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We study infinite two-player win/lose games $(A,B,W)$ where $A,B$ are finiteand $W \subseteq (A \times B)^\omega$. At each round Player 1 and Player 2concurrently choose one action in $A$ and $B$, respectively. Player 1 wins iffthe generated sequence is in $W$. Each history $h \in (A \times B)^*$ induces agame $(A,B,W_h)$ with $W_h := \{\rho \in (A \times B)^\omega \mid h \rho \inW\}$. We show the following: if $W$ is in $\Delta^0_2$ (for the usualtopology), 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-memorywinning 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. Ourresult can be translated to games on finite graphs: e.g. finite-memorydeterminacy of multi-energy games is a direct corollary, whereas it does notfollow from recent general results on finite memory strategies.

Finite-memory strategies in two-player infinite games