Finite-memory strategies in two-player infinite games

TL;DR

This paper proves that in certain infinite two-player games with specific topological and order conditions, winning strategies can be implemented with finite memory, extending understanding of strategy complexity in such games.

Contribution

It establishes conditions under which finite-memory winning strategies exist in infinite games with complex winning sets, using inductive set descriptions and topological properties.

Findings

Finite-memory strategies exist under $ ext{Delta}^0_2$ conditions.

Well partial order on histories ensures finite-memory strategies.

Results apply to finite graph games like multi-energy games.

Abstract

We study infinite two-player win/lose games $(A,B,W)$ where $A,B$ are finite and $W \subseteq (A \times B)^\omega$. At each round Player 1 and Player 2 concurrently choose one action in $A$ and $B$, respectively. Player 1 wins iff the generated sequence is in $W$. Each history $h \in (A \times B)^*$ induces a game $(A,B,W_h)$ with $W_h := \{\rho \in (A \times B)^\omega \mid h \rho \in W\}$. We show the following: if $W$ is in $\Delta^0_2$ (for the usual topology), if the inclusion relation induces a well partial order on the $W_h$'s, and if Player 1 has a winning strategy, then she has a finite-memory winning strategy. Our proof relies on inductive descriptions of set complexity, such as the Hausdorff difference hierarchy of the open sets. Examples in $\Sigma^0_2$ and $\Pi^0_2$ show some tightness of our result. Our result can be translated to games on finite graphs: e.g. finite-memory determinacy of multi-energy games is a direct corollary, whereas it does not follow from recent general results on finite memory strategies.