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, it assigns no power to dummy players and free riders, and it can easily be computed from the minimal winning and blocking coalitions. 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：我们基于玩家与其他玩家的合作定义了简单单调博弈中玩家关键性的概念，这种合作旨在形成获胜联盟或打破获胜联盟，且所有参与玩家均扮演重要角色。我们将此与Beisbart提出的差异关键性概念进行比较——后者将博弈力度量为其他玩家留下的机会。我们证明，所提出的概念满足Young引入的强单调性扩展，不为傀儡玩家与搭便车者分配权力，且可通过最小获胜联盟与最小阻断联盟轻松计算。分析表明，现有群体关键性度量无法在仅保留机会性度量的同时衡量重要玩家。为此，我们提出群体机会测试以调和两种视角。