The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

TL;DR

This paper investigates the computational difficulty of finding Nash equilibria in simple stochastic multiplayer games, revealing undecidability in certain cases and establishing complexity bounds for restricted strategy types.

Contribution

It proves undecidability for pure-strategy Nash equilibria with probability 1 and explores complexity bounds for positional and stationary strategies.

Findings

Undecidability of pure-strategy equilibria with probability 1

NP lower bound for positional strategies

PSPACE upper bound for stationary strategies

Abstract

We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. 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 pure-strategy Nash equilibrium of G where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is restricting 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.