Minimal retentive sets in tournaments

Many problems in multiagent decision making can be addressed using tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a non-empty subset of the alternatives. For a given tournament solution S, Schwartz calls a set of alternatives S-retentive if it satisfies a natural stability criterion with respect to S. He then recursively defines the tournament equilibrium set (TEQ) as ̊ TEQ , the union of all inclusion-minimal TEQ-retentive sets. Due to its unwieldy recursive definition, preciously little is known about TEQ . Assuming a well-known conjecture about TEQ , we show that most desirable properties of tournament solutions are inherited from S to S̊. We thus obtain an infinite hierarchy of efficiently computable tournament solutions that “approximate” TEQ (which is computationally intractable) while maintaining most of its desirable properties. This hierarchy contains well-known tournament solutions such as the top cycle (TC ) and the minimal covering set (MC ). We prove a weaker version of the conjecture mentioned above, which establishes T̊C as an attractive new tournament solution.