The Banzhaf power index on convex geometries

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.  1998 Elsevier Science B.V. All rights reserved.