A Note on the Equilibrium Existence Problem in Discontinuous Games

In this note, we prove an equilibrium existence theorem for games with discontinuous payoffs and convex and compact strategy spaces. It generalizes the classical result of Reny (1999), as well as the recent paper of McLennan, Monteiro, and Tourky (2011). Our conditions are simple and easy to verify. Importantly, examples of spatial location models show that our conditions allow for economically meaningful payoff discontinuities, that are not covered by other conditions in the literature.     

On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games

A game is better-reply secure if for every nonequilibrium strategy x^* and every payoff vector limit u^* resulting from strategies approaching x^*, some player i has a strategy yielding a payoff strictly above u_i^* even if the others deviate slightly from x^*. If strategy spaces are compact and convex, payoffs are quasiconcave in the owner's strategy, and the game is better-reply secure, then a pure strategy Nash equilibrium exists. Better-reply security holds in many economic games. It also permits new results on the existence of symmetric and mixed strategy Nash equilibria.     

Reciprocal Upper Semicontinuity and Better Reply Secure Games: A Comment

A convex, compact, and possibly discontinuous better reply secure game has a Nash equilibrium. We introduce a very weak notion of continuity that can be used to establish that a game is better reply secure and we show that this notion of continuity is satisfied by a large class of games.     

Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information

This paper analyzes a class of games of incomplete information where each agent has private information about her own type, and the types are drawn from an atomless joint probability distribution. The main result establishes existence of pure strategy Nash equilibria (PSNE) under a condition we call the single crossing condition (SCC), roughly described as follows: whenever each opponent uses a nondecreasing strategy (in the sense that higher types choose higher actions), a player's best response strategy is also nondecreasing. When the SCC holds, a PSNE exists in every finite‐action game. Further, for games with continuous payoffs and a continuum of actions, there exists a sequence of PSNE to finite‐action games that converges to a PSNE of the continuum‐action game. These convergence and existence results also extend to some classes of games with discontinuous payoffs, such as first‐price auctions, where bidders may be heterogeneous and reserve prices are permitted. Finally, the paper characterizes the SCC based on properties of utility functions and probability distributions over types. Applications include first‐price, multi‐unit, and all‐pay auctions; pricing games with incomplete information about costs; and noisy signaling games.     

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 metric semilattices and type spaces to be partially ordered probability spaces. Our proof is based on 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.     

广义不确定性下广义博弈中NS均衡的存在性

把不确定性引入广义博弈的研究之中,在此博弈中,局中人策略之间存在相互影响,局中人的策略可以改变不确定参数的变化范围,而且局中人的支付函数和策略可行反应映射都受到不确定参数的作用,此类型博弈定义为广义不确定性下的广义博弈问题。进一步定义出此类型博弈中的NS均衡,并且凭借Fan-Glicksberg不动点定理,证明此均衡点的存在性。最后给出算例验证其可行性。