IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v318y2022i2d10.1007_s10479-021-04199-6.html
   My bibliography  Save this article

Minimal winning coalitions and orders of criticality

Author

Listed:
  • Michele Aleandri

    (Luiss University)

  • Marco Dall’Aglio

    (Luiss University)

  • Vito Fragnelli

    (University of Eastern Piedmont)

  • Stefano Moretti

    (Université PSL)

Abstract

In this paper, we analyze the order of criticality in simple games, under the light of minimal winning coalitions. The order of criticality of a player in a simple game is based on the minimal number of other players that have to leave so that the player in question becomes pivotal. We show that this definition can be formulated referring to the cardinality of the minimal blocking coalitions or minimal hitting sets for the family of minimal winning coalitions; moreover, the blocking coalitions are related to the winning coalitions of the dual game. Finally, we propose to rank all the players lexicographically accounting the number of coalitions for which they are critical of each order, and we characterize this ranking using four independent axioms.

Suggested Citation

  • Michele Aleandri & Marco Dall’Aglio & Vito Fragnelli & Stefano Moretti, 2022. "Minimal winning coalitions and orders of criticality," Annals of Operations Research, Springer, vol. 318(2), pages 787-803, November.
    • Handle: RePEc:spr:annopr:v:318:y:2022:i:2:d:10.1007_s10479-021-04199-6
    DOI: 10.1007/s10479-021-04199-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-021-04199-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10479-021-04199-6?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. Johnston, R. J., 1995. "The Conflict over Qualified Majority Voting in the European Union Council of Ministers: An Analysis of the UK Negotiating Stance Using Power Indices," British Journal of Political Science, Cambridge University Press, vol. 25(2), pages 245-254, April.
    2. 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.
    3. Lloyd S. Shapley & Mark Burgin, 2000. "Enhanced Banzhaf Power Index and It's Mathematical Properties," UCLA Economics Working Papers 797, UCLA Department of Economics.
    4. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    5. Pradeep Dubey & Abraham Neyman & Robert James Weber, 1981. "Value Theory Without Efficiency," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 122-128, February.
    6. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    7. Weber, Robert J., 1994. "Games in coalitional form," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 36, pages 1285-1303, Elsevier.
    8. Francesc Carreras, 2009. "Protectionism and blocking power indices," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 70-84, July.
    9. Marco Dall’Aglio & Vito Fragnelli & Stefano Moretti, 2019. "Indices of Criticality in Simple Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 21(01), pages 1-21, March.
    10. Marco Dall'Aglio & Vito Fragnelli & Stefano Moretti, 2016. "Orders of criticality in voting games," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 26(2), pages 53-67.
    Full references (including those not matched with items on IDEAS)

    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. Ori Haimanko, 2019. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(3), pages 755-768, September.
    2. Ori Haimanko, 2020. "Generalized Coleman-Shapley indices and total-power monotonicity," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 299-320, March.
    3. Annick Laruelle & Federico Valenciano, 2005. "A critical reappraisal of some voting power paradoxes," Public Choice, Springer, vol. 125(1), pages 17-41, July.
    4. Houy, Nicolas & Zwicker, William S., 2014. "The geometry of voting power: Weighted voting and hyper-ellipsoids," Games and Economic Behavior, Elsevier, vol. 84(C), pages 7-16.
    5. Francesc Carreras & María Albina Puente, 2012. "Symmetric Coalitional Binomial Semivalues," Group Decision and Negotiation, Springer, vol. 21(5), pages 637-662, September.
    6. Haimanko, Ori, 2018. "The axiom of equivalence to individual power and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 108(C), pages 391-400.
    7. Michele Aleandri & Marco Dall'Aglio, 2022. "With a little help from my friends: essentiality vs opportunity in group criticality," Papers 2207.03565, arXiv.org, revised Feb 2024.
    8. Ori Haimanko, 2019. "The Banzhaf Value and General Semivalues for Differentiable Mixed Games," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 767-782, August.
    9. Carreras, Francesc & Freixas, Josep & Puente, Maria Albina, 2003. "Semivalues as power indices," European Journal of Operational Research, Elsevier, vol. 149(3), pages 676-687, September.
    10. Carreras, Francesc & Giménez, José Miguel, 2011. "Power and potential maps induced by any semivalue: Some algebraic properties and computation by multilinear extensions," European Journal of Operational Research, Elsevier, vol. 211(1), pages 148-159, May.
    11. Freixas, Josep & Pons, Montserrat, 2008. "Circumstantial power: Optimal persuadable voters," European Journal of Operational Research, Elsevier, vol. 186(3), pages 1114-1126, May.
    12. McQuillin, Ben & Sugden, Robert, 2018. "Balanced externalities and the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 81-92.
    13. Sascha Kurz, 2016. "The inverse problem for power distributions in committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(1), pages 65-88, June.
    14. Laruelle, Annick & Valenciano, Federico, 2008. "Noncooperative foundations of bargaining power in committees and the Shapley-Shubik index," Games and Economic Behavior, Elsevier, vol. 63(1), pages 341-353, May.
    15. Borkowski, Agnieszka, 2003. "Machtverteilung Im Ministerrat Nach Dem Vertrag Von Nizza Und Den Konventsvorschlagen In Einer Erweiterten Europaischen Union," IAMO Discussion Papers 14887, Institute of Agricultural Development in Transition Economies (IAMO).
    16. Saari, Donald G. & Sieberg, Katri K., 2001. "Some Surprising Properties of Power Indices," Games and Economic Behavior, Elsevier, vol. 36(2), pages 241-263, August.
    17. Meinhardt, Holger Ingmar, 2021. "Disentangle the Florentine Families Network by the Pre-Kernel," MPRA Paper 106482, University Library of Munich, Germany.
    18. Gomez, Daniel & Gonzalez-Aranguena, Enrique & Manuel, Conrado & Owen, Guillermo & del Pozo, Monica & Tejada, Juan, 2003. "Centrality and power in social networks: a game theoretic approach," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 27-54, August.
    19. O'Neill, Barry & Peleg, Bezalel, 2008. "Lexicographic composition of simple games," Games and Economic Behavior, Elsevier, vol. 62(2), pages 628-642, March.
    20. Dubey, Pradeep & Einy, Ezra & Haimanko, Ori, 2005. "Compound voting and the Banzhaf index," Games and Economic Behavior, Elsevier, vol. 51(1), pages 20-30, April.

    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:annopr:v:318:y:2022:i:2:d:10.1007_s10479-021-04199-6. 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.