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Ranking sets of interacting objects via semivalues

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  • Roberto Lucchetti
  • Stefano Moretti
  • Fioravante Patrone

Abstract

In this paper, we address the problem of how to extend a ranking over single objects to another ranking over all possible collections of objects, taking into account the fact that objects grouped together can have mutual interaction. An answer to this issue is provided using game theory and, specifically, the fact that an extension (i.e. a total preorder on the set of all subsets of objects) must be aligned with some probabilistic value, in the sense that the ranking of the objects (according to some probabilistic value computed on a numerical representation of the extension) must also preserve the primitive preorder on the singletons, no matter which utility function is used to represent the extension. We characterize families of aligned extensions, we focus on their geometric properties and we provide algorithms to verify their alignments. We also show that the framework introduced in this paper may be used to study a new class of extension problems, which integrate some features dealing with risk and complete uncertainty within the class of preference extension problems known in the literature with the name of sets as final outcomes. Copyright Sociedad de Estadística e Investigación Operativa 2015

Suggested Citation

  • Roberto Lucchetti & Stefano Moretti & Fioravante Patrone, 2015. "Ranking sets of interacting objects via semivalues," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 567-590, July.
    • Handle: RePEc:spr:topjnl:v:23:y:2015:i:2:p:567-590
    DOI: 10.1007/s11750-014-0357-5
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    Cited by:

    1. Felix Fritz & Stefano Moretti & Jochen Staudacher, 2023. "Social Ranking Problems at the Interplay between Social Choice Theory and Coalitional Games," Mathematics, MDPI, vol. 11(24), pages 1-22, December.
    2. Ramón Flores & Elisenda Molina & Juan Tejada, 2019. "Evaluating groups with the generalized Shapley value," 4OR, Springer, vol. 17(2), pages 141-172, June.
    3. Marcella Maia Urtiga & Danielle Costa Morais & Keith W. Hipel & D. Marc Kilgour, 2017. "Group Decision Methodology to Support Watershed Committees in Choosing Among Combinations of Alternatives," Group Decision and Negotiation, Springer, vol. 26(4), pages 729-752, July.
    4. Ritu Dutta & Rajnish Kumnar & Surajit Borkotokey, 2023. "How to choose a Compatible Committee?," Papers 2308.03507, arXiv.org.
    5. Giulia Bernardi & Roberto Lucchetti & Stefano Moretti, 2019. "Ranking objects from a preference relation over their subsets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 52(4), pages 589-606, April.

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