Equilibria In Bayesian Games Research Articles

Pure-strategy equilibrium in Bayesian potential games with absolutely continuous information

We prove the existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games with absolutely continuous information and a Bayesian potential that is upper semi-continuous in actions for any realization of the players' types. In particular, all Bayesian potential games with finitely many actions and absolutely continuous information possess a pure-strategy Bayesian Nash equilibrium.

On the Existence of Equilibrium in Bayesian Games Without Complementarities

On the Existence of Equilibrium in Bayesian Games Without Complementarities

On the existence of Pareto undominated mixed-strategy Nash equilibrium in normal-form games with infinite actions

We show that each mixed strategy of a normal-form game can be reformulated as a pure strategy of an induced Bayesian game. Moreover, a normal-form game has a Pareto undominated mixed-strategy Nash equilibrium if and only if its induced Bayesian game has a pure-strategy Bayesian Nash equilibrium. By relying on the existence result of Pareto undominated pure-strategy Nash equilibrium in Bayesian games in Fu and Yu (2015), we also show that every normal-form game has a Pareto undominated mixed-strategy Nash equilibrium.

On games without approximate equilibria

This note shows that the work by Simon and Tomkowicz (Israel J Math 227(1):215–231, 2018) answers another outstanding open question in game theory in addition to the non-existence of approximate Harsányi equilibrium in Bayesian games: it shows that strategic form games with bounded and separately continuous payoffs need not possess approximate equilibria.

Nash Equilibrium in Discontinuous Games

We review the discontinuous games literature, with a sharp focus on conditions that ensure the existence of pure and mixed strategy Nash equilibria in strategic form games and of Bayes-Nash equilibria in Bayesian games.

Equilibrium existence in games with a concave Bayesian potential

We establish existence of a pure-strategy Bayesian Nash equilibrium in Bayesian games that have a continuous and concave potential at all states of nature, without assuming absolute continuity of information. As an application, we show that Bayesian Nash equilibrium exists in many well-known games that have semi-quadratic payoffs (including Bertrand and Cournot oligopolies with linear demand), for general information structures.

On the Existence of Equilibrium in Bayesian Games Without Complementarities

On the Existence of Equilibrium in Bayesian Games Without Complementarities

On the Existence of Equilibrium in Bayesian Games Without Complementarities

This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium.

On the Existence of Equilibrium in Bayesian Games Without Complementarities

This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium.

Pure-strategy equilibria in Bayesian games

A general condition called “coarser inter-player information” is introduced and shown to be necessary and sufficient for the validity of several fundamental properties on pure-strategy equilibria in Bayesian games, such as existence, purification from behavioral strategies, and convergence for a sequence of games. Our sufficiency results cover various earlier results on pure-strategy equilibria in Bayesian games as special cases. New applications are presented as illustrative examples, including auctions with externalities and risk-neutral bidders, and Bertrand pricing games with asymmetric information.

On the Existence of Nash Equilibrium in Bayesian Games

We furnish conditions on the primitives of a Bayesian game that guarantee the existence of a Bayes-Nash equilibrium. By allowing for payoff discontinuities in actions, we cover various applications that cannot be handled by extant results.

Distributional Comparative Statics

Distributional comparative statics is the study of how individual decisions and equilibrium outcomes vary with changes in the distribution of economic parameters (income, wealth, productivity, information, etc.). This article develops new tools to address such issues and illustrates their usefulness in applications. The central development is a condition called quasi-concave differences, which implies concavity of the policy function in optimization problems without imposing differentiability or quasi-concavity conditions. The general take-away is that many distributional questions in economics which cannot be solved by direct calculations or the implicit function theorem, can be addressed easily with this article’s methods. Several applications demonstrate this: the article shows how increased uncertainty affects the set of equilibria in Bayesian games; it shows how increased dispersion of productivities affects output in the model of Melitz (2003); and it generalizes Carroll and Kimball (1996)’s result on concave consumption functions to the Aiyagari (1994) setting with borrowing constraints.

