Narrow Results

In this paper we revise the axiomatic foundations and meaning of semivalues as measures of power on the domain of simple games, when these are interpreted as models of voting procedures. In this context we characterize the family of preferences on roles in voting procedures they represent, and each of them in particular. To this end we first characterize the family of semivalues and each of them in particular up to the choice of a zero and a unit of scale. As a result a reinterpretation of semivalues as a class of power indices is proposed and critically discussed.

This paper aims to give a global vision concerning the state of the art of studies on 13 power indices and to establish which of them are more suitable for describing the real situations which are, from time to time, taken into consideration. In such contexts, different comparisons have been developed in terms of properties, axiomatic grounds and so on. This analysis points out various open problems.

This paper presents a review of literature on simple games and highlights various open problems concerning such games; in particular, weighted games and power indices.

The paper presents some issues currently under studying in the field of Cooperative Games. The related open problems are also mentioned.

After a brief description of a representative selection of power indices and a discussion of the notion of power in collective decision making, the paper discusses the modeling of power of an individual or collective agent as identified with the potential or factual effect the decision of this agent has on the outcome. It demonstrates that the distribution of power is crucially dependent on the procedures resorted to, and not just on the distribution of resources and the majority threshold as captured by the standard power measures. Similarly, it is shown that the selection and formulation of the questions to analyze can be highly relevant when we link power and power measures to causality. The concluding section discusses whether power indices are measures that represent power and ratios of power, or whether they are indicators that point out properties of the cooperative game and the underlying decision situation.

The paper presents the problem of choosing the representatives in an assembly when the whole electoral region is subdivided into several electoral districts. Because of the two dimensions, geographical (districts) and political (parties), the problem is called bi-apportionment. The main purpose of the paper is to discuss fairness and proportionality axioms and to describe their implementation.

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

Power indices in simple games measure the relevance of a player through her ability in being critical, i.e. essential for a coalition to win. We introduce new indices that measure the power of a player in being decisive through the collaboration of other players. We study the behavior of these criticality indices to compare the power of different players within a single voting situation, and that of the same player with varying weight across different voting situations. In both cases we establish monotonicity results in line with those of Turnovec [1998]. Finally, we examine which properties characterizing the indices of Shapley–Shubik and Banzhaf are shared by these new indices.

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

Power indices in simple games measure the relevance of a player through her ability in being critical, i.e. essential for a coalition to win. We introduce new indices that measure the power of a player in being decisive through the collaboration of other players. We study the behavior of these criticality indices to compare the power of different players within a single voting situation, and that of the same player with varying weight across different voting situations. In both cases we establish monotonicity results in line with those of Turnovec [1998]. Finally, we examine which properties characterizing the indices of Shapley–Shubik and Banzhaf are shared by these new indices.