The Hoede–Bakker Index Modified to the Shapley–Shubik and Holler–Packel Indices

Abstract

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. The modifications presented in the paper are defined for an arbitrary probability distribution over all inclination vectors. In particular, they concern the situation in which the inclination vectors may be not equally probable. Furthermore, by assuming special probability distributions over all inclination vectors, we construct modifications of the Hoede–Bakker index that coincide with the Shapley–Shubik index and with the Holler–Packel index, respectively. We present a practical example in which the concepts in question are applied to Dutch parties, and a theoretical example in which we show how the modifications can be calculated.