Document Open Access Logo

Arena-Independent Memory Bounds for Nash Equilibria in Reachability Games

Author James C. A. Main

PDF
Thumbnail PDF

File

LIPIcs.STACS.2024.50.pdf
  • Filesize: 0.73 MB
  • 18 pages

Document Identifiers

Author Details

James C. A. Main
  • F.R.S.-FNRS & UMONS - Université de Mons, Belgium

Funding

This work has been supported by the Fonds de la Recherche Scientifique - FNRS under Grant n° T.0188.23 (PDR ControlleRS). The author is a Research Fellow of the Fonds de la Recherche Scientifique - FNRS and member of the TRAIL institute.

Acknowledgements

I thank Thomas Brihaye, Aline Goeminne and Mickael Randour for fruitful discussions and their comments on a preliminary version of this paper.

Cite AsGet BibTex

James C. A. Main. Arena-Independent Memory Bounds for Nash Equilibria in Reachability Games. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.50

Abstract

We study the memory requirements of Nash equilibria in turn-based multiplayer games on possibly infinite graphs with reachability, shortest path and Büchi objectives. We present constructions for finite-memory Nash equilibria in these games that apply to arbitrary game graphs, bypassing the finite-arena requirement that is central in existing approaches. We show that, for these three types of games, from any Nash equilibrium, we can derive another Nash equilibrium where all strategies are finite-memory such that the same players accomplish their objective, without increasing their cost for shortest path games. Furthermore, we provide memory bounds that are independent of the size of the game graph for reachability and shortest path games. These bounds depend only on the number of players. To the best of our knowledge, we provide the first results pertaining to finite-memory constrained Nash equilibria in infinite arenas and the first arena-independent memory bounds for Nash equilibria.

Subject Classification

ACM Subject Classification
  • Theory of computation → Solution concepts in game theory
Keywords
  • multiplayer games on graphs
  • Nash equilibrium
  • finite-memory strategies

