From Local to Global Determinacy in Concurrent Graph Games

Authors Benjamin Bordais, Patricia Bouyer, Stéphane Le Roux

Thumbnail PDF


  • Filesize: 1.01 MB
  • 14 pages

Document Identifiers

Author Details

Benjamin Bordais
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190 Gif-sur-Yvette, France
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

Cite AsGet BibTex

Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. From Local to Global Determinacy in Concurrent Graph Games. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In general, finite concurrent two-player reachability games are only determined in a weak sense: the supremum probability to win can be approached via stochastic strategies, but cannot be realized. We introduce a class of concurrent games that are determined in a much stronger sense, and in a way, it is the largest class with this property. To this end, we introduce the notion of local interaction at a state of a graph game: it is a game form whose outcomes (i.e. a table whose entries) are the next states, which depend on the concurrent actions of the players. By definition, a game form is determined iff it always yields games that are determined via deterministic strategies when used as a local interaction in a Nature-free, one-shot reachability game. We show that if all the local interactions of a graph game with Borel objective are determined game forms, the game itself is determined: if Nature does not play, one player has a winning strategy; if Nature plays, both players have deterministic strategies that maximize the probability to win. This constitutes a clear-cut separation: either a game form behaves poorly already when used alone with basic objectives, or it behaves well even when used together with other well-behaved game forms and complex objectives. Existing results for positional and finite-memory determinacy in turn-based games are extended this way to concurrent games with determined local interactions (CG-DLI).

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Concurrent games
  • Game forms
  • Local interaction


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. From local to global determinacy in concurrent graph games. CoRR, abs/2107.04081, 2021. URL:
  2. Endre Boros, Ondrej Cepek, and Vladimir Gurvich. Separable discrete functions: Recognition and sufficient conditions. Discret. Math., 342(5):1275-1292, 2019. URL:
  3. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. In 31st International Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vienna, Austria (Virtual Conference), pages 24:1-24:22, 2020. URL:
  4. Krishnendu Chatterjee, Marcin Jurdzinski, and Thomas A. Henzinger. Quantitative stochastic parity games. In J. Ian Munro, editor, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 121-130. SIAM, 2004. URL:
  5. Luca de Alfaro, Thomas A. Henzinger, and Orna Kupferman. Concurrent reachability games. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 564-575. IEEE Computer Society, 1998. URL:
  6. Thomas Eiter, Kazuhisa Makino, and Georg Gottlob. Computational aspects of monotone dualization: A brief survey. Discret. Appl. Math., 156(11):2035-2049, 2008. URL:
  7. E. Allen Emerson and Charanjit S. Jutla. Tree automata, mu-calculus and determinacy (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science FOCS, San Juan, Puerto Rico, 1-4 October 1991, pages 368-377, 1991. URL:
  8. Hugh Everett. Recursive games. Annals of Mathematics Studies - Contributions to the Theory of Games, 3:67-78, 1957. Google Scholar
  9. Michael L. Fredman and Leonid Khachiyan. On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms, 21(3):618-628, 1996. URL:
  10. Allan Gibbard. Manipulation of voting schemes: A general result. Econometrica, 41(4):587-601, 1973. URL:
  11. Hugo Gimbert and Florian Horn. Solving simple stochastic tail games. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 847-862. SIAM, 2010. URL:
  12. V.A. Gurvich. The solvability of positional games in pure strategies. USSR Computational Mathematics and Mathematical Physics, 15(2):74-87, 1975. URL:
  13. Eryk Kopczyński. Half-positional Determinacy of Infinite Games. PhD thesis, Warsaw University, 2008. Google Scholar
  14. Donald A Martin. Borel determinacy. Annals of Mathematics, pages 363-371, 1975. Google Scholar
  15. Donald A Martin. A purely inductive proof of Borel determinacy. Recursion theory (Ithaca, NY, 1982), 42:303-308, 1985. Google Scholar
  16. Donald A Martin. The determinacy of Blackwell games. The Journal of Symbolic Logic, 63(4):1565-1581, 1998. Google Scholar
  17. Andrzej Włodzimierz Mostowski. Games with forbidden positions. Preprint - Uniwersytet Gdański. Instytut Matematyki. UG, 1991. Google Scholar
  18. Stéphane Le Roux. From winning strategy to nash equilibrium. Math. Log. Q., 60(4-5):354-371, 2014. URL:
  19. Stéphane Le Roux. Concurrent games and semi-random determinacy. In 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, pages 40:1-40:15, 2018. URL:
  20. Stéphane Le Roux. Time-aware uniformization of winning strategies. In Beyond the Horizon of Computability - 16th Conference on Computability in Europe, CiE 2020, Fisciano, Italy, June 29 - July 3, 2020, Proceedings, pages 193-204, 2020. URL:
  21. Lloyd S. Shapley. Stochastic games. Proceedings of the national academy of sciences, 39(10):1095-1100, 1953. Google Scholar
  22. Wieslaw Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science, 200(1-2):135-183, 1998. Google Scholar
  23. Wieslaw Zielonka. Perfect-information stochastic parity games. In Igor Walukiewicz, editor, Foundations of Software Science and Computation Structures, 7th International Conference, FOSSACS 2004, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2004, Barcelona, Spain, March 29 - April 2, 2004, Proceedings, volume 2987 of Lecture Notes in Computer Science, pages 499-513. Springer, 2004. URL: