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The complexity of stochastic games
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Competitive Markov decision processes
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Finitely additive stochastic games with Borel measurable payoffs
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Some NP-complete geometric problems
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Quantitative solution of omega-regular games
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Model Checking Probabilistic Pushdown Automata
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The complexity of quantitative concurrent parity games
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On the Complexity of Numerical Analysis
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Recursive markov decision processes and recursive stochastic games
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
On the decidability of temporal properties of probabilistic pushdown automata
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Recursive markov chains, stochastic grammars, and monotone systems of nonlinear equations
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Algorithmic verification of recursive probabilistic state machines
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Automata, Probability, and Recursion
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Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
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Termination criteria for solving concurrent safety and reachability games
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Computational Aspects of Equilibria
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Computing relaxed abstract semantics w.r.t. quadratic zones precisely
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On the Complexity of Nash Equilibria and Other Fixed Points
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Solving systems of rational equations through strategy iteration
ACM Transactions on Programming Languages and Systems (TOPLAS)
Exact algorithms for solving stochastic games: extended abstract
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Qualitative reachability in stochastic BPA games
Information and Computation
The complexity of solving reachability games using value and strategy iteration
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
A survey of stochastic ω-regular games
Journal of Computer and System Sciences
Analysis of recursive probabilistic models
ATVA'06 Proceedings of the 4th international conference on Automated Technology for Verification and Analysis
Robustness of structurally equivalent concurrent parity games
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Repeated zero-sum games with budget
Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 2
Playing games with counter automata
RP'12 Proceedings of the 6th international conference on Reachability Problems
Strategy improvement for concurrent reachability and turn-based stochastic safety games
Journal of Computer and System Sciences
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We study Recursive Concurrent Stochastic Games (RCSGs), extending our recent analysis of recursive simple stochastic games [14, 15] 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 ε-optimal randomized Stackless & Memoryless (r-SM) strategies, 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 [19] strategy improvement approach from the finite to an infinite state setting. The proofs in our infinite-state setting are very different however We show that our upper bounds, even for qualitative termination, can not be improved 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