Decision Problems for Nash Equilibria in Stochastic Games

TL;DR

This paper investigates the computational complexity of finding Nash equilibria in stochastic multiplayer games with omega-regular objectives, identifying decidable cases and complexity bounds for various restrictions.

Contribution

It identifies decidable restrictions of the problem and establishes complexity bounds for stationary and pure stationary equilibria in stochastic games.

Findings

Existence of equilibria with binary payoffs is decidable.

Stationary equilibria problems are in PSPACE.

Pure stationary equilibria problems are in NP.

Abstract

We analyse the computational complexity of finding Nash equilibria in stochastic multiplayer games with $\omega$-regular objectives. While the existence of an equilibrium whose payoff falls into a certain interval may be undecidable, we single out several decidable restrictions of the problem. First, restricting the search space to stationary, or pure stationary, equilibria results in problems that are typically contained in PSPACE and NP, respectively. Second, we show that the existence of an equilibrium with a binary payoff (i.e. an equilibrium where each player either wins or loses with probability 1) is decidable. We also establish that the existence of a Nash equilibrium with a certain binary payoff entails the existence of an equilibrium with the same payoff in pure, finite-state strategies.