Introduction to the symposium on discontinuous games

Abstract

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 = (Xi , ui )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 Xi that is a subset of a linear topological space, and has bounded payoff function ui : X → R, where X = ×i∈N Xi . 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 first reduce the problem to a game setting in which standard existence results