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.
翻译:我们研究无限的双玩者赢/球游戏$(A,B,W)$(A,B,W)$(美元)$(A,B美元)和$(W) subseteq(A\time B) ⁇ (omega美元)$(美元)$(oomega美元)。在每一回合玩家1和玩家2同时选择一个以美元和美元(B)为单位(美元)的动作。玩家1如果生成的序列以美元为单位(W美元),则会赢(A,B,W_h)$(美元)美元)的游戏$(美元):= ⁇ (rho)\in(A\timeB)\omega\mid\rho)\rho(美元)\rho(B)\x%(W)$)$(美元) 。我们展示的是:如果W$(Delta)0___B$(通常的表层),如果包容关系导致美元(H_h)相当部分的顺序,如果玩家1游戏有一个赢的策略,那么,她就会赢得一个战略。我们的证据依据的内积战略。我们的证据取决于的精度的精度的精度的精度的精度的精度的精度的精度的精度的精度将显示。