Continuous Positional Payoffs

Author Alexander Kozachinskiy

Thumbnail PDF


  • Filesize: 0.81 MB
  • 17 pages

Document Identifiers

Author Details

Alexander Kozachinskiy
  • Department of Computer Science, University of Warwick, Coventry, UK

Cite AsGet BibTex

Alexander Kozachinskiy. Continuous Positional Payoffs. In 32nd International Conference on Concurrency Theory (CONCUR 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 203, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


What payoffs are positionally determined for deterministic two-player antagonistic games on finite directed graphs? In this paper we study this question for payoffs that are continuous. The main reason why continuous positionally determined payoffs are interesting is that they include the multi-discounted payoffs. We show that for continuous payoffs positional determinacy is equivalent to a simple property called prefix-monotonicity. We provide three proofs of it, using three major techniques of establishing positional determinacy - inductive technique, fixed point technique and strategy improvement technique. A combination of these approaches provides us with better understanding of the structure of continuous positionally determined payoffs as well as with some algorithmic results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
  • Games on graphs
  • positional strategies
  • continuous payoffs


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


  1. Henrik Björklund and Sergei Vorobyov. Combinatorial structure and randomized subexponential algorithms for infinite games. Theoretical Computer Science, 349(3):347-360, 2005. Google Scholar
  2. 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). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  3. Cristian S Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan. Deciding parity games in quasipolynomial time. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 252-263, 2017. Google Scholar
  4. Thomas Colcombet and Damian Niwiński. On the positional determinacy of edge-labeled games. Theoretical Computer Science, 352(1-3):190-196, 2006. Google Scholar
  5. Andrzej Ehrenfeucht and Jan Mycielski. Positional strategies for mean payoff games. International Journal of Game Theory, 8(2):109-113, 1979. Google Scholar
  6. E Allen Emerson and Charanjit S Jutla. Tree automata, mu-calculus and determinacy. In FoCS, volume 91, pages 368-377. Citeseer, 1991. Google Scholar
  7. John Fearnley. Strategy iteration algorithms for games and Markov decision processes. PhD thesis, University of Warwick, 2010. Google Scholar
  8. Hugo Gimbert. Pure stationary optimal strategies in markov decision processes. In Annual Symposium on Theoretical Aspects of Computer Science, pages 200-211. Springer, 2007. Google Scholar
  9. Hugo Gimbert and Wieslaw Zielonka. When can you play positionnaly? In I. Fiala, V. Koubek, and J. Kratochvil, editors, Mathematical Foundations of Computer Science 2004, Lecture Notes in Comp. Sci. 3153, pages 686-697, Prague, Czech Republic, 2004. Springer. URL:
  10. Hugo Gimbert and Wiesław Zielonka. Games where you can play optimally without any memory. In International Conference on Concurrency Theory, pages 428-442. Springer, 2005. Google Scholar
  11. Hugo Gimbert and Wieslaw Zielonka. Applying blackwell optimality: priority mean-payoff games as limits of multi-discounted games. In Logic and automata, pages 331-356, 2008. Google Scholar
  12. Erich Gradel and Wolfgang Thomas. Automata, logics, and infinite games: a guide to current research, volume 2500. Springer Science & Business Media, 2002. Google Scholar
  13. Vladimir A Gurvich, Alexander V Karzanov, and LG Khachivan. Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR Computational Mathematics and Mathematical Physics, 28(5):85-91, 1988. Google Scholar
  14. Nir Halman. Simple stochastic games, parity games, mean payoff games and discounted payoff games are all lp-type problems. Algorithmica, 49(1):37-50, 2007. Google Scholar
  15. Thomas Dueholm Hansen, Peter Bro Miltersen, and Uri Zwick. Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor. Journal of the ACM (JACM), 60(1):1-16, 2013. Google Scholar
  16. Ronald A Howard. Dynamic programming and markov processes. The M.I.T. Press, 1960. Google Scholar
  17. Marcin Jurdziński. Deciding the winner in parity games is in up∩co-up. Information Processing Letters, 68(3):119-124, 1998. Google Scholar
  18. Elon Kohlberg. Invariant half-lines of nonexpansive piecewise-linear transformations. Mathematics of Operations Research, 5(3):366-372, 1980. Google Scholar
  19. Alexander Kozachinskiy. Continuous positional payoffs. arXiv preprint, 2020. URL:
  20. Robert McNaughton. Infinite games played on finite graphs. Annals of Pure and Applied Logic, 65(2):149-184, 1993. Google Scholar
  21. Andrzej W Mostowski. Games with forbidden positions. Technical Report 78, Uniwersytet Gdánski, Instytut Matematyki, 1991. Google Scholar
  22. Martin L Puterman. Markov decision processes: discrete stochastic dynamic programming. John Wiley & Sons, 2014. Google Scholar
  23. Lloyd S Shapley. Stochastic games. Proceedings of the national academy of sciences, 39(10):1095-1100, 1953. Google Scholar
  24. Richard S Sutton, Andrew G Barto, et al. Introduction to reinforcement learning, volume 135. MIT press Cambridge, 1998. Google Scholar
  25. Uri Zwick and Mike Paterson. The complexity of mean payoff games on graphs. Theoretical Computer Science, 158(1-2):343-359, 1996. Google Scholar