Minimal retentive sets in tournaments

Abstract

Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a nonempty subset of the alternatives, play an important role in the mathematical social sciences at large. For any given tournament solution \(S\), there is another tournament solution which returns the union of all inclusion-minimal sets that satisfy \(S\)-retentiveness, a natural stability criterion with respect to \(S\). Schwartz’s tournament equilibrium set (\({ TEQ }\)) is defined recursively as . In this article, we study under which circumstances a number of important and desirable properties are inherited from \(S\) to . We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” \({ TEQ }\), which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding \({ TEQ }\), which establishes —a refinement of the top cycle—as an interesting new tournament solution.