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In this paper we propose a simple model for measuring ‘success’ or ‘decisiveness’ in voting situations. For an assessment of these features two inputs are claimed to be necessary: the voting rule and the voters’ behavior. The voting rule specifies when a proposal is to be accepted or rejected depending on the resulting vote configuration. Voting behavior is summarized by a distribution of probability over the vote configurations. This basic model provides a clear common conceptual basis for reinterpreting different power indices and some related game theoretic notions coherently from a unified point of view. Copyright Springer-Verlag 2005

Every day thousands of decisions are made by all kinds of committees, parliaments, councils and boards by a 'yes–no' voting process. Sometimes a committee can only accept or reject the proposals submitted to it for a decision. On other occasions, committee members have the possibility of modifying the proposal and bargaining an agreement prior to the vote. In either case, what rule should be used if each member acts on behalf of a different-sized group? It seems intuitively clear that if the groups are of different sizes then a symmetric rule (e.g. the simple majority or unanimity) is not suitable. The question then arises of what voting rule should be used. Voting and Collective Decision-Making addresses this and other issues through a study of the theory of bargaining and voting power, showing how it applies to real decision-making contexts.

We provide a new axiomatization of the Shapley-Shubik and the Banzhaf power indices in the domain of simple superadditive games by means of transparent axioms. Only anonymity is shared with the former characterizations in the literature. The rest of the axioms are substituted by more transparent ones in terms of power in collective decision-making procedures. In particular, a clear restatement and a weaker alternative for the transfer axiom are proposed. Only one axiom differentiates the characterization of either index, and these differentiating axioms provide a new point of comparison. In a first step both indices are characterized up to a zero and a unit of scale. Then both indices are singled out by simple normalizing axioms.

Abstract:The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoffvector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.