We study two-player (zero-sum) concurrent mean-payoff games played on a finite-state graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lies in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP ∩ coNP is the long-standing best known bound). We present a variant of the strategy-iteration algorithm by Hoffman and Karp; show that both our algorithm and the classical value-iteration algorithm can approximate the value in exponential time; and identify a subclass where the value-iteration algorithm is a FPTAS. We also show that the exact value can be expressed in the existential theory of the reals, and establish square-root sum hardness for a related class of games.
122 - 133
ICST: International Conference on Software Testing, Verification and Validation
2014-07-08 – 2014-07-11
Chatterjee K, Ibsen-Jensen R. The complexity of ergodic mean payoff games. In: Vol 8573. Springer; 2014:122-133. doi:10.1007/978-3-662-43951-7_11
Chatterjee, K., & Ibsen-Jensen, R. (2014). The complexity of ergodic mean payoff games (Vol. 8573, pp. 122–133). Presented at the ICST: International Conference on Software Testing, Verification and Validation, Copenhagen, Denmark: Springer. https://doi.org/10.1007/978-3-662-43951-7_11
Chatterjee, Krishnendu, and Rasmus Ibsen-Jensen. “The Complexity of Ergodic Mean Payoff Games,” 8573:122–33. Springer, 2014. https://doi.org/10.1007/978-3-662-43951-7_11.
K. Chatterjee and R. Ibsen-Jensen, “The complexity of ergodic mean payoff games,” presented at the ICST: International Conference on Software Testing, Verification and Validation, Copenhagen, Denmark, 2014, vol. 8573, no. Part 2, pp. 122–133.
Chatterjee K, Ibsen-Jensen R. 2014. The complexity of ergodic mean payoff games. ICST: International Conference on Software Testing, Verification and Validation, LNCS, vol. 8573. 122–133.
Chatterjee, Krishnendu, and Rasmus Ibsen-Jensen. The Complexity of Ergodic Mean Payoff Games. Vol. 8573, no. Part 2, Springer, 2014, pp. 122–33, doi:10.1007/978-3-662-43951-7_11.
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