Efficient Local Search in Coordination Games on Graphs

TL;DR

This paper investigates the existence and efficient computation of pure Nash and strong equilibria in coordination games on specific classes of graphs, providing polynomial-time algorithms for certain graph classes and showing limitations in generalizations.

Contribution

It identifies graph classes where finite improvement paths lead to equilibria in polynomial time and demonstrates the optimality of these results.

Findings

Polynomial-time algorithms for equilibria in certain graph classes

Existence of equilibria in DAGs and cliques

Limitations in general graph classes

Abstract

We study strategic games on weighted directed graphs, where the payoff of a player is defined as the sum of the weights on the edges from players who chose the same strategy augmented by a fixed non-negative bonus for picking a given strategy. These games capture the idea of coordination in the absence of globally common strategies. Prior work shows that the problem of determining the existence of a pure Nash equilibrium for these games is NP-complete already for graphs with all weights equal to one and no bonuses. However, for several classes of graphs (e.g. DAGs and cliques) pure Nash equilibria or even strong equilibria always exist and can be found by simply following a particular improvement or coalition-improvement path, respectively. In this paper we identify several natural classes of graphs for which a finite improvement or coalition-improvement path of polynomial length always exists, and, as a consequence, a Nash equilibrium or strong equilibrium in them can be found in polynomial time. We also argue that these results are optimal in the sense that in natural generalisations of these classes of graphs, a pure Nash equilibrium may not even exist.