Perfect-Information Stochastic Games with Generalized Mean-Payoff Objectives

TL;DR

This paper analyzes perfect-information stochastic games with multiple mean-payoff objectives, establishing complexity results and strategy requirements, which advance understanding of synthesizing stochastic reactive systems with quantitative goals.

Contribution

It proves coNP-completeness of the almost-sure and basic decision problems, and characterizes strategy memory requirements for both one-player and two-player cases.

Findings

coNP-complete for almost-sure and general decision problems

Polynomial-time solution for one-player stochastic games with memoryless strategies

Exponential memory needed for two-player stochastic games in general

Abstract

Graph games provide the foundation for modeling and synthesizing reactive processes. In the synthesis of stochastic reactive processes, the traditional model is perfect-information stochastic games, where some transitions of the game graph are controlled by two adversarial players, and the other transitions are executed probabilistically. We consider such games where the objective is the conjunction of several quantitative objectives (specified as mean-payoff conditions), which we refer to as generalized mean-payoff objectives. The basic decision problem asks for the existence of a finite-memory strategy for a player that ensures the generalized mean-payoff objective be satisfied with a desired probability against all strategies of the opponent. A special case of the decision problem is the almost-sure problem where the desired probability is 1. Previous results presented a semi-decision procedure for epsilon-approximations of the almost-sure problem. In this work, we show that both the almost-sure problem as well as the general basic decision problem are coNP-complete, significantly improving the previous results. Moreover, we show that in the case of 1-player stochastic games, randomized memoryless strategies are sufficient and the problem can be solved in polynomial time. In contrast, in two-player stochastic games, we show that even with randomized strategies exponential memory is required in general, and present a matching exponential upper bound. We also study the basic decision problem with infinite-memory strategies and present computational complexity results for the problem. Our results are relevant in the synthesis of stochastic reactive systems with multiple quantitative requirements.