Mathematical Social Sciences 36 (1998) 157173The Banzhaf power index on convex geometries*J.M. Bilbao , A. Jimenez, J.J. LopezE. Superior Ingenieros, Camino de los Descubrimientos, 41092 Sevilla, SpainReceived 21 February 1997; received in revised form 30 September 1997; accepted 30 March 1998AbstractIn 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 ofconvex swings and the number of coalitions such that a player is an extreme point are the basictools to define the convex Banzhaf indices, one normalized and other probabilistic. We obtain afamily of axioms that give rise to the Banzhaf indices. In the last section, we present a method tocalculate the convex Banzhaf indices with the computer program Mathematica, and we apply thisto compute power indices in the Spanish and Catalan parliaments and in the Council of Ministersof the European Union. 1998 Elsevier Science B.V. All rights reserved.Keywords: Voting; Political power; Banzhaf index; Convex geometry1. IntroductionThe analysis of power is central in political science. In general, it is difficult to definethe idea of power, but for the special case of voting power there are mathematical powerindices that have been used. The first such power index was proposed by Shapley andShubik (1954). Another concept for measuring voting power was introduced by Banzhaf(1965), a lawyer, whose work has appeared mainly in law journals, and whose index hasbeen used in arguments in various legal proceedings. In this paper, we introduce theBanzhaf power index for cooperative games in which only certain coalitions are allowedto form. We will study the structure of such allowable coalitions using the theory ofconvex geometries, a notion developed to combinatorially abstract geometric convexity.The ShapleyShubik index and the Shapley value on these structures are studied byBilbao and Edelman (1998), and Edelman (1998). We will define the Banzhaf index andgeneralize it to the Banzhaf value on convex geometries.*Corresponding author. Tel.: 134 95 448 6166; fax: 134 95 448 6166; e-mail: mbilbaomatinc.us.es0165-4896/98/$19.00 1998 Elsevier Science B.V. All rights reserved.PII: S0165-4896(98)00021-3158 J.M. Bilbao et al. / Mathematical Social Sciences 36 (1998) 157 173NLet N be a finite set of n elements and +#2 . Edelman and Jamison (1985)introduced the convex geometry as the pair (N, +) where the following axioms hold:1. 5+, and + is closed under intersection.2. If S+ and SN, then there exists jNS such that S S.S+:S$AhjNDefinition 1. A simple game on a convex geometry +#2 is a set function v:+h0, 1j,such that v(5)50, and v is monotone (v(S)#v(T ), whenever S#T ).The collection of the simple games on + is denoted by V(+). If v satisfiesv(ST 55,then we say that v is superadditive. This collection of games is denoted by V (+). Onsathe set V(+) we define the internal operations meet and join by(v w)(S)5min v(S), w(S),(v w)(S)5max v(S), w(S).hj hjIn V (+) the operation join is not internal. The games 1 and 0 are simple games suchsathat 1(S)51, and 0(S)50, for every nonempty S+.Example. We consider the set of players N5h1, 2,.,2k11j and let + be the2k11convex geometry whose convex sets are the empty set and the intervals i, j5hi,i11,.,j21, jj. The game v, defined byv(S)51, if uSu$k 11.J.M. Bilbao et al. / Mathematical Social Sciences 36 (1998) 157 173 159v(S)50, if uSu#k,for all S+ , is the majority simple game on the convex coalitions for the policy2k11order (see Edelman, 1998).Example. A graph G5(N, E) is connected if any two vertices can be joined by a path.A maximal connected subgraph of G is a component of G.Acutvertex is one whoseremoval increases the number of components, and a bridge is an edge with the sameproperty. A graph is 2-connected if it is connected, has at least 3 vertices and containsno cutvertex. A subgraph B of a graph G is a block of G if either B is a bridge or else itis a maximal 2-connected subgraph of G. A graph G is a block graph if every block is acomplete graph. The block graphs are denoted by cycle-complete graphs in van denNouweland and Borm (1991). Let G5(N, E) be a connected block graph and let usconsider the collection+5 S#N:(S, E(S) is a connected subgraph of G .hjEdelman and Jamison (1985) showed that + is a convex geometry.2. Fundamental conceptsConvex coalitions S+ with v(S)51 are called winning and convex coalitions withv(S)50 losing. In the classical theory (see Dubey and Shapley, 1979) a swing for playeri is a pair of coalitions (S, Si) such that S is winning and Si is not. This concept allowsus to define convex swings for games on convex geometries.Definition 2. For a player i and a game v, we say that the pair (S, Si) is a convex swingif S+, iex(S), v(S)51, v(Si)50. The number of convex swings of player i isdenoted by cs (v), and the total number of convex swi