Computing all solutions of Nash equilibrium problems with discrete strategy sets

TL;DR

This paper introduces a comprehensive branching algorithm and a fast Jacobi-type method for computing all solutions of Nash equilibrium problems with discrete strategy sets, filling a gap in existing solution techniques.

Contribution

It presents the first general branching algorithm for discrete strategy Nash problems, along with pruning, set-shrinking procedures, and a fast method for finding equilibria.

Findings

Algorithms perform well in practice

The branching method effectively computes the entire solution set

The Jacobi-type method quickly finds an equilibrium

Abstract

The Nash equilibrium problem is a widely used tool to model non-cooperative games. Many solution methods have been proposed in the literature to compute solutions of Nash equilibrium problems with continuous strategy sets, but, besides some specific methods for some particular applications, there are no general algorithms to compute solutions of Nash equilibrium problems in which the strategy set of each player is assumed to be discrete. We define a branching method to compute the whole solution set of Nash equilibrium problems with discrete strategy sets. This method is equipped with a procedure that, by fixing variables, effectively prunes the branches of the search tree. Furthermore, we propose a preliminary procedure that by shrinking the feasible set improves the performances of the branching method when tackling a particular class of problems. Moreover, we prove existence of equilibria and we propose an extremely fast Jacobi-type method which leads to one equilibrium for a new class of Nash equilibrium problems with discrete strategy sets. Our numerical results show that all proposed algorithms work very well in practice.