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Satisfactory Interval-Valued Cores of Interval-Valued Cooperative Games

  • Deng-Feng Li
Chapter

Abstract

The aim of this chapter is to develop an effective nonlinear programming method for computing interval-valued cores of interval-valued cooperative games. In this chapter, we define satisfactory degrees (or ranking indexes) of comparing intervals with the features of inclusion and/or overlap relations and discuss their important properties. Hereby we construct satisfactory crisp equivalent forms of interval-valued inequalities. Based on the concept of interval-valued cores, we derive the auxiliary nonlinear programming models for computing interval-valued cores of interval-valued cooperative games and propose corresponding bisection algorithm, which can always provide global optimal solutions. The developed models and method can provide cooperative chances under the situation of inclusion and/or overlap relations between interval-type coalitions’ values in which the Moore’s interval ranking method (or order relation between intervals) may not assure that an interval-valued core exists. The proposed method is a generalization of that based on the Moore’s interval ranking relation. The feasibility and applicability of the models and method proposed in this chapter are illustrated with a numerical example.

Keywords

Interval-valued cooperative game Core Interval-valued core Interval ranking Mathematical programming Bisection method 

References

  1. 1.
    Li D-F. Fuzzy multiobjective many-person decision makings and games. Beijing: National Defense Industry Press; 2003 (in Chinese).Google Scholar
  2. 2.
    Owen G. Game theory. 2nd ed. New York: Academic Press; 1982.Google Scholar
  3. 3.
    Branzei R, Branzei O, Alparslan Gök SZ, Tijs S. Cooperative interval games: a survey. Cent Eur J Oper Res. 2010;18:397–411.CrossRefGoogle Scholar
  4. 4.
    Gillies DB. Solutions to general non-zero-sum games. In: Tucker AW, Luce RD, editors. Contributions to theory of games IV, Annals of mathematical studies, vol. 40. Princeton: Princeton University Press; 1959. p. 47–85.Google Scholar
  5. 5.
    Driessen T. Cooperation games: solutions and application. Netherlands: Kluwer Academic Publisher; 1988.CrossRefGoogle Scholar
  6. 6.
    Shapley L. Cores of convex games. Int J Game Theory. 1971;1:11–26.CrossRefGoogle Scholar
  7. 7.
    Branzei R, Alparslan-Gök SZ, Branzei O. Cooperation games under interval uncertainty: on the convexity of the interval undominated cores. Cent Eur J Oper Res. 2011;19:523–32.CrossRefGoogle Scholar
  8. 8.
    Alparslan-Gök SZ, Branzei R, Tijs SH. Cores and stable sets for interval-valued games, vol. 1. Center for Economic Research, Tilburg University; 2008. p. 1–14.Google Scholar
  9. 9.
    Alparslan-Gök SZ, Branzei O, Branzei R, Tijs S. Set-valued solution concepts using interval-type payoffs for interval games. J Math Econ. 2011;47:621–6.CrossRefGoogle Scholar
  10. 10.
    Shapley LS. On balanced sets and cores. Naval Res Logist Quart. 1967;14:453–60.CrossRefGoogle Scholar
  11. 11.
    Han W-B, Sun H, Xu G-J. A new approach of cooperative interval games: the interval core and Shapley value revisited. Oper Res Lett. 2012;40:462–8.CrossRefGoogle Scholar
  12. 12.
    Moore R. Methods and applications of interval analysis. Philadelphia: SIAM Studies in Applied Mathematics; 1979.CrossRefGoogle Scholar
  13. 13.
    Li D-F. Linear programming approach to solve interval-valued matrix games. Omega. 2011;39(6):655–66.CrossRefGoogle Scholar
  14. 14.
    Sengupta A, Pal TK. Theory and methodology on comparing interval numbers. Eur J Oper Res. 2000;127:28–43.CrossRefGoogle Scholar
  15. 15.
    Ishihuchi H, Tanaka M. Multiobjective programming in optimization of the interval objective function. Eur J Oper Res. 1990;48:219–25.CrossRefGoogle Scholar
  16. 16.
    Li D-F, Nan J-X, Zhang M-J. Interval programming models for matrix games with interval payoffs. Optim Methods Softw. 2012;27:1–16.CrossRefGoogle Scholar
  17. 17.
    Zadeh L. Fuzzy sets. Inform Control. 1965;8:338–56.CrossRefGoogle Scholar
  18. 18.
    Dubois D, Prade H. Fuzzy sets and systems: theory and applications. New York: Academic Press; 1980.Google Scholar
  19. 19.
    Collins WD, Hu C-Y. Interval matrix games. In: Hu C-Y, Kearfott RB, Korvinet AD, et al., editors. Knowledge processing with interval and soft computing. London: Springer; 2008. p. 168–72.Google Scholar
  20. 20.
    Collins WD, Hu C-Y. Studying interval valued matrix games with fuzzy logic. Soft Comput. 2008;12(2):147–55.CrossRefGoogle Scholar
  21. 21.
    Nayak PK, Pal M. Linear programming technique to solve two person matrix games with interval pay-offs. Asia Pac J Oper Res. 2009;26(2):285–305.CrossRefGoogle Scholar
  22. 22.
    Li D-F. Notes on “linear programming technique to solve two person matrix games with interval pay-offs”. Asia Pac J Oper Res. 2011;28(6):705–37.CrossRefGoogle Scholar
  23. 23.
    Zimmermann H-J. Fuzzy set theory and its application. 2nd ed. Dordrecht: Kluwer Academic Publishers; 1991.CrossRefGoogle Scholar
  24. 24.
    Li D-F. Lexicographic method for matrix games with payoffs of triangular fuzzy numbers. Int J Uncertain Fuzziness Knowl Based Syst. 2008;16(3):371–89.CrossRefGoogle Scholar
  25. 25.
    Gillies DB. Some theorems on n-person games. PhD thesis. Princeton: Princeton University Press; 1953.Google Scholar
  26. 26.
    Branzei R, Dimitrov D, Tijs S. Models in cooperative game theory. Game theory and mathematical methods. Berlin: Springer; 2008.Google Scholar
  27. 27.
    Alparslan-Gök SZ, Branzei R, Tijs S. Big boss interval games. Institute of Applied Mathematics, METU and Tilburg University, Center for Economic Research, The Netherlands, CentER DP 47 (preprint no. 103); 2008.Google Scholar
  28. 28.
    Sikorski K. Bisection is optimal. Numer Math. 1982;40:111–7.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Deng-Feng Li
    • 1
  1. 1.School of Economics and ManagementFuzhou UniversityFuzhouChina

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