Abstract In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal ? -retentive set (? stands for tournament equilibrium set). A weak variant of Schwartz’s Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tournaments that satisfy Schwartz’s Conjecture and Brandt’s Conjecture. Moreover, we prove that ? can be calculated in polynomial time in several infinite classes of tournaments. Furthermore, our results reveal some structures that are forbidden in every counterexample to Schwartz’s Conjecture.