### 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 pseudo-polynomial 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).

Original language | English |
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Title of host publication | Automata, Languages and Programming |

Subtitle of host publication | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 |

Editors | Luca Aceto, Monika Henzinger, Jiří Sgall |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 147-158 |

Number of pages | 12 |

ISBN (Electronic) | 978-3-642-22006-7 |

ISBN (Print) | 978-3-642-22005-0 |

DOIs | |

Publication status | Published - 2011 |

Event | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland Duration: 4 Jul 2011 → 8 Jul 2011 Conference number: 38 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer Verlag |

Volume | 6755 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 |
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Abbreviated title | ICALP |

Country | Switzerland |

City | Zurich |

Period | 4/07/11 → 8/07/11 |

### Keywords

- METIS-278810
- EWI-20543
- IR-78128

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## Cite this

Boros, E., Elbassioni, K., Fouz, M., Gurvich, V., Makino, K., & Manthey, B. (2011). Stochastic mean payoff games: smoothed analysis and approximation schemes. In L. Aceto, M. Henzinger, & J. Sgall (Eds.),

*Automata, Languages and Programming: 38th International Colloquium on Automata, Languages and Programming, ICALP 2011*(pp. 147-158). (Lecture Notes in Computer Science; Vol. 6755). Berlin: Springer. https://doi.org/10.1007/978-3-642-22006-7_13