The Value 1 Problem Under Finite-memory Strategies for Concurrent Mean-payoff Games

TL;DR

This paper studies the problem of computing states with value 1 in concurrent mean-payoff games under finite-memory strategies, providing a polynomial-time algorithm and insights into stationary strategies and their complexity.

Contribution

It introduces a polynomial-time algorithm for the value 1 problem, shows finite-memory strategies can be stationary, and establishes a double exponential bound on strategy patience.

Findings

Polynomial-time algorithm for the value 1 problem

Finite-memory strategies are equivalent to stationary strategies when value 1 is achievable

Double exponential bound on the patience of stationary strategies

Abstract

We consider concurrent mean-payoff games, a very well-studied class of two-player (player 1 vs player 2) zero-sum games on finite-state graphs where every transition is assigned a reward between 0 and 1, and the payoff function is the long-run average of the rewards. The value is the maximal expected payoff that player 1 can guarantee against all strategies of player 2. We consider the computation of the set of states with value 1 under finite-memory strategies for player 1, and our main results for the problem are as follows: (1) we present a polynomial-time algorithm; (2) we show that whenever there is a finite-memory strategy, there is a stationary strategy that does not need memory at all; and (3) we present an optimal bound (which is double exponential) 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).