Bounds on the disparity and separation of tournament solutions

A tournament solution is a function that maps a tournament, i.e., a directed graph representing an asymmetric and connex relation on a finite set of alternatives, to a non-empty subset of the alternatives. Tournament solutions play an important role in social choice theory, where the binary relation is typically defined via pairwise majority voting. If the number of alternatives is sufficiently small, different tournament solutions may return overlapping or even identical choice sets. For two given tournament solutions, we define the disparity index as the order of the smallest tournament for which the solutions differ and the separation index as the order of the smallest tournament for which the corresponding choice sets are disjoint. Isolated bounds on both values for selected tournament solutions are known from the literature. In this paper, we address these questions systematically using an exhaustive computer analysis. Among other results, we provide the first tournament in which the bipartisan set and the Banks set are not contained in each other.