Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

Dan Butnariu, Tomáš Kroupa

Research output: Contribution to journalArticlepeer-review


In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a "cumulative value" that is the "sum" of all coalitional allocations for whose computation we provide an explicit formula.

Original languageEnglish
Pages (from-to)288-299
Number of pages12
JournalEuropean Journal of Operational Research
Issue number1
StatePublished - 1 Apr 2008

Bibliographical note

Funding Information:
The authors are grateful to a referee whose comments helped to improve an earlier version of this paper. The work of Tomáš Kroupa was supported by the Grant No. 1M0572 of the Ministry of Education, Youth and Sports of the Czech Republic and by a post-doctoral fellowship within the Department of Mathematics of the University of Haifa, Israel.


  • Coalition
  • Cumulative value
  • Fuzzy coalition
  • Shapley value
  • n-Person cooperative game

ASJC Scopus subject areas

  • Computer Science (all)
  • Modeling and Simulation
  • Management Science and Operations Research
  • Information Systems and Management
  • Industrial and Manufacturing Engineering


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