Abstract.
Procedures designed to select alternatives on the basis of the results of pairwise contests between them have received much attention in literature. The particular case of tournaments has been studied in depth. More recently weak tournaments and valued generalizations thereof have been investigated.The purpose of this paper is to investigate to what extent these choice procedures may be meaningfully used to define ranking procedures via their repeated use, i.e. when the equivalence classes of the ranking are determined by successive applications of the choice procedure. This is what we call “ranking by choosing”. As could be expected, such ranking procedures raise monotonicity problems. We analyze these problems and show that it is nevertheless possible to isolate a large class of well-behaved choice procedures for which failures of monotonicity are not overly serious. The hope of finding really attractive ranking by choosing procedures is however shown to be limited. Our results are illustrated on the case of tournaments.
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I am grateful to Hervé Raynaud and Jean-Claude Vansnick for stimulating discussions. Jean-François Laslier, Thierry Marchant, Bernard Monjardet, Xavier Juret and an anonymous referee made very helpful comments on earlier drafts of this text. Special thanks go to Patrice Perny who introduced me to the subject and to Olivier Hudry for his help with Slater orders. The usual caveat applies. Part of this work was accomplished while I was visiting the Service de Mathématiques de la Gestion at the Université Libre de Bruxelles (Brussels, Belgium). I gratefully acknowledge the warm hospitality of the Service de Mathématique de la Gestion as well as the support from the Belgian Fonds National de la Recherche Scientifique and the Brussels-Capital Region through a “Research in Brussels” action grant.
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Bouyssou, D. Monotonicity of ‘ranking by choosing’: A progress report. Soc Choice Welfare 23, 249–273 (2004). https://doi.org/10.1007/s00355-003-0250-x
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DOI: https://doi.org/10.1007/s00355-003-0250-x