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

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  • 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.

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  • 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.
    • Handle: RePEc:arx:papers:2207.03565
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    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.
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    6. 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.
    7. 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.
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