A new Deegan-Packel inspired power index in games with restricted cooperation

Abstract

We propose a new power index, which we call the essential coalitions index. The new index is fit to analyze influence in the formation of stable coalitions to run a government or a company board. Within the field of power indices, it extends the Deegan-Packel power index to situations with restricted cooperation; more specifically, to the class of games introduced by Amer and Carreras in [2]. In general, these are not simple games. We will use the essential coalitions as an analogue to the minimal winning coalitions of a simple game, since they generalize some relevant properties. Similarly to the index that inspires it, we will first define the new index in terms of three reasonable assumptions, resembling those used in [5] for the Deegan-Packel index. Then, we formally characterize the index, using suitable modifications of the properties introduced in [2] to characterize the Shapley value in restricted games. Finally, through numeric examples, we compare the essential coalitions index to the similarly inspired, albeit more constrained, probabilístic Deegan-Packel index. We will see that, in the latter’s domain, the two indices only differ in their normalization.