Minimal retentive sets in tournaments

Journal article

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 S, there is another tournament solution Ṡ which returns the union of all inclusion-minimal sets that satisfy S-retentiveness, a natural stability criterion with respect to S. Schwartz’s tournament equilibrium set (TEQ) is defined recursively as TEQ = TĖQ. In this article, we study under which circumstances a number of important and desirable properties are inherited from S to Ṡ. We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” TEQ, which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding TEQ, which establishes ṪC —a refinement of the top cycle — as an interesting new tournament solution.

Authors: Brandt, F; Brill, M; Fischer, F; Harrenstein, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer-Verlag
• Journal: Social Choice and Welfare
• Volume: 42
• Issue: 3
• Pages: 551-574
• Publication date: 2013-06-07
• DOI: 10.1007/s00355-013-0740-4
• EISSN: 1432-217X
• ISSN: 0176-1714