The Extension of Combinatorial Solutions for Cooperative Games

  • Jiang-Xia Nan
  • Li-Xiao Wei
  • Mao-Jun ZhangEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1082)


In this paper, a new model to solve the social selfish coefficient \( \alpha \in \left[ {0,1} \right] \) for the \( \alpha \)-egalitarian Shapley values and a new convex combinations of single-value in terms of a coalition forming weight coefficient \( \beta \in \left[ {0,1} \right] \), which is called the SCE value for cooperative games are presented. The efficiency, linearity, symmetry and \( \alpha \)-dummy player property of the SCE value are proved. By proposing a procedural interpretation, we define the \( \beta \) as the coalition forming weight (or possibility) coefficient and find a new way of assigning the grand coalition profit among all players is coincided with the SCE value which verify the SCE value’s validity, applicability and superiority.


Cooperative games Egalitarian Shapley value \( \alpha \)-CIS value SCE value 



The authors would like to thank the associate editor and also appreciate the constructive suggestions from the anonymous referees. This research was supported by the key Program of National Natural Science Foundation of China (No. 71231003), the Natural Science Foundation of China (Nos. 71461005 and 71561008). The Innovation Project of Guet Graduate Education (No. 2019YCXS082).


  1. 1.
    Shapley, L.S.: A value for n-person games. Ann. Math. Stud. 28(7), 307–318 (1953)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Driessen, T.S.H., Funaki, Y.: Coincidence of and collinearity between game theoretic solutions. Oper. Res. Spektrum 13(1), 15–30 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Joosten, R.: Dynamics, equilibria and values. Ph.D. dissertation, Maastricht University (1996)Google Scholar
  4. 4.
    Casajus, A., Huettner, F.: Null players, solidarity, and the egalitarian Shapley values. J. Math. Econ. 49(1), 58–61 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brink, R., Funaki, Y., Ju, Y.: Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values. Soc. Choice Welfare 40(3), 693–714 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Wang, W.N., Sun, H., Xu, G.J., Hou, D.S.: Procedural interpretation and associated consistency for the egalitarian Shapley values. Oper. Res. Lett. 45(2), 164–169 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chun, Y., Park, B.: Population solidarity, population fair-ranking, and the egalitarian value. Int. J. Game Theory 41(2), 255–270 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dragan, I., Driessen, T., Funaki, Y.: Collinearity between the Shapley value and the egalitarian division rules for cooperative games. Oper. Res. Spektrum 18(2), 97–105 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brink, R., Chun, Y., Funaki, Y., Park, B.: Consistency, population solidarity, and egalitarian solutions for TU-Games. Theor. Decis. 81(3), 427–447 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Xu, G.J., Dai, H., Shi, H.B.: Axiomatizations and a Noncooperative Interpretation of the α-CIS Value. Asia-Pac. J. Oper. Res. 32(5), 1550031 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hou, D.S., Sun, P.F., Xu, G.J., Driessen, T.: Compromise for the complaint: an optimization approach to the ENSC value and the CIS value. J. Oper. Res. Soc. 63(3), 1–9 (2017)Google Scholar
  12. 12.
    Hu, X.F.: A new axiomatization of a class of equal surplus division values for TU games. RAIRO Oper. Res. 52(3), 935–942 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Brink, R., Funaki, Y.: Axiomatizations of a class of equal surplus sharing solutions for TU-games. Theor. Decis. 67(3), 303–340 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ju, Y., Borm, P., Ruys, P.: The consensus value: a new solution concept for cooperative games. Soc. Choice Welfare 28(4), 685–703 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hou, D.S., Sun, H., Xu, G.J.: Compromise for the complaint: process and optimization approach to the alpha-CIS value. Working paperGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and ComputationGuilin University of Electronic TechnologyGuilinChina

Personalised recommendations