Social Choice and Welfare

, Volume 42, Issue 3, pp 551–574 | Cite as

Minimal retentive sets in tournaments

  • Felix Brandt
  • Markus Brill
  • Felix Fischer
  • Paul Harrenstein
Original Paper

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix Brandt
    • 1
  • Markus Brill
    • 1
  • Felix Fischer
    • 2
  • Paul Harrenstein
    • 3
  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany
  2. 2.Statistical LaboratoryUniversity of CambridgeCambridgeUK
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUK

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