## Abstract

We study subgame phi-maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, phi denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame f-maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by phi. First, we provide necessary and sufficient conditions for a strategy to be a subgame phi-maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame f-maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function phi* with the following property: if a player has a subgame phi*-maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame phi-maxmin strategy for every positive tolerance function f is equivalent to the existence of a subgame maxmin strategy.

Original language | English |
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Pages (from-to) | 704-737 |

Number of pages | 34 |

Journal | Dynamic Games and Applications |

Volume | 11 |

Issue number | 4 |

Early online date | 2 Mar 2021 |

DOIs | |

Publication status | Published - Dec 2021 |

## Keywords

- Stochastic games
- Zero-sum games
- Subgame phi-maxmin strategies