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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> </mrow> </math> </EquationSource> </InlineEquation>, there is another tournament solution [InlineEquation not available: see fulltext.] which returns the union of all inclusion-minimal sets that satisfy <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> </mrow> </math> </EquationSource> </InlineEquation>-retentiveness, a natural stability criterion with respect to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> </mrow> </math> </EquationSource> </InlineEquation>. Schwartz’s tournament equilibrium set (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${ TEQ }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">TEQ</mi> </mrow> </math> </EquationSource> </InlineEquation>) 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>S</mi> </mrow> </math> </EquationSource> </InlineEquation> to [InlineEquation not available: see fulltext.]. We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$${ TEQ }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">TEQ</mi> </mrow> </math> </EquationSource> </InlineEquation>, which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$${ TEQ }$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">TEQ</mi> </mrow> </math> </EquationSource> </InlineEquation>, 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