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Roderick Bloem, Krishnendu Chatterjee, and Barbara Jobstmann. Graph games and reactive synthesis. In Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem, editors, Handbook of Model Checking, pages 921-962. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-10575-8_27.
  2. Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Pure Nash equilibria in concurrent deterministic games. Log. Methods Comput. Sci., 11(2), 2015. URL: https://doi.org/10.2168/LMCS-11(2:9)2015.
  3. Patricia Bouyer, Stéphane Le Roux, Youssouf Oualhadj, Mickael Randour, and Pierre Vandenhove. Games where you can play optimally with arena-independent finite memory. Log. Methods Comput. Sci., 18(1), 2022. URL: https://doi.org/10.46298/lmcs-18(1:11)2022.
  4. Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. Characterizing omega-regularity through finite-memory determinacy of games on infinite graphs. TheoretiCS, 2, 2023. URL: https://doi.org/10.46298/theoretics.23.1.
  5. Thomas Brihaye, Véronique Bruyère, Aline Goeminne, Jean-François Raskin, and Marie van den Bogaard. The complexity of subgame perfect equilibria in quantitative reachability games. Log. Methods Comput. Sci., 16(4), 2020. URL: https://lmcs.episciences.org/6883.
  6. Thomas Brihaye, Véronique Bruyère, Aline Goeminne, and Nathan Thomasset. On relevant equilibria in reachability games. J. Comput. Syst. Sci., 119:211-230, 2021. URL: https://doi.org/10.1016/j.jcss.2021.02.009.
  7. Thomas Brihaye, Julie De Pril, and Sven Schewe. Multiplayer cost games with simple Nash equilibria. In Sergei N. Artëmov and Anil Nerode, editors, Logical Foundations of Computer Science, International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings, volume 7734 of Lecture Notes in Computer Science, pages 59-73. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-35722-0_5.
  8. Thomas Brihaye, Aline Goeminne, James C. A. Main, and Mickael Randour. Reachability games and friends: A journey through the lens of memory and complexity (invited talk). In Patricia Bouyer and Srikanth Srinivasan, editors, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2023, December 18-20, 2023, IIIT Hyderabad, Telangana, India, volume 284 of LIPIcs, pages 1:1-1:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.FSTTCS.2023.1.
  9. Véronique Bruyère. Computer aided synthesis: A game-theoretic approach. In Émilie Charlier, Julien Leroy, and Michel Rigo, editors, Developments in Language Theory - 21st International Conference, DLT 2017, Liège, Belgium, August 7-11, 2017, Proceedings, volume 10396 of Lecture Notes in Computer Science, pages 3-35. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-62809-7_1.
  10. Véronique Bruyère, Noémie Meunier, and Jean-François Raskin. Secure equilibria in weighted games. In Thomas A. Henzinger and Dale Miller, editors, Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS '14, Vienna, Austria, July 14 - 18, 2014, pages 26:1-26:26. ACM, 2014. URL: https://doi.org/10.1145/2603088.2603109.
  11. Rodica Condurache, Emmanuel Filiot, Raffaella Gentilini, and Jean-François Raskin. The complexity of rational synthesis. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 121:1-121:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ICALP.2016.121.
  12. Julie De Pril. Equilibria in Multiplayer Cost Games. PhD thesis, UMONS, 2013. URL: http://math.umons.ac.be/staff/ancien/DePril.Julie/thesis_Julie_DePril.pdf.
  13. E. Allen Emerson and Charanjit S. Jutla. The complexity of tree automata and logics of programs. In FOCS, pages 328-337. IEEE Computer Society, 1988. URL: https://doi.org/10.1109/SFCS.1988.21949.
  14. Nathanaël Fijalkow, Nathalie Bertrand, Patricia Bouyer-Decitre, Romain Brenguier, Arnaud Carayol, John Fearnley, Hugo Gimbert, Florian Horn, Rasmus Ibsen-Jensen, Nicolas Markey, Benjamin Monmege, Petr Novotný, Mickael Randour, Ocan Sankur, Sylvain Schmitz, Olivier Serre, and Mateusz Skomra. Games on graphs. CoRR, abs/2305.10546, 2023. URL: https://doi.org/10.48550/arXiv.2305.10546.
  15. James Friedman. A non-cooperative equilibrium for supergames. Review of Economic Studies, 38(1):1-12, 1971. URL: https://EconPapers.repec.org/RePEc:oup:restud:v:38:y:1971:i:1:p:1-12.
  16. David Gale and Frank M Stewart. Infinite games with perfect information. Contributions to the Theory of Games, 2(245-266):2-16, 1953. Google Scholar
  17. Hugo Gimbert and Wieslaw Zielonka. Games where you can play optimally without any memory. In CONCUR 2005 - Concurrency Theory, 16th International Conference, CONCUR 2005, San Francisco, CA, USA, August 23-26, 2005, Proceedings, pages 428-442, 2005. URL: https://doi.org/10.1007/11539452_33.
  18. Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research [outcome of a Dagstuhl seminar, February 2001], volume 2500 of Lecture Notes in Computer Science. Springer, 2002. URL: https://doi.org/10.1007/3-540-36387-4.
  19. Stéphane Le Roux and Arno Pauly. Extending finite-memory determinacy to multi-player games. Inf. Comput., 261:676-694, 2018. URL: https://doi.org/10.1016/j.ic.2018.02.024.
  20. James C. A. Main. Arena-independent memory bounds for Nash equilibria in reachability games. CoRR, abs/2310.02142, 2023. URL: https://doi.org/10.48550/ARXIV.2310.02142.
  21. René Mazala. Infinite games. In Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors, Automata, Logics, and Infinite Games: A Guide to Current Research [outcome of a Dagstuhl seminar, February 2001], volume 2500 of Lecture Notes in Computer Science, pages 23-42. Springer, 2001. URL: https://doi.org/10.1007/3-540-36387-4_2.
  22. John F. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1):48-49, 1950. URL: https://doi.org/10.1073/pnas.36.1.48.
  23. Martin J. Osborne and Ariel Rubinstein. A course in game theory. The MIT Press, 1994. Google Scholar
  24. Mickael Randour. Automated synthesis of reliable and efficient systems through game theory: A case study. In Proc. of ECCS 2012, Springer Proceedings in Complexity XVII, pages 731-738. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-00395-5_90.
  25. Michael Ummels. Rational behaviour and strategy construction in infinite multiplayer games. In S. Arun-Kumar and Naveen Garg, editors, FSTTCS 2006: Foundations of Software Technology and Theoretical Computer Science, 26th International Conference, Kolkata, India, December 13-15, 2006, Proceedings, volume 4337 of Lecture Notes in Computer Science, pages 212-223. Springer, 2006. URL: https://doi.org/10.1007/11944836_21.
  26. Michael Ummels. The complexity of Nash equilibria in infinite multiplayer games. In Roberto M. Amadio, editor, Foundations of Software Science and Computational Structures, 11th International Conference, FOSSACS 2008, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2008, Budapest, Hungary, March 29 - April 6, 2008. Proceedings, volume 4962 of Lecture Notes in Computer Science, pages 20-34. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-78499-9_3.
  27. Michael Ummels and Dominik Wojtczak. The complexity of Nash equilibria in limit-average games. In Joost-Pieter Katoen and Barbara König, editors, CONCUR 2011 - Concurrency Theory - 22nd International Conference, CONCUR 2011, Aachen, Germany, September 6-9, 2011. Proceedings, volume 6901 of Lecture Notes in Computer Science, pages 482-496. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-23217-6_32.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact