Complexity of Efficient Outcomes in Binary-Action Polymatrix Games with Implications for Coordination Problems

TL;DR

This paper explores the computational complexity of finding efficient solutions in binary-action polymatrix coordination games, revealing that some objectives are efficiently solvable in pure-coordination but NP-hard in anti-coordination games.

Contribution

It introduces the Maximum Weighted Digraph Partition problem and provides a complexity dichotomy for various coordination objectives in these games.

Findings

Objectives (i) and (ii) are efficiently solvable in pure-coordination games.

Objectives (i) and (ii) are NP-hard in anti-coordination games.

Objective (iii) is NP-hard even in simple pure-coordination games.

Abstract

We investigate the difficulty of finding economically efficient solutions to coordination problems on graphs. Our work focuses on two forms of coordination problem: pure-coordination games and anti-coordination games. We consider three objectives in the context of simple binary-action polymatrix games: (i) maximizing welfare, (ii) maximizing potential, and (iii) finding a welfare-maximizing Nash equilibrium. We introduce an intermediate, new graph-partition problem, termed Maximum Weighted Digraph Partition, which is of independent interest, and we provide a complexity dichotomy for it. This dichotomy, among other results, provides as a corollary a dichotomy for Objective (i) for general binary-action polymatrix games. In addition, it reveals that the complexity of achieving these objectives varies depending on the form of the coordination problem. Specifically, Objectives (i) and (ii) can be efficiently solved in pure-coordination games, but are NP-hard in anti-coordination games. Finally, we show that objective (iii) is NP-hard even for simple non-trivial pure-coordination games.