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Games Where You Can Play Optimally Without Any Memory

  • Hugo Gimbert
  • Wiesław Zielonka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3653)

Abstract

Reactive systems are often modelled as two person antagonistic games where one player represents the system while his adversary represents the environment. Undoubtedly, the most popular games in this context are parity games and their cousins (Rabin, Streett and Muller games). Recently however also games with other types of payments, like discounted or mean-payoff [5,6], previously used only in economic context, entered into the area of system modelling and verification. The most outstanding property of parity, mean-payoff and discounted games is the existence of optimal positional (memoryless) strategies for both players. This observation raises two questions: (1) can we characterise the family of payoff mappings for which there always exist optimal positional strategies for both players and (2) are there other payoff mappings with practical or theoretical interest and admitting optimal positional strategies. This paper provides a complete answer to the first question by presenting a simple necessary and sufficient condition on payoff mapping guaranteeing the existence of optimal positional strategies. As a corollary to this result we show the following remarkable property of payoff mappings: if both players have optimal positional strategies when playing solitary one-player games then also they have optimal positional strategies for two-player games.

Keywords

Optimal Strategy Preference Relation Stochastic Game Simple Cycle Tree Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hugo Gimbert
    • 1
  • Wiesław Zielonka
    • 1
  1. 1.Université Paris 7 and CNRS, LIAFA, case 7014Paris Cedex 05France

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