What are Strategies in Delay Games? Borel Determinacy for Games with Lookahead

TL;DR

This paper studies delay games with Borel winning conditions, proving their determinacy with fixed lookahead evolution, analyzing universal strategies, and exploring decidability for omega-regular and omega-context-free conditions.

Contribution

It introduces the concept of universal strategies in delay games, establishes their determinacy, and investigates decidability issues for complex winning conditions.

Findings

Delay games with Borel conditions are determined with fixed lookahead.

Universal strategies can be evolution-independent under certain conditions.

Decidability results are provided for omega-regular and omega-context-free conditions.

Abstract

We investigate determinacy of delay games with Borel winning conditions, infinite-duration two-player games in which one player may delay her moves to obtain a lookahead on her opponent's moves. First, we prove determinacy of such games with respect to a fixed evolution of the lookahead. However, strategies in such games may depend on information about the evolution. Thus, we introduce different notions of universal strategies for both players, which are evolution-independent, and determine the exact amount of information a universal strategy needs about the history of a play and the evolution of the lookahead to be winning. In particular, we show that delay games with Borel winning conditions are determined with respect to universal strategies. Finally, we consider decidability problems, e.g., "Does a player have a universal winning strategy for delay games with a given winning condition?", for omega-regular and omega-context-free winning conditions.