<p>We propose a new power index, which we call the <i>essential coalitions index</i>. The index is defined on the class of simple games with cooperation restricted by a cooperation index, which encompasses other restriction models, including Myerson’s communication games. In general, the games we study are not simple themselves. We introduce the essential coalitions of restricted games as an analogue to the minimal winning coalitions of a simple game; in particular, we show they provide a representation of the game. Their namesake index is defined to distribute payments based on the agents’ presence in essential coalitions. In terms of its characterization, the new index differs from the Shapley value of the restricted game in only one property. On the other hand, under a suitable normalization, the payoff distribution of the essential coalitions index generalizes that given by the Deegan-Packel index. Furthermore, this normalization admits an intuitive probabilistic interpretation, which we apply to the analysis of power distribution in a legislative chamber.</p>