We study infinite two-player win/lose games (A,B,W) where A,B are finite and W ⊆ (A×B)^ω. 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 ∈ (A×B)^* induces a game (A,B,W_h) with W_h : = {ρ ∈ (A×B)^ω ∣ h ρ ∈ W}. We show the following: if W is in Δ⁰₂ (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 Σ⁰₂ and Π⁰₂ 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.

We study pure Nash equilibria in infinite-duration games on graphs, with partial visibility of actions but communication (based on a graph) among the players. We show that a simple communication mechanism consisting in reporting the deviator when seeing it and propagating this information is sufficient for characterizing Nash equilibria. We propose an epistemic game construction, which conveniently records important information about the knowledge of the players. With this abstraction, we are able to characterize Nash equilibria which follow the simple communication pattern via winning strategies. We finally discuss the size of the construction, which would allow efficient algorithmic solutions to compute Nash equilibria in the original game.