With a little help from my friends: essentiality vs opportunity in group criticality
Michele Aleandri, Marco Dall'Aglio
TL;DR
The paper develops $g$-criticality, a group-level extension of criticality for simple monotone games, emphasizing essential coalitions and cooperative formation. It contrasts $g$-criticality with Beisbart’s $d$-criticality (opportunity left by others) and with prior inside/outside criticalities, and provides a computation framework based on minimal winning and blocking coalitions. The authors prove that $g$-criticality enjoys strong monotonicity and null-player immunity, relate average ranks to minimal essential criticality, and demonstrate how to compute the measures in practice. An electoral application to the Italian Parliament (2013, 2018, 2022) illustrates that group power distributions can diverge markedly from seat shares and evolve with coalition structure, motivating a reconciled, coalition-aware notion of opportunity measure.
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.