Chapter 2, Problem 79E

a. Give an example of a weighted voting system with four players and such that the Shapley-Shubik power index of $P_1$ is $\frac{3}{4}$ .

b. Show that in any weighted voting system with four players a player cannot have a Shapley-Shubik power index of more than $\frac{3}{4}$ unless he or she is a dictator.

c. Show that in any weighted voting system with N players a player cannot have a Shapley-Shubik power index of more than $\left(N-1\right)/N$ unless he or she is a dictator.

d. Give an example of a weighted voting system with N players and such that $P_1$ has a Shapley-Shubik power index of $\left(N-1\right)/N$.