The Shapley value of cooperative games under fuzzy settings: a survey

We survey the recent developments in the studies of cooperative games under fuzzy environment. The basic problems of a cooperative game in both crisp and fuzzy contexts are to find how the coalitions form vis-á-vis how the coalitions distribute the worth. One of the fuzzification processes assumes that the coalitions thus formed are fuzzy in nature having only partial participations of the players. A second group of researchers fuzzify the worths of the coalitions while a few others assume that both the coalitions and the worths are fuzzy quantities. Among the various solution concepts of a cooperative game, the Shapley value is the most popular one-point solution concept which is characterized by a set of rational axioms. We confine our study to the developments of the Shapley value in fuzzy setting and try to highlight the respective characterizations. keywords : Cooperative game, Fuzzy coalitions, Vague expectation, Shapley mapping. ∗In memory of Milan Mareš whose sudden demise is a great loss to the Game Theory and Fuzzy Quantities Communities and in honor of L. Shapley, the 2012 Nobel Laureate in Economics. †Corresponding Author. [email protected] 1