Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games

Radzik (1991) showed that two-player games on compact intervals of the real line have " { equilibria for all " > 0, provided that payo® functions are upper semicontinuous and strongly quasi-concave. In an attempt to generalize this theorem, Ziad (1997) stated that the same is true for n-player games on compact, convex subsets of Rm, m ¸ 1 provided that we strengthen the upper semicontinuity condition. We show that: 1. the action spaces need to be polyhedral in order for Ziad's ap- proach to work, 2. Ziad's strong upper semicontinuity condition is equivalent to some form of quasi-polyhedral concavity of players' value func- tions in simple games, and 3. Radzik's Theorem is a corollary of (the corrected) Ziad's result.

Nash equilibrium existence for some discontinuous games

Answering to an open question of Herings et al. (see [3]), one extends their fixed point theorem to mappings defined on convex compact subset of Rn, and not only polytopes. Such extension is important in non-cooperative game theory, where typical strategy sets are convex and compact. An application in game theory is given.Discontinuous game, Nash equilibrium, fixed point theorem.

Polyhedral Convexity and the Existence of Approximate Equilibria in Discontinuous Games

Radzik (1991) showed that two-player games on compact intervals of the real line have ε – equilibria for all ε > 0, provided that payoff functions are upper semicontinuous and strongly quasi-concave. In an attempt to generalize this theorem, Ziad (1997) stated that the same is true for n-player games on compact, convex subsets of Rm, m ≥ 1 provided that we strengthen the upper semicontinuity condition. We show that: 1. the action spaces need to be polyhedral in order for Ziad’s approach to work, 2. Ziad’s strong upper semicontinuity condition is equivalent to some form of quasi-polyhedral concavity of players’ value functions in simple games, and 3. Radzik’s Theorem is a corollary of (the corrected) Ziad’s result.N/

Multidimensional Political Competition with Non-Common Beliefs

This paper extends a probabilistic voting model with a multidimensional policy space, allowing candidates to have different prior probability distributions of the distribution of voters' ideal policies. In this model, we show that a platform pair is a Nash equilibrium if and only if both candidates choose a common generalized median of expected ideal policies. Thus, the existence of a Nash equilibrium requires not only that each candidate's belief have an expected generalized median, which is already a knife-edge condition, but also that the two medians coincide. We also study limits of ε-equilibria of Radner (1980) as ε → 0, which we call "limit equilibria." Limit equilibria are policy pairs that approximate choices by the candidates who almost perfectly optimize. We show that a policy pair is a limit equilibrium if and only if both candidates choose the same policy around which they form "opposite expectations" in a certain sense. For a limit equilibrium to exist (equivalently, for ε-equilibria to exist for all ε > 0), it is sufficient, though not necessary, that either candidate has an expected generalized median.

Existence of equilibria in countable games: an algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games

Nash equilibrium existence for some discontinuous games

URL des Documents de travail :http://ces.univ-paris1.fr/cesdp/CESFramDP2007.htmDocuments de travail du Centre d'Economie de la Sorbonne 2007.69 - ISSN : 1955-611XAnswering to an open question of Herings et al. (see [3]), one extends their fixed point theorem to mappings defined on convex compact subset of Rn, and not only polytopes. Such extension is important in non-cooperative game theory, where typical strategy sets are convex and compact. An application in game theory is given.Nous répondons dans cet article à une question de Herings et al. concernant l'extension d'un de leur théorème de point fixe ; ceux-ci ont traité le cas de fonctions définies sur des polytopes et nous traitons le cas plus général d'ensembles convexes compactes d'un espace Euclidien. Une telle extension peut être importante pour la théorie des jeux non-coopérative, car les espaces typiques de stratégies des joueurs sont des ensembles convexes compactes. Nous donnons justement une application à l'existence d'équilibres de Nash dans des jeux discontinus

Externalities in economies with endogenous sharing rules

International audienceEndogenous 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

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

This paper studies the existence of equilibrium solution concepts in a large class of economic models with discontinuous payoff functions. The issue is well understood for Nash equilibria, thanks to Reny's better‐reply security condition (Reny, 1999) and its recent improvements (Barelli and Meneghel, 2013, McLennan et al.., 2011, Reny 2009, 2011). We propose new approaches, related to Reny's work, and obtain tight conditions for the existence of approximate equilibria and of sharing rule solutions in pure and mixed strategies (Simon and Zame, 1990). As byproducts, we prove that many auction games with correlated types admit an approximate equilibrium, and that many competition models have a sharing rule solution