A bi-convex optimization problem to compute Nash equilibrium in n-player games and an algorithm

TL;DR

This paper introduces a biconvex optimization framework for computing Nash equilibria in n-player general-sum games, demonstrating convergence of a projected gradient method through theoretical analysis and simulations.

Contribution

It formulates Nash equilibrium computation as a biconvex optimization problem and proves convergence of a projected gradient descent algorithm to a partial optimum.

Findings

The optimization problem's global minima correspond to Nash equilibria.

The projected gradient descent algorithm converges to a partial optimum.

Simulation results show convergence to Nash equilibrium strategies.

Abstract

In this paper we present optimization problems with biconvex objective function and linear constraints such that the set of global minima of the optimization problems is the same as the set of Nash equilibria of a n-player general-sum normal form game. We further show that the objective function is an invex function and consider a projected gradient descent algorithm. We prove that the projected gradient descent scheme converges to a partial optimum of the objective function. We also present simulation results on certain test cases showing convergence to a Nash equilibrium strategy.