Skip to main content
SpringerLink
Log in

Robust Multiple Objective Game Theory

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

Abstract

In this paper, we propose a distribution-free model instead of considering a particular distribution for multiple objective games with incomplete information. We assume that each player does not know the exact value of the uncertain payoff parameters, but only knows that they belong to an uncertainty set. In our model, the players use a robust optimization approach for each of their objective to contend with payoff uncertainty. To formulate such a game, named “robust multiple objective games” here, we introduce three kinds of robust equilibrium under different preference structures. Then, by using a scalarization method and an existing result on the solutions for the generalized quasi-vector equilibrium problems, we obtain the existence of these robust equilibria. Finally, we give an example to illustrate our model and the existence theorems. Our results are new and fill the gap in the game theory literature.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Harsanyi, J.: Games with incomplete information played by ‘Bayesian’ players, Part I, II, III. Manag. Sci. 14 (3, 5, 7) (1967 and 1968)

  4. Luo, G.M., An, X., Xia, J.Y.: Robust optimization with applications to game theory. Appl. Anal. 88, 1183–1195 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 175, 335–353 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Nishimura, R., Hayashi, S., Fukushima, M.: Robust Nash equilibria in N-person noncooperative games: uniqueness and reformulation. Pac. J. Optim. 5, 237–259 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Aghassi, M., Bertsimas, D.: Robust game theory. Math. Program., Ser. B 107, 231–273 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769–805 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ben-Tal, A., Nemirovski, A.: Selected topics in robust convex optimization. Math. Program. 112, 125–158 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Blackwell, D.: An analog of the minimax theorem for vector payoffs. Pac. J. Math. 6, 1–8 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shapley, L.S.: Equilibrium points in games with vector payoffs. Nav. Res. Logist. Q. 6, 57–61 (1959)

    Article  MathSciNet  Google Scholar 

  13. Ghose, D., Prasad, U.R.: Solution concepts in two-person multicriteria games. J. Optim. Theory Appl. 63, 167–189 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yuan, X.Z., Tarafdar, E.: Non-compact Pareto equilibria for multiobjective games. J. Math. Anal. Appl. 204, 156–163 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ding, X.P.: Pareto equilibria of muliticriteria games without compactness, continuity, and concavity. Appl. Math. Methods 17, 847–854 (1996)

    MATH  Google Scholar 

  16. Ding, X.P.: Existence of Pareto equilibria for constrained multiobjective games in H-space. Comput. Math. Appl. 39, 125–134 (2000)

    Article  MATH  Google Scholar 

  17. Borm, P., van Megen, F., Tijs, S.: A perfectness concept for multicriteria games. Math. Methods Oper. Res. 49, 401–412 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Aggarwal, S., Bhatia, D.: Vector-valued constrained games and multiobjective symmetric duality. Asia-Pac. J. Oper. Res. 8, 106–118 (1991)

    MathSciNet  MATH  Google Scholar 

  19. Zhao, J.: The equilibrium of a multiple objective game. Int. J. Game Theory 20, 171–182 (1991)

    Article  MATH  Google Scholar 

  20. Zeleny, M.: Games with multiple payoffs. Int. J. Game Theory 4, 179–191 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, S.Y.: Existence of a Pareto equilibrium. J. Optim. Theory Appl. 79, 373–384 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, H., Yu, J.: Essential components of the set of weakly Pareto-Nash equilibrium points. Appl. Math. Lett. 15, 553–560 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 32, 451–466 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Borm, P.E.M., Tijs, S.H., van den Aarssen, J.C.M.: Pareto equilibrium in multiobjective games. Methods Oper. Res. 60, 303–312 (1990)

    Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant number: 70871126; 90924009; 71101160) and Program for New Century Excellent Talents in University of Ministry of Education of China (Grant number: NCET-10-0843).

Author information

Authors and Affiliations

  1. School of Economics and Business Administration, Chongqing University, Chongqing, 400030, China

    H. Yu & H. M. Liu

  2. Chongqing Key Laboratory of Logistics, Chongqing University, Chongqing, 400044, China

    H. Yu

Authors
  1. H. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. M. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Yu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, H., Liu, H.M. Robust Multiple Objective Game Theory. J Optim Theory Appl 159, 272–280 (2013). https://doi.org/10.1007/s10957-012-0234-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0234-z

Keywords

Search

Navigation