On the existence of approximate equilibria and sharing rule solutions in discontinuous games

Bich, Philippe; Laraki, Rida

doi:10.3982/te2081

Cited by 17 publications

(20 citation statements)

References 31 publications

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“…This extends the sharing rule existence result in [5] (Theorem 2). The proof is a direct consequence of Theorem 2.…”

Section: Generalized Games With Discontinuous Payoffssupporting

confidence: 83%

“…Such pair (x * , v * ) refines the Reny solution concept introduced in [5]. When u i is continuous for every i ∈ N , x * is a Nash equilibrium and v * = u(x * ) is the associated payoff vector.…”

Section: Appendix A: Proof Of Theoremmentioning

confidence: 93%

“…In strategic games where B i (x −i ) = X i for every x −i ∈ X −i and every i ∈ I, we get the existence of a solutionà la Simon-Zame in pure strategies. This was an open question in Jackson et al [10] and was solved recently in Bich and Laraki [5] by using as a tool Reny 's [13] better-reply security condition. But we shall see that adding externalities makes the proof more complex.…”

Section: Economies With Endogenous Sharing Rulesmentioning

confidence: 99%

“…This is used to construct in step 4 a sharing rule solution of G that satisfies some desirable properties. Finally, such a sharing rule solution is used to build a solution of G. This methodology follows the one developed in [5] and [4] to prove existence of Nash, approximate and sharing rule equilibria in discontinuous games. But it is more complicated because of the externalities.…”

Section: Appendix A: Proof Of Theoremmentioning

confidence: 99%

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Externalities in economies with endogenous sharing rules

Bich

Laraki

2017

Econ Theory Bull

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Endogenous sharing rules were introduced by Simon and Zame [16] to model payoff indeterminacy in discontinuous games. They prove the existence in every compact strategic game of a mixed Nash equilibrium and an associated sharing rule. We extend their result to economies with externalities [1] where, by definition, players are restricted to pure strategies. We also provide a new interpretation of payoff indeterminacy in Simon and Zame's model in terms of preference incompleteness.

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“…This extends the sharing rule existence result in [5] (Theorem 2). The proof is a direct consequence of Theorem 2.…”

Section: Generalized Games With Discontinuous Payoffssupporting

confidence: 83%

Section: Appendix A: Proof Of Theoremmentioning

confidence: 93%

Section: Economies With Endogenous Sharing Rulesmentioning

confidence: 99%

Section: Appendix A: Proof Of Theoremmentioning

confidence: 99%

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Externalities in economies with endogenous sharing rules

Bich

Laraki

2017

Econ Theory Bull

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“…Then, this surrogate game is point secure with respect to player set I = {1} as defined in Reny's symposium paper if and only if (x, y) is transfer continuous. Scalzo relates quasi-Nash equilibrium to the concept of Reny equilibrium introduced by Bich and Laraki (2014) who connect the latter equilibrium concept to the endogenous sharing rules of Simon and Zame (1990). Both equilibrium concepts share the property that they exist so long as a form of quasi-concavity of payoffs holds; in the case of Reny equilibrium, the condition is quasi-concavity itself, i.e., so long as each player's payoff function is own-strategy quasi-concave, a Reny equilibrium exists.…”

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confidence: 99%

Introduction to the symposium on discontinuous games

Reny

2016

Econ Theory

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The literature on the existence of Nash equilibrium in discontinuous games has blossomed since the seminal contribution of Dasgupta and Maskin (1986). The present symposium brings together a number of recent developments. Throughout this introduction, it is assumed that any game G = (X i , u i ) i∈N under consideration has a finite player set, N , that each player i ∈ N has a nonempty, compact, and convex set of pure strategies X i that is a subset of a linear topological space, and has bounded payoff function u i : X → R, where X = × i∈N X i . Not all papers in the symposium are always so restrictive as this. Indeed, some authors occasionally do not require strategy sets to be convex nor do they always require the existence of utility representations of the players' preferences over strategy profiles. Such exceptions will be noted in this introduction only when absolutely necessary. Also, "Nash equilibrium" will always mean pure strategy Nash equilibrium. Each paper included in this symposium issue is briefly discussed below, and the papers have been arranged in alphabetical order.The paper by Guilherme Carmona (2014) entitled "Reducible equilibrium properties: comments on recent existence results," is focused on connecting a rather wide variety of existence results in the literature by way of a common proof technique. Carmona's paper shows that the various sufficient conditions for existence can all be understood as allowing the existence problem to be reduced to a "simpler" existence problem in which the relevant best-reply correspondences are well behaved, i.e., upper hemicontinuous, nonempty valued, and convex valued, on a domain that is amenable to fixed point analysis, i.e., nonempty, compact and convex. While some early results in the literature take a related route (e.g., Dasgupta and Maskin 1986; Reny 1999, both can be applied), the recent literature's more sophisticated techniques are rather further removed from such a reduction principle, although, in the end, one cannot get away from an eventual application of Kakutani-Fan-Glicksberg. In any case, the unifying perspective offered here is both welcome and illuminating.The paper by Guilherme Carmona and Konrad Podczeck (2015) entitled, "Existence of Nash equilibrium in ordinal games with discontinuous preferences," provides several new Nash equilibrium existence results for discontinuous games that generalize many in the literature, including the main result in the paper by Reny included in this symposium issue. The authors introduce the notions of target point security and target correspondence security which generalize Reny's point security and correspondence security conditions, respectively. The authors allow the players' preferences to be given by binary relations that need not be complete or transitive. This can be especially fruitful when preferences are those of a group of individuals. Another difference between point security and target security is, very roughly, that when a strategy profile, x, is not a Nash equilibrium, point security re...

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Legitimate equilibrium

2021

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We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)).

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On the existence of approximate equilibria and sharing rule solutions in discontinuous games

Cited by 17 publications

References 31 publications

Externalities in economies with endogenous sharing rules

Externalities in economies with endogenous sharing rules

Introduction to the symposium on discontinuous games

Legitimate equilibrium

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