Q:

Reference request for a generalization of the Johnston power index

Background: Recall that the Johnston power index is a variation of the more well-known Banzhaf power index. While the Banzhaf power allocates 1 point to each time a player is in a winning coalition, the Johnston power index weighs by the size of each coalition, allocating $1/k$ points to that player where $k$ is the number of players in that coalition. The essential logic of the Johnston index is that it takes into account that larger coalitions are more likely to break down and thus should be worth less. 
But, one doesn't need to assign exactly $1/k$, one could use another function of $k$. Even more generally, one might want to also include how many players are critical in a given coalition (since a critical player dropping out matters in some sense more than a non-critical player dropping out). In that context, define a generalized Johnston power index with function $f(c,k)$ to be the same as the Johnston index but one instead for each player when they are critical in a winning coalition one assigns to that player $f(c,k)$ points where $c$ is the number of critical player and $k$ is the number of total players in that coalition. For a few years now, I've done a few problems with students involving these generalized Johnston power indices. But it occurred to me that I'm not actually aware of any source or citation for them. Does somewhere know of somewhere where this idea is written up, possibly under another name? 

A:

I am afraid you have been misleading your students slightly - for the Johnston index, it should be divided not by the total number of players in the coalition, but by the number of critical players - which, as you say, is a very logical thing to be concerned about.
There is certainly room for generalization of it, and I don't believe I've seen a specific generalization of the Johnston power index described.
There are a few papers that describe generalized classes of power indices, but generalizations tend to focus on the game theoretic framework (applicability to settings other than voting).

