Computing the Equilibria of Bimatrix Games using Dominance Heuristics

TL;DR

This paper introduces a graph-based heuristic method for computing Nash equilibria in general-sum bimatrix games by analyzing elementary cycles in a bipartite directed graph representation.

Contribution

It formulates bimatrix games as bipartite graphs and links elementary cycles to Nash equilibria, providing a new heuristic for equilibrium computation especially in sparse graphs.

Findings

Elementary cycles correspond to Nash equilibria.

The heuristic improves efficiency for sparse graphs.

Supports pre-processing for equilibrium algorithms.

Abstract

We propose a formulation of a general-sum bimatrix game as a bipartite directed graph with the objective of establishing a correspondence between the set of the relevant structures of the graph (in particular elementary cycles) and the set of the Nash equilibria of the game. We show that finding the set of elementary cycles of the graph permits the computation of the set of equilibria. For games whose graphs have a sparse adjacency matrix, this serves as a good heuristic for computing the set of equilibria. The heuristic also allows the discarding of sections of the support space that do not yield any equilibrium, thus serving as a useful pre-processing step for algorithms that compute the equilibria through support enumeration.