The cost of getting local monotonicity

Committees with yes-no-decisions are commonly modeled as simple games and the ability of a member to influence the group decision is measured by so-called power indices. For a weighted game we say that a power index satisfies local monotonicity if a player who controls a large share of the total weight vote does not have less power than a player with a smaller voting weight. In (Holler, 1982) Manfred Holler introduced the Public Good index. In its unnormalized version, i.e., the raw measure, it counts the number of times that a player belongs to a minimal winning coalition. Unlike the Banzhaf index, it does not count the remaining winning coalitions in which the player is crucial. Holler noticed that his index does not satisfy local monotonicity, a fact that can be seen either as a major drawback (Felsenthal & Machover, 1998, 221 ff.)or as an advantage (Holler & Napel 2004). In this paper we consider a convex combination of the two indices and require the validity of local monotonicity. We prove that the cost of obtaining it is high, i.e., the achievable new indices satisfying local monotonicity are closer to the Banzhaf index than to the Public Good index. All these achievable new indices are more solidary than the Banzhaf index, which makes them as very suitable candidates to divide a public good. As a generalization we consider convex combinations of either: the Shift index, the Public Good index, and the Banzhaf index, or alternatively: the Shift Deegan-Packel, Deegan-Packel, and Johnston indices.Peer Reviewe