Equilibrium convergence in large games

TL;DR

This paper establishes a general closed graph property for Nash equilibria in large games, showing that limits of equilibria in finite-player approximations can be realized in the large game, with applications to equilibrium selection.

Contribution

It introduces a new closed graph property for randomized Nash equilibria in large games, extending previous results and enabling better understanding of equilibrium limits.

Findings

Limits of equilibria in finite games can be induced by equilibria in the large game.

The result extends previous pure strategy equilibrium properties to randomized strategies.

Application to equilibrium selection demonstrates practical relevance.

Abstract

This paper presents a general closed graph property for (randomized strategy) Nash equilibrium correspondence in large games. In particular, we show that for any large game with a convergent sequence of fiinite-player games, the limit of any convergent sequence of Nash equilibria of the corresponding finite-player games can be induced by a Nash equilibrium of the large game. Such a result goes beyond earlier results on the closed graph property for pure strategy Nash equilibrium correspondence in large games in multiple aspects. An application on equilibrium selection in large games is also presented.