The Banzhaf power index on convex geometries

Abstract

In this paper, we introduce the Banzhaf power indices for simple games on convex geometries. We define the concept of swing for these structures, obtaining convex swings. The number of convex swings and the number of coalitions such that a player is an extreme point are the basic tools to define the convex Banzhaf indices, one normalized and other probabilistic. We obtain a family of axioms that give rise to the Banzhaf indices. In the last section, we present a method to calculate the convex Banzhaf indices with the computer program Mathematica, and we apply this to compute power indices in the Spanish and Catalan parliaments and in the Council of Ministers of the European Union. 

A coalitional value for games on convex geometries with a coalition structure

“…Similar to the properties of the Banzhaf value for the traditional games, Bilbao et al [38] gave the following properties, where…”

“…To both consider these two aspects, this paper researches a special kind of games on feasible coalitions with a coalition structure. Following the works of Alonso-Meijde and Fiestras-Janeiro [18], Bilbao and Edelman [37] and Bilbao et al [38], this paper defines a coalitional payoff index called the generalized symmetric coalitional Banzhaf value. To ensure the rationality of this coalitional value, an axiomatic system is given, by which its existence and uniqueness can be shown.…”

“…When there is only one coalition in , then hierarchical strength in coalitions degenerates to hierarchical strength for games on convex geometries presented by Bilbao and Edelman [37], and hierarchical strength on unions degenerates to relative extremal power for simple games on convex geometries proposed by Bilbao et al [38]. When all subsets of M and that of eachB k ∈ are both feasible, then quasi coalitional total power property degenerates to coalitional total power property for games with a coalition structure introduced by Alonso-Meijde and Fiestras-Janeiro [18].…”

“…Furthermore, the authors studied the generalized Owen value for this kind of games. Based on the symmetric coalitional Banzhaf value [18] for games with a coalition structure, the Shapley value [37] and the Banzhaf value [38] for games on convex geometries, we here research another importance payoff index of games on convex geometries with a coalition structure called the generalized symmetric coalitional Banzhaf value. The existence and uniqueness of this coalitional value are proved by the established axiomatic system.…”

Power Indices When Players Can Commit to Reject Coalitions

“…The structure of communication (Myerson, 1977(Myerson, , 1980Faigle and Kern, 1992) or physical or ideological position (Bilbao, Jiménez, and López, 1998) may put restrictions on the set of feasible coalitions. Would all coalitions form in the absence of such restrictions?…”