Stochastic mean payoff games: smoothed analysis and approximation schemes

  • Authors:

  • Endre Boros;Khaled Elbassioni;Mahmoud Fouz;Vladimir Gurvich;Kazuhisa Makino;Bodo Manthey

  • Affiliations:

  • RUTCOR, Rutgers University;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Universität des Saarlandes, Fachrichtung Informatik, Germany;RUTCOR, Rutgers University;University of Tokyo, Graduate School of Information Science and Technology;University of Twente, Department of Applied Mathematics

  • Venue:
  • ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
  • Year:
  • 2011

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Abstract

In this paper, we consider two-player zero-sum stochastic mean payoff games with perfect information modeled by a digraph with black, white, and random vertices. These BWR-games games are polynomially equivalent with the classical Gillette games, which include many well-known subclasses, such as cyclic games, simple stochastic games, stochastic parity games, and Markov decision processes. They can also be used to model parlor games such as Chess or Backgammon. It is a long-standing open question if a polynomial algorithm exists that solves BWR-games. In fact, a pseudo-polynomial algorithm for these games with an arbitrary number of random nodes would already imply their polynomial solvability. Currently, only two classes are known to have such a pseudo-polynomial algorithm: BW-games (the case with no random nodes) and ergodic BWR-games (in which the game's value does not depend on the initial position) with constant number of random nodes. In this paper, we show that the existence of a pseudopolynomial algorithm for BWR-games with constant number of random vertices implies smoothed polynomial complexity and the existence of absolute and relative polynomial-time approximation schemes. In particular, we obtain smoothed polynomial complexity and derive absolute and relative approximation schemes for BW-games and ergodic BWR-games (assuming a technical requirement about the probabilities at the random nodes).