Multilinear formulations for computing Nash equilibrium of multi-player matrix games

TL;DR

This work addresses the challenge of computing Nash equilibria in $n$-player finite games by developing an optimization-based framework that centers on a $n$-player multilinear feasibility program. It also introduces mixed-integer multilinear formulations and continuous/feasibility variants to explore a spectrum of solution strategies. Empirical results show the multilinear feasibility program often outperforms Gambit’s global Newton, simplicial subdivision, and logit (QRE) methods on larger games, while the mixed-integer approaches do not offer the same advantages; continuous/non-binary variants provide additional options. The proposed formulations thus offer a scalable and principled alternative for equilibrium computation in multi-player settings and help clarify when different formulation choices are advantageous.

Abstract

We present multilinear and mixed-integer multilinear programs to find a Nash equilibrium in multi-player noncooperative games. We compare the formulations to common algorithms in Gambit, and conclude that a multilinear feasibility program finds a Nash equilibrium faster than any of the methods we compare it to, including the quantal response equilibrium method, which is recommended for large games. Hence, the multilinear feasibility program is an alternative method to find a Nash equilibrium in multi-player games, and outperforms many common algorithms. The mixed-integer formulations are generalisations of known mixed-integer programs for two-player games, however unlike two-player games, these mixed-integer programs do not give better performance than existing algorithms.

Theorems & Definitions (12)

• definition 1
• theorem 1
• corollary 1
• proposition 1
• proposition 2
• proposition 3
• proposition 4
• proof : Proof of \ref{['multilinform']}
• proof : Proof of \ref{['mip1']}
• proof : Proof of \ref{['mip2']}