IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2207.03565.html

With a little help from my friends: essentiality vs opportunity in group criticality

Author

Listed:
  • Michele Aleandri
  • Marco Dall'Aglio

Abstract

We define a notion of the criticality of a player for simple monotone games based on cooperation with other players, either to form a winning coalition or to break a winning one, with an essential role for all the players involved. We compare it with the notion of differential criticality given by Beisbart that measures power as the opportunity left by other players. We prove that our proposal satisfies an extension of the strong monotonicity introduced by Young, assigns no power to null players and does not reward free riders, and can easily be computed from the minimal winning and blocking coalitions. An application to the Italian elections is presented. Our analysis shows that the measures of group criticality defined so far cannot weigh essential players while only remaining an opportunity measure. We propose a group opportunity test to reconcile the two views.

Suggested Citation

  • 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 Jul 2024.
    • Handle: RePEc:arx:papers:2207.03565
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2207.03565
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Annick Laruelle & Federico Valenciano, 2005. "Assessing success and decisiveness in voting situations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(1), pages 171-197, January.
    2. Claus Beisbart, 2010. "Groups can make a difference: voting power measures extended," Theory and Decision, Springer, vol. 69(3), pages 469-488, September.
    3. 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.
    4. R J Johnston, 1978. "On the Measurement of Power: Some Reactions to Laver," Environment and Planning A, , vol. 10(8), pages 907-914, August.
    5. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
    6. Robert J. Weber, 1979. "Subjectivity in the Valuation of Games," Cowles Foundation Discussion Papers 515, Cowles Foundation for Research in Economics, Yale University.
    7. 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.
    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. Annick Laruelle & Federico Valenciano, 2005. "A critical reappraisal of some voting power paradoxes," Public Choice, Springer, vol. 125(1), pages 17-41, July.
    2. Aleandri, M. & Dall’Aglio, M., 2025. "With a little help from my friends: Essentiality vs opportunity in group criticality," Mathematical Social Sciences, Elsevier, vol. 133(C), pages 13-22.
    3. Josep Freixas & Montserrat Pons, 2017. "Using the Multilinear Extension to Study Some Probabilistic Power Indices," Group Decision and Negotiation, Springer, vol. 26(3), pages 437-452, May.
    4. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Theory and Decision, Springer, vol. 69(1), pages 69-96, July.
    5. Michel Grabisch & Agnieszka Rusinowska, 2007. "Influence Indices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00142479, HAL.
      • Michel Grabisch & Agnieszka Rusinowska, 2007. "Influence Indices," Post-Print halshs-00142479, HAL.
      • Agnieszka Rusinowska & Michel Grabisch, 2007. "Influence Indices," Working Papers 0705, Groupe d'Analyse et de Théorie Economique Lyon St-Etienne (GATE Lyon St-Etienne), Université de Lyon.
    6. Wilms, Ingo, 2020. "Dynamic programming algorithms for computing power indices in weighted multi-tier games," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 175-192.
    7. Annick Laruelle & Federico Valenciano, 2001. "Shapley-Shubik and Banzhaf Indices Revisited," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 89-104, February.
    8. 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.
    9. 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.
    10. M. Albizuri & Annick Laruelle, 2013. "An axiomatization of success," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 145-155, June.
    11. Annick Laruelle & Federico Valenciano, 2008. "Potential, value, and coalition formation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 16(1), pages 73-89, July.
    12. 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.
    13. László Á. Kóczy, 2016. "Power Indices When Players can Commit to Reject Coalitions," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 33(1), pages 77-91, August.
    14. Claus Beisbart & Stephan Hartmann, 2010. "Welfarist evaluations of decision rules under interstate utility dependencies," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(2), pages 315-344, February.
    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. Omer Edhan, 2012. "Representations Of Positive Projections On Lipschitz Vector," Discussion Paper Series dp624, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    17. Freixas, Josep & Pons, Montserrat, 2017. "Decisiveness indices are semiindices: Addendum," Economics Letters, Elsevier, vol. 150(C), pages 146-148.
    18. Friedman, Jane & Parker, Cameron, 2018. "The conditional Shapley–Shubik measure for ternary voting games," Games and Economic Behavior, Elsevier, vol. 108(C), pages 379-390.
    19. Stefan Napel & Mika Widgrén, 2011. "Strategic versus non-strategic voting power in the EU Council of Ministers: the consultation procedure," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(3), pages 511-541, September.
    20. Michel Grabisch & Agnieszka Rusinowska, 2008. "Measuring influence among players with an ordered set of possible actions," Working Papers 0801, Groupe d'Analyse et de Théorie Economique Lyon St-Etienne (GATE Lyon St-Etienne), Université de Lyon.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:arx:papers:2207.03565. 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: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    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.