Stable Solutions to the Social Choice Problem: Dominant, Weakly Stable, Uncovered Sets and Their Extensions

Abstract

Social Choice problem is defined for a finite set of alternatives A, over which a finite number of agents have preferences. Solution to the problem is a rule determining alternatives, which are the “best” to the group. The Condorcet winner is regarded as the best choice but it is absent in general case. Attempts were made to extend the set of chosen alternatives up to a certain always non-empty subset of A, defined through majority relation. Being different incarnations of an idea of optimal choice this solutions enable one to compare and evaluate social choice procedures. And when there is a connection of a solution set with a particular voting game, it enables one to make predictions with respect to choice results. Here for such class of majority relation as tournaments three solutions are considered: the uncovered set, the minimal weakly stable set and the minimal dominant set (top cycle). A criterion to determine whether an alternative belongs to a minimal weakly stable set is found. It relates the minimal weakly stable set to the uncovered set. The idea of stability is employed to generalize the notions of weakly stable and uncovered sets. Concepts of k-stable alternatives and k-stable sets are introduced and their mutual relations are explored. A concept of the minimal dominant set is generalized. It is shown that the classes of k-stable alternatives and k-stable sets and dominant sets constitute a system of reference based on difference in degrees of stability.