Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We prove a new property of dual hypergraphs and derive from it Nash-solvability of the corresponding (tight) game forms. This result is known since 1975, but its new proof is much simpler.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Abdou, J.: Tight and effectively rectangular game forms: A Nash solvable class. Games Econ. Behav. 23(1), 1–11 (1998)

    Article  MathSciNet  Google Scholar 

  2. Abdou, J.: Rectangularity and tightness: A normal form characterization of perfect information extensive game forms. Math. Oper. Res. 23(3), 553–567 (1998)

    Article  MathSciNet  Google Scholar 

  3. Abdou, J., Keiding, H.: Effectivity Functions in Social Choice. Springer, New York (1991)

    Book  Google Scholar 

  4. Berge, C.: Hypergraphs, Combinatorics of finite sets. North-Holland Mathematical Library, vol. 45, pp. 1–255 (1989)

  5. Boros, E., Cepek, O., Gurvich, V., Makino, K.: Recognizing distributed approval voting forms and correspondences. arXiv:2010.15730 (2020)

  6. Boros, E., Gurvich, V., Milanic, M., Oudalov, V., Vicic, J.: A three-person deterministic graphical game without Nash equilibria. Discret. Appl. Math. 243, 21–38 (2018)

    Article  MathSciNet  Google Scholar 

  7. Crama, Y., Hammer, P.L.: Boolean Functions: Theory, Algorithms, and Applications. Cambridge University Press, Cambridge (2011)

    Book  Google Scholar 

  8. Danilov, V.I.: Private communications (1988)

  9. Edmonds, J., Fulkerson, D.R.: Bottleneck extrema. J. Comb. Theory 8, 299–306 (1970)

    Article  MathSciNet  Google Scholar 

  10. Ekel, E., Fudenberg, D., Levine, D.: Payoff information and self-confirming equilibrium. J. Econ. Theory 89(2), 165–185 (1999)

    Article  MathSciNet  Google Scholar 

  11. Fredman, M., Khachiyan, L.: On the complexity of dualization of monotone disjunctive normal forms. J. Algorithms 21, 618–628 (1996)

    Article  MathSciNet  Google Scholar 

  12. Gurvich, V.: On theory of multi-step games. USSR Comput. Math. Math. Phys. 13(6), 143–161 (1973)

    Article  Google Scholar 

  13. Gurvich, V.: Solution of positional games in pure strategies. USSR Comput. Math. Math. Phys. 15(2), 74–87 (1975)

    Article  Google Scholar 

  14. Gurvich, V.: Equilibrium in pure strategies. Soviet Math. Dokl. 38(3), 597–602 (1989)

    MathSciNet  Google Scholar 

  15. Gurvich, V., Khachiyan, L.: On Generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions. Discret. Appl. Math. 96–97, 363–373 (1999)

    Article  MathSciNet  Google Scholar 

  16. Gurvich, V., Naumova, M.: Polynomial algorithms computing two lexicographically safe Nash equilibria in finite two-person games with tight game forms given by oracles. In: ArXiv e-prints arXiv:2108.05469 (2021)

  17. Kolpin, V.W.: A note on tight extensive game forms. Int. J. Game Theory 17, 187–191 (1988)

    Article  MathSciNet  Google Scholar 

  18. Mojzisch, A., Schulz-Hardt, S.: Knowing others’ preferences degrades the quality of group decisions. J. Pers. Soc. Psychol. 98(5), 794–808 (2010)

    Article  PubMed  Google Scholar 

  19. Moulin, H.: The Strategy of Social Choice. North-Holland Publishing Company (1983)

  20. Nash, J.: Equilibrium points in n-person games. Proc. Nat. Acad. Sci. 36(1), 48–49 (1950)

    Article  ADS  MathSciNet  PubMed  PubMed Central  CAS  Google Scholar 

  21. Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)

    Article  MathSciNet  Google Scholar 

  22. Peleg, B.: Game Theoretic Analysis of Voting in Committees. Cambridge University Press, Cambridge (1984)

    Book  Google Scholar 

  23. Roth, A.E.: Two-sided matching with incomplete information about others’ preferences. Games Econ. Behav. 1(2), 191–209 (1989)

    Article  MathSciNet  Google Scholar 

  24. Schipper, B.C.: Discovery and equilibrium in games with unawareness. J. Econ. Theory 198(1–44), 105365 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Endre Boros and to an anonymous reviewer for many helpful remarks and suggestions. The article was prepared within the framework of the HSE University Basic Research Program.

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russian Federation

    Vladimir Gurvich

  2. RUTCOR, Rutgers University, Piscataway, NJ, USA

    Vladimir Gurvich

  3. Rutgers Business School, Rutgers University, Piscataway, NJ, USA

    Mariya Naumova

Authors
  1. Vladimir Gurvich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariya Naumova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mariya Naumova.

Ethics declarations

Conflicts of interest

The authors have no relevant financial or non-financial interests to disclose, and no competing interests to declare that are relevant to the content of this paper.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gurvich, V., Naumova, M. Lexicographically maximal edges of dual hypergraphs and Nash-solvability of tight game forms. Ann Math Artif Intell 92, 49–57 (2024). https://doi.org/10.1007/s10472-022-09820-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-022-09820-3

Keywords

Mathematics subject classification (2010)

Search

Navigation