Continuum and Finite-Player Noncooperative Models of Competition

Abstract

The anonymous interaction of large numbers of economic agents is a kind of noncooperative situation which is markedly different from small-numbers strategic conflict. The mathematical model of a nonatomic game, or a game with a continuum of players, has been introduced as a model for these many-agent situations on the basis that its equilibria should closely approximate those of games with large finite numbers of players. This paper contains a precise definition of what it means for a nonatomic game to be the limit of a sequence of finite-player games, and a theorem which states when the limit of equilibria of finite-player games will be an equilibrium of the nonatomic limit game. This is analogous to theorems prompted by Edgeworth's conjecture in core theory. It is derived from a general set of sufficient conditions for the graph of a noncooperative equilibrium correspondence to be closed.