The Complexity of Nash Equilibria in Limit-Average Games

TL;DR

This paper investigates the computational complexity of finding Nash equilibria in limit-average games, revealing undecidability in some cases and decidability in others, with implications for game theory and algorithm design.

Contribution

It establishes undecidability results for the existence of Nash equilibria in certain classes of concurrent and turn-based games, and identifies decidable cases with complexity analysis.

Findings

Existence of Nash equilibrium in randomized strategies is undecidable.

Existence of Nash equilibrium in pure strategies is decidable.

Constrained existence problem is undecidable for both concurrent and turn-based games.

Abstract

We study the computational complexity of Nash equilibria in concurrent games with limit-average objectives. In particular, we prove that the existence of a Nash equilibrium in randomised strategies is undecidable, while the existence of a Nash equilibrium in pure strategies is decidable, even if we put a constraint on the payoff of the equilibrium. Our undecidability result holds even for a restricted class of concurrent games, where nonzero rewards occur only on terminal states. Moreover, we show that the constrained existence problem is undecidable not only for concurrent games but for turn-based games with the same restriction on rewards. Finally, we prove that the constrained existence problem for Nash equilibria in (pure or randomised) stationary strategies is decidable and analyse its complexity.