Minimal retentive sets in tournaments

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 $$S$$S, there is another tournament solution  which returns the union of all inclusion-minimal sets that satisfy $$S$$S-retentiveness, a natural stability criterion with respect to $$S$$S. Schwartz’s tournament equilibrium set ($${ TEQ }$$TEQ) is defined recursively as . In this article, we study under which circumstances a number of important and desirable properties are inherited from $$S$$S to . We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” $${ TEQ }$$TEQ, which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding $${ TEQ }$$TEQ, which establishes —a refinement of the top cycle—as an interesting new tournament solution.