EXAMPLE 23 Power Indices of the Scholarship Committee

The dean has veto power. Therefore, she will be the pivotal voter in any permutation where she appears last. If she is second to last in a permutation, she will still be the pivotal voter, because among the three voters before her, there will be either both professors or both financial-aid officers. In the middle position, she will be pivotal if and only if both professors or both aid officers come first. Adding this up, we have permutations in which the dean is in fourth or fifth position. There are four permutations of the form Prof, Prof, Dean, Aid, Aid, because the professors and the aid officers can be in either order, and another four of the form Aid, Aid, Dean, Prof, Prof. The dean is not the pivotal voter when she is first or second because at least three people have to approve a scholarship. We conclude that the dean is pivotal in permutations in all. Her Shapley-Shubik power index is therefore . Each of the other participants is equally powerful, and they share the remaining of the power. Thus each professor and each aid officer has a Shapley-Shubik power index of .

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To compute the Banzhaf index, we will make a list of winning coalitions. There are seven of them:

The dean has veto power, so she is a critical voter in each of them. Her Banzhaf power index is therefore 7.

Professor is a critical voter in , , and . His Banzhaf power index is 3. The remaining participants have the same power.