Combinatorics of Correlated Equilibria

TL;DR

This paper explores the combinatorial structure of correlated equilibrium polytopes in games, introducing a stratification method and analyzing their algebraic boundaries, with specific results for small games.

Contribution

It introduces a stratification approach for correlated equilibrium polytopes and characterizes their algebraic boundaries, providing a detailed analysis for small game sizes.

Findings

The full-dimensionality region is a semialgebraic set.

For (2 x n)-games, the algebraic boundary is composed of coordinate hyperplanes and binomial hypersurfaces.

There is a unique maximal combinatorial type for generic (2 x 3)-games.

Abstract

We study the correlated equilibrium polytope $P_G$ of a game $G$ from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of $P_G$, and show that for $(2 \times n)$-games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic $(2 \times 3)$-games.