash. References Aumann, R., 1959. Acceptable points in general cooperative n-person games. In: Contributions to the Theory of Games IV. Princeton University Press, Princeton.

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