Non-Cooperative Games: Equilibrium Existence
Philip J. Reny Department of Economics, University of Chicago August 2005
Abstract This entry in The New Palgrave Dictionary of Economics, Second Edition, provides a brief overview of equilibrium existence results for continuous and discontinuous non-cooperative games. JEL Classi…cation Number: C7
1. Introduction
Nash equilibrium is the central notion of rational behavior in noncooperative game theory.1 Our purpose here is to discuss various conditions under which a strategic form game possesses at least one Nash equilibrium. Strategic settings arising in economics are often naturally modeled as games with in…nite strategy spaces. For example, models of price and spatial competition (Bertrand (1883), Hotelling (1929)), quantity competition (Cournot (1838)), auctions (Milgrom and Weber (1982)), patent races (Fudenberg et. al. (1983)), etc., typically allow players to choose any one of a continuum of actions. The analytic convenience of the continuum from both an equilibrium characterization and comparative statics point of view is perhaps the central reason for the prevalence and usefulness of in…nite-action games. Because of this, our treatment will permit both …nite-action and in…nite-action games. Games with possibly in…nite strategy spaces can be divided into two categories: those with continuous payo¤s and those with discontinuous payo¤s. Cournot oligopoly models and Bertrand price-competition models with di¤erentiated products, as well as all …nite-action games, are important examples of continuous
See Osborne (2005) and Osborne and Rubinstien (1994) for a discussion of Nash equilibrium, including motivation and interpretation.
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games, while Bertrand price-competition with homogeneous products, auctions, and Hotelling spatial competition, are important examples in which payo¤s are discontinuous. Equilibrium existence results for both continuous and discontinuous games will be reviewed here. We begin with some notation. A strategic form game, G = (Si ; ui )N ; consists of a positive …nite number, N; i=1 of players, and for each player i 2 f1; :::; N g; a non empty set of pure strategies, Si ; and a payo¤ function ui : S ! R; where S = N Si : The notation s i i=1 and S i have their conventional meanings: s i = (s1 ; :::; si 1 ; si+1 ; :::; sN ) and S i = j6=i Sj : Throughout, we shall assume that each Si is a subset of some metric space and that if any …nite number of sets are each endowed with a topology, then the product of those sets is endowed with the product topology.

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