Chapter 2, Problem 54E

A law firm has six partners: a senior partner $\left(P_1\right)$ with 5 votes and five junior partners $\left(P_2\text{ through }{P}_7\right)$ with 1 vote each, the quota is a simple majority of the votes. (This law firm operates as the weighted voting system [6: 5, 1, 1, 1, 1, 1].)

a. In how many sequential coalitions is the senior partner $P_1$ the pivotal player? (Hint: First note that $P_1$ is the pivotal player in all sequential coalitions except those in which he is the first player.)

b. Using your answer in (a), find the Shapley-Shubik power index of the senior partner $P_1$.

c. Using your answer in (b), find the Shapley-Shubik power distribution on in this law firm.