China-Dutch GTA 2016, China GTA 2016: Game Theory and Applications pp 303-317

# Interval-Valued Least Square Prenucleolus of Interval-Valued Cooperative Games with Fuzzy Coalitions

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

## Abstract

In this paper, an important solution concept of interval-valued (IV) cooperative games with fuzzy coalitions, called the IV least square prenucleolus, is proposed. Firstly, we determine the fuzzy coalitions’ values by using Choquet integral and hereby obtain the IV cooperative games with fuzzy coalitions in Choquet integral forms. Then, we develop a simplified method to compute the IV least square prenucleolus of a special subclass of IV cooperative games with fuzzy coalitions in Choquet integral forms. In this method, we give some weaker coalition size monotonicity-like conditions, which can always ensure that the least square prenucleolus of our defined cooperative games with fuzzy coalitions in Choquet integral form are monotonic and non-decreasing functions of fuzzy coalitions’ values. Hereby, the lower and upper bounds of the proposed IV least square prenucleolus can be directly obtained via utilizing the lower and upper bounds of the IV coalitions values, respectively. In addition, we investigate some important properties of the IV least square prenucleolus. The feasibility and applicability of the method proposed in this paper are illustrated with numerical examples.

### Keywords

Game theory Interval-valued cooperative game Fuzzy game Least square prenucleolus Choquet integral

### References

1. Aubin, J.P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam (1980)Google Scholar
2. Aubin, J.P.: Cooperative fuzzy games. Math. Oper. Res. 61, 1–13 (1981)
3. Alparslan Gök, S.Z.: On the interval Shapley value. Optimization 63(5), 747–755 (2014)
4. 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)
5. Butnariu, D.: Stability and Shapley value for an $$n$$-persons fuzzy game. Fuzzy Sets Syst. 4, 63–72 (1980)
6. Borkotokey, S.: Cooperative games with fuzzy coalitions and fuzzy characteristic functions. Fuzzy Sets Syst. 159, 138–151 (2008)
7. Branzei, R., Branzei, O., Alparslan G$$\rm \ddot{o}$$k, S.Z., Tijs, S.: Cooperative interval games: a survey. Central Europ. J. Oper. Res.18, 397–411 (2010)Google Scholar
8. Branzei, R., Dimitrov, D., Tijs, S.: Shapley-like values for interval bankruptcy games. Econ. Bull. 3, 1–8 (2003)Google Scholar
9. Driessen, T.S.H., Radzik, T.: A weighted pseudo-potential approach to values for TU-games. Int. Trans. Oper. Res. 9, 303–320 (2002)
10. Hong, F.X., Li, D.F.: Nonlinear programming approach for IV $$n$$-person cooperative games. Oper. Res. Int. J. (2016). doi:
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. 40, 462–468 (2012)
12. Li, D.F.: Models and Methods of Interval-Valued Cooperative Games in Economic Management. Springer, Cham (2016)
13. Liu, J.Q., Liu, X.D.: Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games. Fuzzy Sets Syst. 188, 88–101 (2012)
14. Li, D.F., Ye, Y.F.: Interval-valued least square prenucleolus of interval-valued cooperative games and a simplified method. Oper. Res. Int. J. (2016). doi:
15. Lin, J., Zhang, Q.: The least square B-nucleolus for fuzzy cooperative games. J. Intell. Fuzzy Syst. 30, 279–289 (2016)
16. Moore, R.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Philadelphia (1979)
17. Meng, F.Y., Chen, X.H., Tan, C.Q.: Cooperative fuzzy games with interval characteristic functions. Oper. Res. Int. J. 16, 1–24 (2016)
18. Palanci, O., Alparslan Gök, S.Z., Weber, G.W.: An axiomatization of the interval Shapley value and on some interval solution concepts. Contrib. Game Theory Manage. 8, 243–251 (2015)
19. 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 Theory 25, 113–134 (1996)
20. Sagara, N.: Cores and Weber sets for fuzzy extensions of cooperative games. Fuzzy Sets Syst. 272, 102–114 (2015)
21. Sakawa, M., Nishizaki, I.: A lexicographical solution concept in a $$n$$-person cooperative fuzzy game. Fuzzy Sets Syst. 61(3), 265–275 (1994)
22. Tijs, S., Branzei, R., Ishihara, S., Muto, S.: On cores and stable sets for fuzzy games. Fuzzy Sets Syst. 146, 285–296 (2004)
23. Tan, C.Q., Jiang, Z.Z., Chen, X.H., Ip, W.H.: A Banzhaf Function for a Fuzzy Game. IEEE Trans. Fuzzy Syst. 22(6), 1489–1502 (2014)
24. Tsurumi, M., Tanino, T., Inuiguchi, M.: A Shapley function on a class of cooperative fuzzy games. Eur. J. Oper. Res. 129(3), 596–618 (2001)
25. Yu, X.H., Zhang, Q.: The fuzzy core in games with fuzzy coalitions. J. Comput. Appl. Math. 230, 173–186 (2009)
26. Zadeh, L.A.: Fuzzy Sets. Inform. Control. 8(3), 338–353 (1965)Google Scholar