The Complexity of Nash Equilibria in Stochastic Multiplayer Games

TL;DR

This paper investigates the computational difficulty of finding Nash equilibria in stochastic multiplayer games with omega-regular objectives, revealing undecidability in general and identifying decidable cases with efficient solutions.

Contribution

It establishes undecidability results for general cases and provides complexity bounds for restricted strategy classes, also identifying a polynomial-time solvable special case.

Findings

Deciding existence of certain Nash equilibria is undecidable.

Positional and stationary strategies lead to NP and PSPACE complexity bounds.

A special case with parity objectives admits polynomial-time solutions.

Abstract

We analyse the computational complexity of finding Nash equilibria in turn-based stochastic multiplayer games with omega-regular objectives. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a Nash equilibrium of G where Player 0 wins with probability 1? Moreover, this problem remains undecidable when restricted to pure strategies or (pure) strategies with finite memory. One way to obtain a decidable variant of the problem is to restrict the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively. Finally, we single out a special case of the general problem that, in many cases, admits an efficient solution. In particular, we prove that deciding the existence of an equilibrium in which each player either wins or loses with probability 1 can be done in polynomial time for games where the objective of each player is given by a parity condition with a bounded number of priorities.