Abstract
We study infinite games played by arbitrarily many players on a directed graph. Equilibrium states capture rational behaviour in these games. Instead of the well-known notion of a Nash equilibrium, we focus on the notion of a subgame perfect equilibrium. We argue that the latter one is more appropriate for the kind of games we study, and we show the existence of a subgame perfect equilibrium in any infinite game with ω-regular winning conditions.
As, in general, equilibria are not unique, it is appealing to compute one with a maximal payoff. This problem corresponds naturally to the problem of deciding given a game and two payoff vectors whether the game has an equilibrium with a payoff in between the given thresholds. We show that this problem is decidable for games with ω-regular winning conditions played on a finite graph and analyse its complexity. Moreover, we establish that any subgame perfect equilibrium of a game with ω-regular winning conditions played on a finite graph can be implemented by finite-state strategies.
Finally, we consider logical definability. We state that if we fix the number of players together with an ω-regular winning condition for each of them and two payoff vectors the property that a game has a subgame perfect equilibrium with a payoff in between the given thresholds is definable in the modal μ-calculus.
This research has been partially supported by the European Community Research Training Network “Games and Automata for Synthesis and Validation” (games).
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Ummels, M. (2006). Rational Behaviour and Strategy Construction in Infinite Multiplayer Games. In: Arun-Kumar, S., Garg, N. (eds) FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2006. Lecture Notes in Computer Science, vol 4337. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11944836_21
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DOI: https://doi.org/10.1007/11944836_21
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