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 [InlineEquation not available: see fulltext.] 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 [InlineEquation not available: see fulltext.]. In this article, we study under which circumstances a number of important and desirable properties are inherited from $$S$$ S to [InlineEquation not available: see fulltext.]. 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 [InlineEquation not available: see fulltext.]—a refinement of the top cycle—as an interesting new tournament solution. Copyright Springer-Verlag Berlin Heidelberg 2014