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.

翻译：我们定义了简单单调博弈中玩家关键性的一种概念，该概念基于玩家与其他玩家的合作：要么形成获胜联盟，要么打破获胜联盟，且所有参与者均发挥重要作用。我们将其与Beisbart提出的差异性关键性概念进行比较，后者将权力衡量为其他玩家留下的机会。我们证明，我们的提议满足Young引入的强单调性的扩展，赋予无效玩家零权力，不奖励搭便车者，并且可以轻松地从最小获胜联盟和最小阻碍联盟中计算得出。我们将其应用于意大利选举的分析。我们的分析表明，迄今为止定义的群组关键性度量无法权衡必要玩家，而仅仅是一种机会度量。我们提出一项群组机会测试来调和这两种观点。