Existential Theory of the Reals Completeness of Stationary Nash Equilibria in Perfect Information Stochastic Games

TL;DR

This paper proves that determining the existence of stationary Nash equilibria in various classes of perfect information stochastic games is complete for the existential theory of the reals, highlighting computational complexity in game theory.

Contribution

It establishes the ETR-completeness of stationary Nash equilibrium existence in acyclic and cyclic perfect information stochastic games, extending known results to new game classes.

Findings

Existential theory of the reals completeness for acyclic games.

Efficient backward induction for acyclic games with terminal rewards.

Deciding equilibrium existence in cyclic games is also ETR-complete.

Abstract

We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is Existential Theory of the Reals complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic acyclic games with non-negative terminal rewards. We further extend our results to the existence of Nash equilibria where a single player is surely winning. Combining our result with known gadget games without any stationary Nash equilibrium, we obtain that for cyclic games, just deciding existence of any stationary Nash equilibrium is Existential Theory of the Reals complete. This holds for reach-a-set games, stay-in-a-set games, and for deterministic recursive games.