Generalized form solutions of cooperative game with fuzzy coalition structure

  • Xiaohui YuEmail author
  • Zhiping Du
  • Qiang Zhang
  • Zhengxing Zou
  • Zhen Zhou
  • Jinhui Pang


A fuzzy coalition structure is a partition of player set under a certain participation level. In this study, a kind of cooperative games with fuzzy coalition structure (i.e., generalized fuzzy game) is proposed, which can be seen as an extension of game in Owen’s coalition structure. The generalized fuzzy game is defined by partition function, which could be used to express several cooperative games with fuzzy coalition structure (i.e., fuzzy coalition structure games). For the fuzzy coalition structure, the generalized cooperative game could denote the coalition interaction under different participation ratios for players. The properties of generalized fuzzy game have been proven, such as inheritance, supperadditivity and convexity. In order to get unified solution for the generalized fuzzy game, fuzzy Owen value is also extended based on the consistent formula of generalized fuzzy game. It is proved that the fuzzy Owen value is a unique value for the generalized fuzzy game based on symmetric within fuzzy coalition, symmetric across fuzzy coalitions, fuzzy null player, linearity and efficiency. Finally, the fuzzy Owen value is represented by crisp one in order to simplify fuzzy solution computation.


Fuzzy coalition structure Partition function Generalized form Owen value 



This study was funded by National Natural Science Foundation of China (Grant Nos. 71801016, 71771025, 71874112, 71772016, 71871002), by Youth fund project for Humanities and social sciences research of Ministry of Education (grant number 17YJC630203), and General Projects of Social Science Program of Beijing Education Commission (SM201910037007).

Compliance with ethical standards

Conflict of interest

All the authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with animals performed by any of the authors. All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed consent

Informed consent was obtained from all individual participants included in the study.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of LogisticsBeijing Wuzi UniversityBeijingChina
  2. 2.School of Management and EconomicsBeijing Institute of TechnologyBeijingChina
  3. 3.School of ManagementCapital Normal UniversityBeijingChina
  4. 4.School of ComputerBeijing Institute of TechnologyBeijingChina

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