Computing Possible and Necessary Equilibrium Actions (and Bipartisan Set Winners)

TL;DR

This paper investigates the computational complexity of designing and analyzing incomplete multiagent games, showing that determining equilibrium actions and winners is NP-hard, and provides a MILP approach for certain game classes.

Contribution

It formalizes the complexity of incomplete game design and equilibrium analysis, and introduces a MILP method for weak tournament games.

Findings

Deciding equilibrium actions in incomplete games is NP-hard.

Hardness persists even in symmetric and tournament games.

A MILP formulation is effective for weak tournament games.

Abstract

In many multiagent environments, a designer has some, but limited control over the game being played. In this paper, we formalize this by considering incompletely specified games, in which some entries of the payoff matrices can be chosen from a specified set. We show that it is NP-hard for the designer to make these choices optimally, even in zero-sum games. In fact, it is already intractable to decide whether a given action is (potentially or necessarily) played in equilibrium. We also consider incompletely specified symmetric games in which all completions are required to be symmetric. Here, hardness holds even in weak tournament games (symmetric zero-sum games whose entries are all -1, 0, or 1) and in tournament games (symmetric zero-sum games whose non-diagonal entries are all -1 or 1). The latter result settles the complexity of the possible and necessary winner problems for a social-choice-theoretic solution concept known as the bipartisan set. We finally give a mixed-integer linear programming formulation for weak tournament games and evaluate it experimentally.