Given a coalition of an n-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, nondecreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.
In this article, we complete the proof that the extended Shapley value has w-consistent property proposed by Hsiao, Yeh and Mo [4]. Then we suggest an axiomatization which is the parallel of Hart and Mas-Colell's [1] axiomatization of the Shapley value by applying the w-consistency property.
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