Social Decision Functions and Strongly Decisive Sets

Properties of the strongly decisive sets (some preference for x over y along with no preference for y over x allows coalitional enforcement of x over y) associated with a social decision function are investigated. The collection of such sets does not have the superset preserving property of filters, but is characterized by properties defining a target. A 1-1 and onto mapping is exhibited between the class of targets and a certain class of social decision functions, showing that such functions are completely characterized by the structure of their strongly decisive sets. The "ring" structure of targets is shown to be closely related to known results on veto hierarchies

Linear Problems (with Extended Range) Have Linear Optimal Algorithms

Abstract. Let F1 and F2 be normed linear spaces and S: F0→F2 a linear operator on a balanced subset F0 of F2. If N denotes a finite dimensional linear information operator on F0, it is known that there need not be a linear algorithm φ:N(F0)→F2 which is optimal in the sense that ||φ:(N(f))-S(f)|| is minimized. We show that the linear problem defined by S and N can be regarded as having a linear optimal algorithm if we allow the range of φ to be extended in a natural way. The result depends upon imbedding F2 isometrically in the space of continuous functions on a compact Hausdorff space X. This is done by making use of a consequence of the classical Banach-Alaoglu theorem

Power Structure and Cardinality Restrictions for Paretian Social Choice Rules

Let f be a multiple-valued Paretian social choice rule for n voters and an outcome set X. The preventing sets for f are shown to form an acyclic majority when |X| ̅n, and a filter when f also satisfies a binary independence condition. These results are then shown to yield inequalities relating |X|, n, and certain preventing sets. In particular, if every coalition of q voters constitutes a preventing set, then |X|≤[(n-1)/(n-q)]. Other n-q inequalities are obtained if strong equilibria are present for every preference profile

An Axiomatized Family of Power Indices for Simple n-Person Games

In this probabilistic generalization of the Deegan-Packel power index, a new family of power indices based on the notions of minimal winning coalitions and equal division of payoffs is developed. These indices are axiomatically characterized and compared to other similarly characterized indices. Additionally, a dual family of minimal blocking coalition indices and their characterization axioms is presented

Continuous Social Decision Procedures

Classical social decision procedures are supposed to map lists of preference orderings into binary relations which describe society's "preferences." But when there are infinitely many alternatives the resulting plethora of possible preference orderings make it impossible to differentiate "nearby" preference relations. If the preference information used to make social decisions is imperfect, society may wish to implement a continuous social decision procedure (SDP) so that nearby preference configurations will map into nearby social preference relations. It is shown here that a continuity requirement can severely restrict the admissible behavior of a social decision procedure. Furthermore a characterization of continuous SDPs is presented which facilitates the examination of such procedures and their relation to various voting mechanisms

Methods for Comparison of Markov Processes by Stochastic Dominance

A technique is developed for proving existence and obtaining bounds for the concentration of a stationary distribution for a given Markov process on the basis of comparisons, via stochastic dominance, with a different Markov process, having a known stationary distribution

Social Decision Functions and Strongly Decisive Sets

Properties of the strongly decisive sets (some preference for x over y along with no preference for y over x allows coalitional enforcement of x over y) associated with a social decision function are investigated. The collection of such sets does not have the superset preserving property of filters, but is characterized by properties defining a target. A 1-1 and onto mapping is exhibited between the class of targets and a certain class of social decision functions, showing that such functions are completely characterized by the structure of their strongly decisive sets. The "ring" structure of targets is shown to be closely related to known results on veto hierarchies

Limiting Distributions for Continuous State Markov Voting Models

This paper proves the existence of a stationary distribution for a class of Markov voting models. We assume that alternatives to replace the current status quo arise probabilistically, with the probability distribution at time t+1 having support set equal to the set of alternatives that defeat, according to some voting rule, the current status quo at time t. When preferences are based on Euclidean distance, it is shown that for a wide class of voting rules, a limiting distribution exists. For the special case of majority rule, not only does a limiting distribution always exist, but we obtain bounds for the concentration of the limiting distribution around a centrally located set. The implications are that under Markov voting models, small deviations from the conditions for a core point will still leave the limiting distribution quite concentrated around a generalized median point. Even though the majority relation is totally cyclic in such situations, our results show that such chaos is not probabilistically significant