Strategy Iteration using Non-Deterministic Strategies for Solving Parity Games

This article introduces the idea of non-deterministic strategies for parity games: In a non-deterministic strategy a player restricts himself to some nonempty subset of possible actions at a given node, instead of limiting himself to exactly one action. We show that a strategy-improvement algorithm by by Björklund, Sandberg, and Vorobyov [3] can easily be adapted to the more general setting of nondeterministic strategies. Further, we show that applying the heuristic of “all profitable switches” (cf. [1]) leads to choosing a “locally optimal” successor strategy in the setting of non-deterministic strategies, thereby obtaining an easy proof of an algorithm by Schewe [13]. In contrast to [3], we present our algorithm directly for parity games which allows us to compare it to the algorithm by Jurdzinski and Vöge [15]: We show that the valuations used in both algorithm coincide on parity game arenas in which one player can “surrender”. Thus, our algorithm can also be seen as a generalization of the one by Jurdzinski and Vöge to non-deterministic strategies. Finally, using non-deterministic strategies allows us to show that the number of improvement steps is bound from above by O(1.724). For strategy-improvement algorithms, this bound was previously only known to be attainable by using randomization (cf. [1]).