A new Shapley value for games with fuzzy coalitions

This paper deals with cooperative games over fuzzy coalitions. In these situations there is a continuous set of fuzzy coalitions instead of a finite set of them (as in the classical case), the unit square in an n-dimensional space. There exist in the literature two different extensions of the known Shapley value for crisp games to games with fuzzy coalitions: the crisp Shapley value and the diagonal value. The first value only uses a finite information in the set of fuzzy coalitions, the vertices of the square. While the second one uses a neighbourhood of the diagonal of the square. We propose a new extension of the Shapley value improving the crisp Shapley value for games with fuzzy coalitions. This new version uses the faces of the square, namely an infinity quantity of information. We analyze several properties of the new value, we endow it with an axiomatization and we study the behavior when it is applied to known fuzziness of crisp games. (C) 2019 Elsevier B.V. All rights reserved.