The proportional Shapley value and applications

We study a non linear weighted Shapley value (Shapley, 1953b) for cooperative games with transferable utility, in which the weights are endogenously given by the players' stand-alone worths. We call it the proportional Shapley value since it distributes the Harsanyi dividend (Harsanyi, 1959) of all coalitions in proportion to the stand-alone worths of its members. We show that this value recommends appealing payoff distributions in several applications among which a land production economy introduced in Shapley and Shubik (1967). Although the proportional Shapley value does not satisfy the classical axioms of linearity and consistency (Hart and Mas-Colell, 1989), the main results provide comparable axiomatic characterizations of our value and the Shapley value by means of weak versions of these two axioms. These characterizations rely on another result, which states that there exists a unique extension of a value defined on games that are additive except, possibly, for the grand coalition to the set of all games in the much larger class we consider. Moreover, our value inherits several well-known properties of the weighted Shapley values.