Solution Concepts and Algorithms for Infinite Multiplayer Games

We survey and discuss several solution concepts for infinite turn-based multiplayer games with qualitative (i.e. win-lose) objectives of the players. These games generalise in a natural way the common model of games in verification which are two-player, zero-sum games with ω-regular winning conditions. The generalisation is in two directions: our games may have more than two players, and the objectives of the players need not be completely antagonistic. The notion of a Nash equilibrium is the classical solution concept in game theory. However, for games that extend over time, in particular for games of infinite duration, Nash equilibria are not always satisfactory as a notion of rational behaviour. We therefore discuss variants of Nash equilibria such as subgame perfect equilibria and secure equilibria. We present criteria for the existence of Nash equilibria and subgame perfect equilibria in the case of arbitrarily many players and for the existence of secure equilibria in the two-player case. In the second part of this paper, we turn to algorithmic questions: For each of the solution concepts that we discuss, we present algorithms that decide the existence of a solution with certain requirements in a game with parity winning conditions. Since arbitrary ω-regular winning conditions can be reduced to parity conditions, our algorithms are also applicable to games with arbitrary ω-regular winning conditions.