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

TL;DR

This paper introduces a new measure of player criticality in simple monotone games based on cooperation, contrasting it with existing opportunity-based measures, and applies it to analyze Italian elections.

Contribution

It proposes a novel group criticality measure that captures players' essential roles, extending existing opportunity-based measures, and provides a method to reconcile the two perspectives.

Findings

The new measure satisfies strong monotonicity and null player properties.

It can be computed from minimal winning and blocking coalitions.

Application to Italian elections demonstrates the measure's practical relevance.

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.