The General Nucleolus of n-Person Cooperative Games

  • Qianqian Kong
  • Hao Sun
  • Genjiu Xu
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 758)


In this paper, we define the concept of the general nucleolus whose objective function is limited to the player complaint, to reflect the profit distribution more intuitively on the space of n-person cooperative games. An algorithm for calculating the general nucleolus under the case of linear complaint functions is given so that we can get an accurate allocation to pay for all players. A system of axioms are proposed to characterize the general nucleolus axiomatically and the Kohlberg Criterion is also given to characterize it in terms of balanced collections of coalitions. Finally, we prove the equivalence relationship of the general nucleolus, the least square general nucleolus and the p-kernel to normalize the different assignment criteria.


Player complaint General nucleolus Kohlberg Criterion Least square p-kernel 



The research has been supported by the National Natural Science Foundation of China (Grant Nos. 71571143 and 71671140) and sponsored by the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Grant No. Z2017043).


  1. Davis, M., Maschler, M.: The kernel of a cooperative game. Naval Research Logistics (NRL) 12(3), 223–259 (1965)MathSciNetCrossRefMATHGoogle Scholar
  2. Kohlberg, E.: On the nucleolus of a characteristic function game. SIAM J. Appl. Math. 20(1), 62–66 (1971)MathSciNetCrossRefMATHGoogle Scholar
  3. Kong, Q., Sun, H., Xu, G.: The general prenucleolus of n-person cooperative fuzzy games. Fuzzy Sets Syst. (2017). doi: 10.1016/j.fss.2017.08.005
  4. Maschler, M., Potters, J.A., Tijs, S.H.: The general nucleolus and the reduced game property. Int. J. Game Theory 21(1), 85–106 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. Molina, E., Tejada, J.: The least square nucleolus is a general nucleolus. Int. J. Game Theory 29(1), 139–142 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. Owen, G.: A generalization of thekohlberg criterion. Int. J. Game Theory 6(4), 249–255 (1977)CrossRefMATHGoogle Scholar
  7. 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(1), 113–134 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. Sakawa, M., Nishizaki, I.: A lexicographical solution concept in an n-person cooperative fuzzy game. Fuzzy Sets Syst. 61(3), 265–275 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17(6), 1163–1170 (1969)MathSciNetCrossRefMATHGoogle Scholar
  10. Sun, H., Hao, Z., Xu, G.: Optimal solutions for tu-games with decision approach. Preprint, Northwestern Polytechnical University, Xian, Shaanxi, China (2015)Google Scholar
  11. Sun, P., Hou, D., Sun, H., Driessen, T.: Optimization implementation and characterization of the equal allocation of nonseparable costs value. J. Optim. Theory Appl. 173(1), 336–352 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. Vanam, K.C., Hemachandra, N.: Some excess-based solutions for cooperative games with transferable utility. Int. Game Theory Rev. 15(04), 1340029 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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