Minimal retentive sets in tournaments

Brandt, F., Brill, M., Fischer, F. and Harrenstein, P. (2014) Minimal retentive sets in tournaments. Social Choice and Welfare, 42(3), pp. 551-574. (doi: 10.1007/s00355-013-0740-4)

Abstract

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  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 . In this article, we study under which circumstances a number of important and desirable properties are inherited from S to . 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 —a refinement of the top cycle—as an interesting new tournament solution.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fischer, Dr Felix
• Authors: Brandt, F., Brill, M., Fischer, F., and Harrenstein, P.
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: Social Choice and Welfare
• Publisher: Springer
• ISSN: 0176-1714
• ISSN (Online): 1432-217X
• Published Online: 07 June 2013
• Copyright Holders: Copyright © 2013 Springer-Verlag
• First Published: First published in Social Choice and Welfare 42(3): 551-574
• Publisher Policy: Reproduced in accordance with the publisher copyright policy