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A tournament can be represented as a set of candidates and the results from pairwise comparisons of the candidates. In our setting, candidates may form coalitions. The candidates can choose to fix who wins the pairwise comparisons within their coalition. A coalition is winning if it can guarantee that a candidate from this coalition will win each pairwise comparison. This approach divides all coalitions into two groups and is, hence, a simple game. We show that each minimal winning coalition consists of a certain uncovered candidate and its dominators. We then apply solution concepts developed for simple games and consider the desirability relation and the power indices which preserve this relation. The tournament solution, defined as the maximal elements of the desirability relation, is a good way to select the strongest candidates. The Shapley–Shubik index, the Penrose–Banzhaf index, and the nucleolus are used to measure the power of the candidates. We also extend this approach to the case of weak tournaments. [ABSTRACT FROM AUTHOR]

Simple games are abstract representations of voting systems and other group-decision procedures. A stable set —or von Neumann–Morgenstern solution—of a simple game represents a “standard of behavior” that satisfies certain internal and external stability properties. Compound simple games are built out of component games, which are, in turn, “players” of a quotient game. I describe a method to construct fair —or symmetry-preserving—stable sets of compound simple games from fair stable sets of their quotient and components. This method is closely related to the composition theorem of Shapley (1963c) , and contributes to the answer of a question that he formulated: What is the set G of simple games that admit a fair stable set? In particular, this method shows that the set G includes all simple games whose factors —or quotients in their “unique factorization” of Shapley (1967) —are in G , and suggests a path to characterize G . [ABSTRACT FROM AUTHOR]

We present an axiomatic characterization of the Owen-Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent. [ABSTRACT FROM AUTHOR]

A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games. Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering. [ABSTRACT FROM AUTHOR]

Linear simple games with consensus are considered. These games are obtained by intersecting a linear simple game (a game where the desirability order for players is total) and a symmetric (k-out-of-n) game. We investigate the behavior of the Shapley–Shubik power index when passing from a linear game with consensus level q1 to the same game with consensus level q2>q1. We also introduce a “range notion”, which intuitively represents the egalitarianism of the Shapley–Shubik index, obtain an upper bound on this measure, and characterize when it is achieved. Finally, the theory developed here is illustrated with several real-world voting systems. [Copyright &y& Elsevier]