Behavioral perfect equilibrium in Bayesian games

We develop the notion of perfect Bayesian Nash Equilibrium—perfect BNE—in general Bayesian games. We test perfect BNE against the criteria laid out by Kohlberg and Mertens (1986). We show that, for a focal class of Bayesian games, perfect BNE exists. Moreover, when payoffs are continuous, perfect BNE is limit undominated for almost every type.We illustrate the use of perfect BNE in the context of a second-price auction with interdependent values. Perfect BNE selects the unique pure strategy equilibrium in continuous strategies that separates types. Moreover, when valuations become independent, the equilibrium converges to the classical truthful dominant strategy equilibrium.We also show that less intuitive equilibria in which types are pooled are ruled out by our selection criterion. We further argue that standard selection criteria for second-price auctions have no bite here. Bidders have no dominant strategies, and the separating equilibrium is not sincere.

Existence of pure-strategy equilibria in Bayesian games: a sharpened necessity result

In earlier work, the authors showed that a pure-strategy Bayesian-Nash equilibria in games with uncountable action sets and atomless private information spaces may not exist if the information space of each player is not saturated. This paper sharpens this result by exhibiting a failure of the existence claim for a game in which the information space of only one player is not saturated. The methodology that enables this extension of the necessity theory is novel relative to earlier work, and its conceptual underpinnings may have independent interest.

Aggregating the Single Crossing Property

The single crossing property plays a crucial role in economic theory, yet there are important instances where the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum or convex combination of two functions with the single crossing property need not have that property. We introduce a new condition characterizing when the single crossing property is stable under aggregation, and also identify sufficient conditions for the preservation of the single crossing property under multidimensional aggregation. We use our results to establish properties of objective functions (convexity, logsupermodularity), the monotonicity of optimal decisions under uncertainty, and the existence of monotone equilibria in Bayesian games.

On the Existence of Monotone Pure-Strategy Equilibria in Bayesian Games

We extend and strengthen both Athey’s (2001) and McAdams’(2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilatttices and can handle both a weaker form of quasisupermodularity than is employed by McAdams and a weaker single-crossing property than is required by both Athey and McAdams. Our proof — which is based upon contractibility rather than convexity of best reply sets — demonstrates that the only role of single-crossing is to help ensure the existence of monotone best replies. Finally, we do not require the Milgrom-Weber (1985) absolute continuity condition on the joint distribution of types.

On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games

We generalize Athey's (2001) and McAdams' (2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilattices and type spaces to be partially ordered probability spaces. Our proof is based upon contractibility rather than convexity of best reply sets. Several examples illustrate the scope of the result, including new applications to multi-unit auctions with risk-averse bidders.

On purification of equilibrium in Bayesian games and expost Nash equilibrium

Treating games of incomplete information, we demonstrate that the existence of an ex post stable strategy vector implies the existence of an approximate Bayesian equilibrium in pure strategies that is also expost stable. Through examples we demonstrate the ‘bounds obtained on the approximation’ are tight.

Approximation of Nash equilibria in Bayesian games

AbstractWe define a new concept of constrained strategic equilibrium (CSE) for Bayesian games. We show that a sequence of CSEs approximates an equilibrium under standard conditions. We also provide an algorithm to implement the CSE approximation method numerically in a broad class of Bayesian games, including games without analytically tractable solutions. Finally, we illustrate the flexibility of the CSE approximation with a series of auction examples, including a complex multi‐unit auction. Copyright © 2008 John Wiley & Sons, Ltd.

Sequential equilibria in Bayesian games with communication

We study the effects of communication in Bayesian games when the players are sequentially rational but some combinations of types have zero probability. Not all communication equilibria can be implemented as sequential equilibria. We define the set of strong sequential communication equilibria (SSCE) and characterize it. SSCE differs from the concept of sequential communication equilibrium (SCE) defined by Myerson [Myerson, R.B., 1986. Multistage games with communication. Econometrica 54, 323–358] in that SCE allows the possibility of trembles by the mediator. We show that these two concepts coincide when there are three or more players, but the set of SSCE may be strictly smaller than the set of SCE for two-player games.