Equilibrium payoffs in finite games

TL;DR

This paper characterizes the structure of equilibrium payoffs in finite games, showing that Nash equilibrium payoffs form finite unions of rectangles and exploring the relationship with correlated equilibrium payoffs.

Contribution

It provides a complete characterization of Nash equilibrium payoffs in bimatrix games and extends results to correlated equilibrium and n-player games.

Findings

Nash equilibrium payoffs are exactly finite unions of rectangles.

For any finite union of rectangles and containing polytope, a game exists with these as equilibrium payoffs.

Results are robust to payoff perturbations.

Abstract

We study the structure of the set of equilibrium payoffs in finite games, both for Nash equilibrium and correlated equilibrium. A nonempty subset of R^2 is shown to be the set of Nash equilibrium payoffs of a bimatrix game if and only if it is a finite union of rectangles. Furthermore, we show that for any nonempty finite union of rectangles U and any polytope P in R^2 containing U, there exists a bimatrix game with U as set of Nash equilibrium payoffs and P as set of correlated equilibrium payoffs. The n-player case and the robustness of this result to perturbation of the payoff matrices are also studied.