On Finding Pure Nash Equilibria of Discrete Preference Games and Network Coordination Games

TL;DR

This paper investigates the computational complexity of finding pure Nash equilibria in discrete preference and network coordination games, providing new algorithms and complexity results for specific graph structures and cost functions.

Contribution

It introduces conditions for polynomial-time algorithms in grid graphs and analyzes complexity for games with submodular and symmetric cost functions.

Findings

Best response dynamics estimate for discrete preference games.

Polynomial-time algorithm conditions for grid graphs.

Complexity analysis for submodular cost functions.

Abstract

This paper deals with the complexity of the problem of computing a pure Nash equilibrium for discrete preference games and network coordination games beyond $O(\log n)$-treewidth and tree metric spaces. First, we estimate the number of iterations of the best response dynamics for a discrete preference game on a discrete metric space with at least three strategies. Second, we present a sufficient condition that we have a polynomial-time algorithm to find a pure Nash equilibrium for a discrete preference game on a grid graph. Finally, we discuss the complexity of finding a pure Nash equilibrium for a two-strategic network coordination game whose cost functions satisfy submodularity. In this case, if every cost function is symmetric, the games are polynomial-time reducible to a discrete preference game on a path metric space.