Finite-Memory Strategies in Two-Player Infinite Games

Authors Patricia Bouyer, Stéphane Le Roux, Nathan Thomasset

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Author Details

Patricia Bouyer
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France
Stéphane Le Roux
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France
Nathan Thomasset
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France

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Patricia Bouyer, Stéphane Le Roux, and Nathan Thomasset. Finite-Memory Strategies in Two-Player Infinite Games. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Verification by model checking
  • Two-player win/lose games
  • Infinite trees
  • Finite-memory winning strategies
  • Well partial orders
  • Hausdorff difference hierarchy


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