EXAMPLE 11 A Dictator

Consider the five-voter system . We’ll name the voters and , where stands for dictator. A set of voters is a winning coalition if and only if belongs to it. We can find the number of subsets of by applying the multiplication principle. Construct a subset by deciding if is in or out, two possibilities, and do the same for , , and . The number of subsets is . All of the winning coalitions consist of combined with one of these 16 subsets. in each winning coalition, is the only critical voter. Thus, has a Banzhaf index of 16, and I each of the other participants has a Banzhaf index of 0.