Power and the Shapley value

This chapter reviews some results from the literature in which power indices related to the Shapley value are developed. First, a class of power indices for so-called effectivity functions is axiomatically characterized, based on [1]. An effectivity function describes for each coalition of players each set of alternatives such that the coalition can make sure that the final alternative is in that set. As a special case, the Owen-Shapley spatial power index as proposed in [2] is obtained. Second, following [3], a class of power indices for situations in which subsets of players control other players is described. Examples of such situations include financial structures in which firms and other shareholders exercise control through shares in each other. Formally, such a situation is described by a collection of simple games: for each player there is a simple game of which the winning coalitions control that player. Third, following [4], a class of power indices is considered where relations between the players are determined via a directed graph.