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Recursive stochastic games with positive rewards. (English) Zbl 1153.91328
Aceto, Luca (ed.) et al., Automata, languages and programming. 35th international colloquium, ICALP 2008, Reykjavik, Iceland, July 7–11, 2008. Proceedings, Part I. Berlin: Springer (ISBN 978-3-540-70574-1/pbk). Lecture Notes in Computer Science 5125, 711-723 (2008).
Summary: We study the complexity of a class of Markov decision processes and, more generally, stochastic games, called 1-exit Recursive Markov Decision Processes (1-RMDPs) and Simple Stochastic Games (1-RSSGs) with strictly positive rewards. These are a class of finitely presented countable-state zero-sum stochastic games, with total expected reward objective. They subsume standard finite-state MDPs and Condon’s simple stochastic games and correspond to optimization and game versions of several classic stochastic models, with rewards. Such stochastic models arise naturally as models of probabilistic procedural programs with recursion, and the problems we address are motivated by the goal of analyzing the optimal/pessimal expected running time in such a setting.
We give polynomial time algorithms for 1-exit Recursive Markov Decision Processes (1-RMDPs) with positive rewards. Specifically, we show that the exact optimal value of both maximizing and minimizing 1-RMDPs with positive rewards can be computed in polynomial time (this value may be $$\infty )$$. For two-player 1-RSSGs with positive rewards, we prove a “stackless and memoryless” determinacy result, and show that deciding whether the game value is at least a given value $$r$$ is in $$\text{NP} \cap \text{coNP}$$. We also prove that a simultaneous strategy improvement algorithm converges to the value and optimal strategies for these stochastic games. We observe that 1-RSSG positive reward games are “harder” than finite-state SSGs in several senses.
For the entire collection see [Zbl 1142.68001].

##### MSC:
 91A15 Stochastic games, stochastic differential games 90C40 Markov and semi-Markov decision processes 68Q25 Analysis of algorithms and problem complexity
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