Recursive Concurrent Stochastic Games

TL;DR

This paper analyzes recursive concurrent stochastic games, characterizing their values, establishing PSPACE decidability for termination, and demonstrating strategy optimality and determinacy in an infinite-state setting.

Contribution

It extends analysis of recursive stochastic games to concurrent settings, providing new decidability results, strategy improvement techniques, and determinacy proofs for single-exit recursive concurrent stochastic games.

Findings

Characterized the value of 1-RCSG termination as a least fixed point of nonlinear minimax equations.

Proved PSPACE decidability for the quantitative termination problem.

Established that player 1 has -optimal randomized strategies, and player 2 has optimal strategies, showing r-SM-determinacy.

Abstract

We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games to a concurrent setting where the two players choose moves simultaneously and independently at each state. For multi-exit games, our earlier work already showed undecidability for basic questions like termination, thus we focus on the important case of single-exit RCSGs (1-RCSGs). We first characterize the value of a 1-RCSG termination game as the least fixed point solution of a system of nonlinear minimax functional equations, and use it to show PSPACE decidability for the quantitative termination problem. We then give a strategy improvement technique, which we use to show that player 1 (maximizer) has \epsilon-optimal randomized Stackless & Memoryless (r-SM) strategies for all \epsilon > 0, while player 2 (minimizer) has optimal r-SM strategies. Thus, such games are r-SM-determined. These results mirror and generalize in a strong sense the randomized memoryless determinacy results for finite stochastic games, and extend the classic Hoffman-Karp strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however, relying on subtle analytic properties of certain power series that arise from studying 1-RCSGs. We show that our upper bounds, even for qualitative (probability 1) termination, can not be improved, even to NP, without a major breakthrough, by giving two reductions: first a P-time reduction from the long-standing square-root sum problem to the quantitative termination decision problem for finite concurrent stochastic games, and then a P-time reduction from the latter problem to the qualitative termination problem for 1-RCSGs.