Exercise 43

Question 11.73

43. Use the Shapley-Shubik model to compare the share of power of a junior member in the systems described in Exercises 41 and 42.

43.

Use , to denote a senior member (either or ) and to represent junior members. In the committee with three junior members, is pivotal in the following permutations:

(12 permutations)
( permutations)
(also 72 permutations)
( permutations)

Thus, the Shapley-Shubik power index of is . By the Shapley-Shubik model, the three senior members hold between them of the power. The remaining 15% is divided between the junior members, so each has a Shapley-Shubik power index of 5%.

With four junior members, would be pivotal in these permutations:

(48 permutations)
( permutations)
( permutations)
(384 permutations)
( permutations)

This comes to a total of 1296 permutations. Dividing by , we obtain the Shapley-Shubik power index of , 25.714%. The three senior members have 77.142% of the power in the Shapley-Shubik model. The remaining 22.858% belongs to the junior members, so each has 5.714% of the voting power in this model. Thus, by including another junior member, the power of each junior member has slightly increased.