Recognizing Members of the Tournament Equilibrium Set is NP-hard

A recurring theme in the mathematical social sciences is how to select the “most desirable” elements given some binary dominance relation on a set of alternatives. Schwartz’s Tournament Equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions that have been proposed so far in this context. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem up to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive tournament solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in the Tournament Equilibrium set is NP-hard.