Advertisement

Models and Algorithms for Least Square Interval-Valued Nucleoli of Cooperative Games with Interval-Valued Payoffs

Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 758)

Abstract

The aim of this paper is to develop a new method for computing least square interval-valued nucleoli of cooperative games with interval-valued payoffs, which usually are called interval-valued cooperative games for short. In this methodology, based on the square excess which can be intuitionally interpreted as a measure of the dissatisfaction of the coalitions, we construct a quadratic programming model for least square interval-valued prenucleolus of any interval-valued cooperative game and obtain its analytical solution, which is used to determine players’ interval-valued imputations via the designed algorithms that ensure the nucleoli always satisfy the individual rationality of players. Hereby the least square interval-valued nucleoli of interval-valued cooperative games are determined in the sense of minimizing the difference of the square excesses of the coalitions. Moreover, we discuss some useful and important properties of the least square interval-valued nucleolus such as its existence and uniqueness, efficiency, individual rationality, additivity, symmetry, and anonymity.

Keywords

Game algorithm Cooperative game Interval computing Quadratic programming Optimization model 

References

  1. 1.
    Branzei, R., Branzei, O., Alparslan Gök, S.Z., Tijs, S.: Cooperative interval games: a survey. CEJOR 18, 397–411 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Li, D.-F.: Models and Methods for Interval-Valued Cooperative Games in Economic Management. Springer, Cham (2016). doi: 10.1007/978-3-319-28998-4 CrossRefMATHGoogle Scholar
  3. 3.
    Branzei, R., Dimitrov, D., Tijs, S.: Shapley-like values for interval bankruptcy games. Econ. Bull. 3, 1–8 (2003)Google Scholar
  4. 4.
    Alparslan Gök, S.Z., Branzei, R., Tijs, S.: The interval Shapley value: an axiomatization. CEJOR 18, 131–140 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Alparslan Gök, S.Z., Palanci, O., Olgun, M.O.: Cooperative interval games: mountain situations with interval data. J. Comput. Appl. Math. 259, 622–632 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kimms, A., Drechsel, J.: Cost sharing under uncertainty: an algorithmic approach to cooperative interval-valued games. Bus. Res. 2, 206–213 (2009)CrossRefGoogle Scholar
  7. 7.
    Mallozzi, L., Scalzo, V., Tijs, S.: Fuzzy interval cooperative games. Fuzzy Sets Syst. 165, 98–105 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Moore, R.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Philadelphia (1979)CrossRefMATHGoogle Scholar
  9. 9.
    Hong, F.-X., Li, D.-F.: Nonlinear programming method for interval-valued n-person cooperative games. Oper. Res. Int. J. (2016). doi: 10.1007/s12351-016-0233-1
  10. 10.
    Branzei, R., Alparslan Gök, S.Z., Branzei, O.: Cooperation games under interval uncertainty: on the convexity of the interval undominated cores. CEJOR 19, 523–532 (2011)CrossRefMATHGoogle Scholar
  11. 11.
    Alparslan Gök, S.Z., Branzei, O., Branzei, R., Tijs, S.: Set-valued solution concepts using interval-type payoffs for interval games. J. Math. Econ. 47, 621–626 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Alparslan Gök, S.Z., Miquel, S., Tijs, S.: Cooperation under interval uncertainty. Math. Methods Oper. Res. 69, 99–109 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Li, D.-F.: Linear programming approach to solve interval-valued matrix games. Omega Int. J. Manag. Sci. 39(6), 655–666 (2011)CrossRefGoogle Scholar
  14. 14.
    Li, D.-F.: Fuzzy Multiobjective Many-Person Decision Makings and Games. National Defense Industry Press, Beijing (2003). (in Chinese)Google Scholar
  15. 15.
    Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17(6), 1163–1170 (1969)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ruiz, L.M., Valenciano, F., Zarzuelo, J.M.: The least square prenucleolus and the least square nucleolus: two values for TU games based on the excess vector. Int. J. Game Theor. 25(1), 113–134 (1996)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Li, D.-F., Hong, F.-X.: Solving constrained matrix games with payoffs of triangular fuzzy numbers. Comput. Math Appl. 64, 432–446 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Verma, T., Kumar, A., Appadoo, S.S.: Modified difference-index based ranking bilinear programming approach to solving bimatrix games with payoffs of trapezoidal intuitionistic fuzzy numbers. J. Intell. Fuzzy Syst. 29, 1607–1618 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li, D.-F.: Decision and Game Theory in Management with Intuitionistic Fuzzy Sets. SFSC, vol. 308. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-40712-3 MATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.College of EngineeringMichigan State UniversityEast LansingUSA

Personalised recommendations