Modifying the Hoede-Bakker index to the Shapley-Shubik and Holler-Packel indices

Abstract

In the paper, we present some modifications of the Hoede-Bakker index defined in a social network in which players may influence each other. Due to influences of the other actors, the final decision of a player may be different from his original inclination. These modifications are defined for an arbitrary probability distribution over all inclination vectors. In particular, they concern a situation in which the inclination vectors may be not equally probable. Furthermore, byassuming special probability distributions over all inclination vectors, we construct modificationsof the Hoede-Bakker index that coincide with the Shapley-Shubik index and with the Holler-Packelindex, respectively.Hoede-Bakker index ; inclination vector ; probability distribution ; Shapley-Shubik index ; Holler-Packel index