Strategy Iteration using Non-Deterministic Strategies for Solving Parity Games

TL;DR

This paper extends strategy iteration algorithms for parity games to non-deterministic strategies, providing new theoretical insights, algorithmic adaptations, and a bound on the number of improvement steps.

Contribution

It adapts existing strategy-improvement algorithms to non-deterministic strategies and establishes a bound on improvement steps, generalizing previous randomized bounds.

Findings

Algorithm adapts to non-deterministic strategies.

Valuations coincide on certain game arenas.

Number of improvement steps bounded by O(1.724^n).

Abstract

This article extends the idea of solving parity games by strategy iteration to non-deterministic strategies: In a non-deterministic strategy a player restricts himself to some non-empty 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 Bjoerklund, Sandberg, and Vorobyov can easily be adapted to the more general setting of non-deterministic strategies. Further, we show that applying the heuristic of "all profitable switches" 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. In contrast to the algorithm by Bjoerklund et al., we present our algorithm directly for parity games which allows us to compare it to the algorithm by Jurdzinski and Voege: 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 Voege 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^n). For strategy-improvement algorithms, this bound was previously only known to be attainable by using randomization.