Kousha Etessami ; Mihalis Yannakakis - Recursive Concurrent Stochastic Games

lmcs:1196 - Logical Methods in Computer Science, November 11, 2008, Volume 4, Issue 4 - https://doi.org/10.2168/LMCS-4(4:7)2008
Recursive Concurrent Stochastic Games

Authors: Kousha Etessami ; Mihalis Yannakakis

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.

Volume: Volume 4, Issue 4
Published on: November 11, 2008
Accepted on: June 25, 2015
Submitted on: May 24, 2007
Keywords: Computer Science - Computer Science and Game Theory,Computer Science - Computational Complexity,G.3,F.2,F.1.1


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