The Complexity of Ergodic Mean-payoff Games

TL;DR

This paper investigates the computational complexity of ergodic mean-payoff games, providing bounds on strategies, complexity classifications, and algorithms for approximation, advancing understanding of these fundamental stochastic games.

Contribution

It establishes exponential bounds on strategy patience, classifies approximation complexity, and connects game values to the existential theory of reals, advancing the theoretical framework.

Findings

Exponential bound on patience of stationary strategies

Approximation problem lies in FNP and is as hard as simple stochastic games decision problem

Value can be expressed in the existential theory of the reals and exhibits square-root sum hardness

Abstract

We study two-player (zero-sum) concurrent mean-payoff games played on a finite-state graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lie in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP intersection coNP is the long-standing best known bound). We present a variant of the strategy-iteration algorithm by Hoffman and Karp; show that both our algorithm and the classical value-iteration algorithm can approximate the value in exponential time; and identify a subclass where the value-iteration algorithm is a FPTAS. We also show that the exact value can be expressed in the existential theory of the reals, and establish square-root sum hardness for a related class of games.