Exercise 7

Question 11.37

7. Voters $A, B, C, D$ use the weighted voting system $[51 : 30, 25, 24, 21]$ .

List all permutations in which $A$ is pivotal.
List all permutations in which $B$ is pivotal.
Calculate the Shapley–Shubik power index of the system.

(a) $A$ is pivotal in these permutations: $B A C D$ , $B A D C$ , $B C A D$ , $B D A C$ , $C A B D$ , $C A D B$ , $C B A D$ , $C D A B$ , $D A B C$ , $D A C B$ , $D B A C$ , and $D C A B$ .

(b) $B$ is pivotal in these permutations: $A B C D$ , $A B D C$ , $C D B A$ , and $D C B A$ .

(c) The Shapley-Shubik index of $A$ , which is pivot in 12 permutations, is $\frac{12}{4!} = \frac{1}{2}$ . Each of the other voters has four pivots and a Shapley-Shubik power index of $\frac{1}{6}$ .