Skip to main content
Log in

Existence of the Equilibrium in Choice

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we prove the existence of the equilibrium in choice for games in choice form. Thus, we add to the research recently appeared in the scientific literature. In fact, our results link the most recent research to the older approaches of the games in normal-form and the qualitative games.

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

Access this article

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

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Stefanescu, A., Ferrara, M., Stefanescu, M.V.: Equilibria of the games in choice form. J. Optim. Theory Appl. 155, 160–172 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Stefanescu, A., Ferrara, M.: Implementation of voting operators. J. Math. Econ. 42, 315–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Patriche, M.: Existence of equilibrium for generalized games in choice form and applications. Optimization 65, 2135–2151 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alcantud, J.C.R., Ferrer, C.A.: Nash equilibria for non-binary choice rules. Int. J. Game Theory 35, 455–464 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yuan, X.Z.: The study of minimax inequalities and applications to economies and variational inequalities. Memoirs Am. Math. Soc. 132, 625 (1988)

    MathSciNet  Google Scholar 

  8. Ferrara, M., Stefanescu, A.: Equilibrium in Choice of Generalized Games. In: Cartier, P., et al. (eds.) Mathematics in the 21th Century. Springer, Berlin (2015)

    Google Scholar 

  9. Wu, X., Shen, S.: A further generalisation of Yannelis-Prabhakar’s continuous selection theorem and its applications. J. Math. Anal. Appl. 197, 61–74 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, X., He, Y.: Best approximation theorem for set-valued mappings without convex values and continuity. Appl Math. Mech. Engl. Ed. 19, 831–836 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Patriche, M.: Equilibrium in abstract economies with weakly convex graph set-valued maps. Math. Rep. 10, 359–373 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Wu, X.: A new fixed point theorem and its applications. Proc. Am. Math. Soc. 125, 1779–1783 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tian, C.G.: Generalizations of the KKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity. J. Math. Anal. Appl. 170, 457–471 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, L.J.: Applications of a fixed point theorem in \(G\) -convex spaces. Nonlinear Anal. 46, 601–608 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ansari, Q.H., Yao, J.-C.: A fixed point theorem and its applications to a system of variational inequalities. Bull. Aust. Math. Soc. 59, 433–442 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is indebted to the editor and the anonymous referees for helpful suggestions and valuable comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Monica Patriche.

Additional information

Communicated by Jean-Pierre Crouzeix.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Patriche, M. Existence of the Equilibrium in Choice. J Optim Theory Appl 175, 158–171 (2017). https://doi.org/10.1007/s10957-016-1047-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-1047-2

Keywords

Mathematics Subject Classification

Navigation