A tournament of order 24 with two disjoint TEQ-retentive sets

A tournament T is a pair (A,≻), whereA is a set of alternatives and ≻ is an asymmetric and complete (and thus irreflexive) binary relation on A, usually referred to as the dominance relation. The dominance relation can be extended to sets of alternatives by writing X ≻ Y when x ≻ y for all x ∈ X and y ∈ Y . For a tournament (A,≻), an alternative x ∈ A, and a subset X ⊆ A of alternatives, we denote by DX(x) = { y ∈ X | y ≻ x } the dominators of x. A tournament solution is a function that maps a tournament to a nonempty subset of its alternatives (see, e.g., Laslier, 1997, for further information). Given a tournament T = (A,≻) and a tournament solution S, a nonempty subset of alternatives X ⊆ A is called S-retentive if S(DA(x)) ⊂ X for all x ∈ X such that DA(x) 6= ∅. Schwartz (1990) defined the tournament equilibrium set (TEQ) of a given tournament T = (A,≻) recursively as the union of all inclusion-minimal TEQ-retentive sets in T . Schwartz conjectured that every tournament contains a unique inclusion-minimal TEQretentive set, which was later shown to be equivalent to TEQ satisfying any one of a number of desirable properties for tournament solutions (Laffond et al., 1993; Houy, 2009a,b; Brandt et al., 2010a; Brandt, 2011b; Brandt and Harrenstein, 2011; Brandt, 2011a). This conjecture was recently disproved by Brandt et al. (2013) who have shown the existence of a counterexample with about 10 alternatives using the probabilistic method. Since