Smith set

Abstract: Social choice rules are often evaluated and compared by inquiring whether they fulfill certain desirable criteria such as the Condorcet criterion, which states that an alternative should always be chosen when more than half of the voters prefer it over any other alternative. Many of these criteria can be formulated in terms of choice sets that single out reasonable alternatives based on the preferences of the voters. In this paper, we consider choice sets whose definition merely relies on the pairwise majority relation. These sets include the Copeland set, the Smith set, the Schwartz set, von Neumann-Morgenstern stable sets (a concept originally introduced in the context of cooperative game theory), the Banks set, and the Slater set. We investigate the relationships between these sets and completely characterize their computational complexity which allows us to obtain hardness results for entire classes of social choice rules. In contrast to most existing work, we do not impose any restrictions on individual preferences, apart from the indifference relation being reflexive and symmetric. This assumption is motivated by the fact that many realistic types of preferences in computational contexts such as incomplete or quasi-transitive preferences may lead to general pairwise majority relations that need not be complete.

Abstract: The problems of controlling an election have been shown NP-complete in general but polynomial-time solvable in single-peaked elections for many voting correspondences. To explore the complexity border, we consider these control problems by adding/deleting votes in elections with bounded single-peaked width k. Single-peaked elections have single-peaked width k=1. We prove that the constructive control problems for Copelandα with 0≤α

Abstract: The Smith equivalence of real representations of a finite group has been studied by many mathematicians, e.g. J. Milnor, T. Petrie, S. Cappell-J. Shaneson, K. Pawalowski-R. Solomon. For a given finite group, let the primary Smith set of the group be the subset of real representation ring consisting of all differences of pairs of prime matched, Smith equivalent representations. The primary Smith set was rarely determined for a nonperfect group G besides the case where the primary Smith set is trivial. In this paper we determine the primary Smith set of an arbitrary Oliver group such that a Sylow 2-subgroup is normal and the nilquotient is isomorphic to the direct product of a finite number of cyclic groups of order 2 or 3. In particular, we answer to a problem posed by T. Sumi.

Abstract: In this article, we analyze the problem of choice based on potentially ill-behaved binary relations that may fail to possess maximal elements. The approach, which is based on Duggan (Soc Choice Welf 28:491–506, 2007), is to consider every maximal subrelation (resp. minimal superrelation) satisfying a regularity condition (acyclicity, consistency, or negative consistency) and to unify the maximal elements of all such relations. Based on this procedure, we present a characterization of the Smith set, the Duggan set, the Schwartz set, and the generalized stable sets solution of an arbitrary binary relation over non-finite sets. Schwartz’s and Van Deemen’s results for asymmetric binary relations stated in the finite case, are also extended to this more general framework. Finally, we give a set theoretical description of the Duggan set and show that the Schwartz set offers a refinement of the Duggan set, and the Schwartz and Duggan sets are nested between the union of generalized stable sets and the Smith set.