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Positional Determinacy with Colored Vertices: A 1-To-2-Player Lift

Authors Raphaël Berthon , Stéphane Le Roux

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LIPIcs.CONCUR.2026.17.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • two-player games
  • one-player games
  • parity objectives

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Abstract

Positional determinacy of vertex-colored parity games was proved in the 1990s, which directly implies positional determinacy of edge-colored parity games. In 2006, it was shown that if a prefix-independent color-based objective ensures that every edge-colored two-player turn-based game is positionally determined, this objective is equivalent to a parity objective. We prove a similar result for vertex-colored games, namely that the following are equivalent for any prefix-independent objective W over a finite set of colors:  
- W is positionally determined on all vertex-colored one-player games. 
- W is positionally determined on all vertex-colored two-player games. 
- W is equivalent to a parity objective on ordrerd pairs of colors.  We prove that finiteness of the color set is required for our equivalence to hold. Beyond this 1-to-2-player lift, the technique that we develop to handle the pairs of colors establishes a promising 2-way correspondence between edge-colored games and vertex-colored games.

Cite As Get BibTex

Raphaël Berthon and Stéphane Le Roux. Positional Determinacy with Colored Vertices: A 1-To-2-Player Lift. In 37th International Conference on Concurrency Theory (CONCUR 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 391, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CONCUR.2026.17

Author Details

Raphaël Berthon
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France
Stéphane Le Roux
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

Funding

  • Berthon, Raphaël: Funded by ANR-22-CE48-0012 (Bisous).
  • Le Roux, Stéphane: Funded by ANR-22-CE48-0012 (Bisous).

References

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