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Weihrauch Degrees of Finding Equilibria in Sequential Games

  • Stéphane Le Roux
  • Arno Pauly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

Abstract

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.

Keywords

Nash Equilibrium Solution Concept Winning Strategy Strategy Profile Sequential Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work benefited from the Royal Society International Exchange Grant IE111233 and the Marie Curie International Research Staff Exchange Scheme Computable Analysis, PIRSES-GA-2011- 294962.

References

  1. 1.
    Akama, Y., Berardi, S., Hayashi, S., Kohlenbach, U.: An arithmetical hierarchy of the law of excluded middle and related principles. In: 19th IEEE Symposium on Logic in Computer Science (LICS 2004), pp. 192–201 (2004)Google Scholar
  2. 2.
    Brattka, V.: Effective Borel measurability and reducibility of functions. Math. Logic Q. 51(1), 19–44 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Logic 1, 73–117 (2011). arXiv:0905.4685 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symbolic Logic 76, 143–176 (2011). arXiv:0905.4679 MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. arXiv 1312.7305 (2013). http://arxiv.org/abs/1312.7305
  6. 6.
    Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass theorem is the jump of weak König’s Lemma. Ann. Pure Appl. Logic 163(6), 623–625 (2012). arXiv:1101.0792 MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cenzer, D., Remmel, J.: Recursively presented games and strategies. Math. Soc. Sci. 24(2–3), 117–139 (1992)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Eguchi, N.: Infinite games in the cantor space over admissible set theories. In: Higuchi, K. (ed.) Proceedings of Computability Theory and Foundations of Mathematics (2014)Google Scholar
  9. 9.
    Friedman, H.: Higher set theory and mathematical practice. Ann. Math. Logic 2(3), 325–357 (1971)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Gale, D., Stewart, F.: Infinite games with perfect information. In: Contributions to the Theory of Games, Annals of Mathematical Studies, vol. 28, pp. 245–266. Princeton University Press (1953)Google Scholar
  11. 11.
    Higuchi, K., Kihara, T.: Inside the muchnik degrees I: discontinuity, learnability and constructivism. Ann. Pure Appl. Logic 165(5), 1058–1114 (2014)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kechris, A.: Classical Descriptive Set Theory. Graduate Texts in Mathematic, vol. 156. Springer, New York (1995)MATHGoogle Scholar
  13. 13.
    Le Roux, S.: Infinite sequential Nash equilibria. Logical Methods Comput. Sci. 9(2), 14 (2013)Google Scholar
  14. 14.
    Le Roux, S.: From winning strategy to Nash equilibrium. Math. Logic Q. 60(4–5), 354–371 (2014). http://dx.doi.org/10.1002/malq.201300034, arXiv 1203.1866MATHCrossRefGoogle Scholar
  15. 15.
    Le Roux, S., Pauly, A.: Infinite sequential games with real-valued payoffs. In: CSL-LICS 2014, pp. 62:1–62:10. ACM (2014). http://doi.acm.org/10.1145/2603088.2603120
  16. 16.
    Le Roux, S., Pauly, A.: Weihrauch degrees of finding equilibria in sequential games. arXiv:1407.5587 (2014)
  17. 17.
    Martin, D.A.: Borel determinacy. Ann. Math. 102(2), 363–371 (1975). http://www.jstor.org/stable/1971035 MATHCrossRefGoogle Scholar
  18. 18.
    Montalbán, A., Shore, R.A.: The limits of determinacy in second-order arithmetic. Proc. London Math. Soc. 104(2), 223–252 (2012). http://plms.oxfordjournals.org/content/104/2/223.abstract MATHCrossRefGoogle Scholar
  19. 19.
    Nemoto, T.: Determinacy of wadge classes and subsystems of second order arithmetic. Math. Logic Q. 55(2), 154–176 (2009). http://dx.doi.org/10.1002/malq.200710081 MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Nemoto, T., MedSalem, M.O., Tanaka, K.: Infinite games in the Cantor space and subsystems of second order arithmetic. Math. Logic Q. 53(3), 226–236 (2007)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Pauly, A.: How incomputable is finding Nash equilibria? J. Univ. Comput. Sci. 16(18), 2686–2710 (2010)MATHMathSciNetGoogle Scholar
  22. 22.
    Pauly, A.: Computable Metamathematics and its Application to Game Theory. Ph.D. thesis, University of Cambridge (2012)Google Scholar
  23. 23.
    Pauly, A., de Brecht, M.: Towards synthetic descriptive set theory: an instantiation with represented spaces. arXiv 1307.1850Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Département d’informatiqueUniversité libre de BruxellesBruxellesBelgique
  2. 2.Clare CollegeUniversity of CambridgeCambridgeUK

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