IDEAS home Printed from https://ideas.repec.org/a/spr/topjnl/v23y2015i2p567-590.html
   My bibliography  Save this article

Ranking sets of interacting objects via semivalues

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s11750-014-0357-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11750-014-0357-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Josep Freixas, 2010. "On ordinal equivalence of the Shapley and Banzhaf values for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 513-527, October.
    2. Puppe, Clemens, 1996. "An Axiomatic Approach to "Preference for Freedom of Choice"," Journal of Economic Theory, Elsevier, vol. 68(1), pages 174-199, January.
    3. Kreps, David M, 1979. "A Representation Theorem for "Preference for Flexibility"," Econometrica, Econometric Society, vol. 47(3), pages 565-577, May.
    4. Bossert Walter & Pattanaik Prasanta K. & Xu Yongsheng, 1994. "Ranking Opportunity Sets: An Axiomatic Approach," Journal of Economic Theory, Elsevier, vol. 63(2), pages 326-345, August.
    5. Barbera, S. & Barrett, C. R. & Pattanaik, Prasanta K., 1984. "On some axioms for ranking sets of alternatives," Journal of Economic Theory, Elsevier, vol. 33(2), pages 301-308, August.
    6. Faruk Gul & Wolfgang Pesendorfer, 2001. "Temptation and Self-Control," Econometrica, Econometric Society, vol. 69(6), pages 1403-1435, November.
    7. Stefano Moretti & Fioravante Patrone, 2008. "Rejoinder on: Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 60-61, July.
    8. Stefano Moretti & Fioravante Patrone, 2008. "Transversality of the Shapley value," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 1-41, July.
    9. Francesc Carreras & Josep Freixas, 2000. "A Note On Regular Semivalues," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 345-352.
    10. Roth, Alvin E., 1985. "The college admissions problem is not equivalent to the marriage problem," Journal of Economic Theory, Elsevier, vol. 36(2), pages 277-288, August.
    11. Carreras, Francesc & Freixas, Josep, 2008. "On ordinal equivalence of power measures given by regular semivalues," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 221-234, March.
    12. Nitzan, Shmuel I. & Pattanaik, Prasanta K., 1984. "Median-based extensions of an ordering over a set to the power set: An axiomatic characterization," Journal of Economic Theory, Elsevier, vol. 34(2), pages 252-261, December.
    13. Kannai, Yakar & Peleg, Bezalel, 1984. "A note on the extension of an order on a set to the power set," Journal of Economic Theory, Elsevier, vol. 32(1), pages 172-175, February.
    14. Hadar, Josef & Russell, William R, 1969. "Rules for Ordering Uncertain Prospects," American Economic Review, American Economic Association, vol. 59(1), pages 25-34, March.
    15. Fishburn, Peter C., 1992. "Signed orders and power set extensions," Journal of Economic Theory, Elsevier, vol. 56(1), pages 1-19, February.
    16. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Ritu Dutta & Rajnish Kumnar & Surajit Borkotokey, 2023. "How to choose a Compatible Committee?," Papers 2308.03507, arXiv.org.
    3. Ramón Flores & Elisenda Molina & Juan Tejada, 2019. "Evaluating groups with the generalized Shapley value," 4OR, Springer, vol. 17(2), pages 141-172, June.
    4. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Barbera, S. & Bossert, W. & Pattanaik, P.K., 2001. "Ranking Sets of Objects," Cahiers de recherche 2001-02, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    2. Walter Bossert, 1998. "Opportunity Sets and the Measurement of Information," Discussion Papers 98/6, University of Nottingham, School of Economics.
    3. Bossert, Walter, 2000. "Opportunity sets and uncertain consequences1," Journal of Mathematical Economics, Elsevier, vol. 33(4), pages 475-496, May.
    4. Bossert, Walter, 1997. "Uncertainty aversion in nonprobabilistic decision models," Mathematical Social Sciences, Elsevier, vol. 34(3), pages 191-203, October.
    5. Baharad, Eyal & Nitzan, Shmuel, 2003. "Essential alternatives and set-dependent preferences--an axiomatic approach," Mathematical Social Sciences, Elsevier, vol. 45(2), pages 121-129, April.
    6. Walter Bossert & Kotaro Suzumura, 2011. "Rationality, external norms, and the epistemic value of menus," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(4), pages 729-741, October.
    7. Ran Spiegler, 2001. "Inferring a linear ordering over a power set," Theory and Decision, Springer, vol. 51(1), pages 31-49, August.
    8. Arlegi, Ritxar & Dimitrov, Dinko, 2016. "Power set extensions of dichotomous preferences," Mathematical Social Sciences, Elsevier, vol. 83(C), pages 20-29.
    9. Ballester, Miguel A. & de Miguel, Juan R. & Nieto, Jorge, 2004. "Set comparisons in a general domain: the Indirect Utility Criterion," Mathematical Social Sciences, Elsevier, vol. 48(2), pages 139-150, September.
    10. Lefgren, Lars J. & Stoddard, Olga B. & Stovall, John E., 2021. "Rationalizing self-defeating behaviors: Theory and evidence," Journal of Health Economics, Elsevier, vol. 76(C).
    11. Klaus Nehring, 2003. "Preference for Flexibility and Freedom of Choice in a Savage Framework," Working Papers 51, University of California, Davis, Department of Economics.
    12. Antoinette Baujard, 2006. "Conceptions of freedom and ranking opportunity sets. A typology," Economics Working Paper Archive (University of Rennes & University of Caen) 200611, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    13. Arlegi, Ricardo, 2007. "Sequentially consistent rules of choice under complete uncertainty," Journal of Economic Theory, Elsevier, vol. 135(1), pages 131-143, July.
    14. Flores-Szwagrzak, Karol, 2022. "Learning by Convex Combination," Working Papers 16-2022, Copenhagen Business School, Department of Economics.
    15. Jorge Alcalde-Unzu & Ricardo Arlegi & Miguel Ballester, 2013. "Uncertainty with ordinal likelihood information," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 397-425, July.
    16. Ballester, Miguel A. & De Miguel, Juan R., 2006. "On freedom of choice and infinite sets: The Suprafinite Rule," Journal of Mathematical Economics, Elsevier, vol. 42(3), pages 291-300, June.
    17. Prasanta Pattanaik & Yongsheng Xu, 1998. "On Preference and Freedom," Theory and Decision, Springer, vol. 44(2), pages 173-198, April.
    18. Giulia Bernardi & Roberto Lucchetti, 2015. "Generating Semivalues via Unanimity Games," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 1051-1062, September.
    19. BOSSERT, Walter & SLINKO, Arkadii, 2004. "Relative Uncertainty and Additively Representable Set Rankings," Cahiers de recherche 2004-13, Universite de Montreal, Departement de sciences economiques.
    20. Ok, Efe A., 1997. "On Opportunity Inequality Measurement," Journal of Economic Theory, Elsevier, vol. 77(2), pages 300-329, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:topjnl:v:23:y:2015:i:2:p:567-590. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.