Indices of criticality in simple games

We generalize the notion of power index for simple games to different orders of criticality, where the order of criticality represents the possibility for players to gain more power over the members of a coalition thanks to the collusion with other players. We study the behaviour of these criticality indices to compare the power of different players within a single voting situation, and that of the same player with varying weight across different voting situations. In both cases we establish monotonicity results in line with those of Turnovec (1998). Finally, we examine which properties characterizing the indices of Shapley-Shubik and Banzhaf are shared by these new indices.