Computing Good Nash Equilibria in Graphical Games

TL;DR

This paper develops efficient algorithms for computing fair Nash equilibria in graphical games with bounded-degree trees, leveraging the best response policy to ensure certain payoffs and approximate social welfare maximization.

Contribution

It introduces algorithms for constructing Nash equilibria with fairness guarantees in graphical games with polynomial-sized best response policies.

Findings

Polynomial-time algorithm for equilibrium construction on bounded-degree trees.

Existence of an FPTAS for approximate social welfare maximization.

Algorithms enable fairness-oriented equilibrium selection.

Abstract

This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the {\em best response policy}, which was proposed by Kearns et al. \cite{kls} as a way to represent all Nash equilibria of a graphical game. In \cite{egg}, it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is a bounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria.