Abstract
We derive versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players. A fuzzy coalition is an n-vector τ = (τi) associating with each player i his “rate of participation” τi ∈ [0, 1] in the fuzzy coalition and the real number v(·) is the coalition worth, assumed to be positively homogeneous. If also v is superadditive, or equivalently concave, the fuzzy game with side payments has a nonempty core. Associated with the coalition-worth function for any ordinary game with side payments is a fuzzy extension thereof, viz., the fuzzy game whose coalition-worth function is the least positively-homogeneous superadditive function majorizing the coalition-worth function of the original game. The fuzzy extension always has a nonempty core. Moreover, if the original game has a nonempty core, it coincides with that of its fuzzy extension. Analogous results are established for games without side payments. An axiomatization of “values” of fuzzy games with side payments is also given. The results are applied to show that the set of Walras equilibria coincides with the fuzzy core of an economy.
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