Q1. Consider the weighted voting system: [11 : 5,5,4,4]?

a. List the winning coalitions of the system. Hint: there are five.

b. Construct a different weighted voting system which is isomorphic to the original system, and in which v₁ and v₂ have different weights from one another. Show the winning coalitions to demonstrate that the systems are isomorphic. Hint: try changing one of the two weights and seeing what happens.

Q2. Recall the Roman "Centuriate Assembly" from problem 2 on the previous homework could be represented by the weighted voting system [97 : 80,30,20,20,20,18,5], where the voters (in order of largest to smallest weight) were the first enlisted class, second enlisted class, third enlisted class, fourth enlisted class, fifth enlisted class, officer class, and unarmed adjuncts?

a. Writing out all of the winning coalitions is rather involved, but one can find that the first enlisted class is critical 120 times, that the unarmed adjuncts are critical 0 times, and that the other five classes are critical 8 times each. Find the Banzhaf index for each of the seven voters?

b. Writing out all of the arrangements is also rather tedious, but one can find that the first enlisted class is pivotal 3360 times, that the unarmed adjuncts are pivotal 0 times, that the other five classes are pivotal the remaining number of times and that they are all pivotal an equal number of times. Find the Shapley-Shubik index for each of the seven voters.

c. Now consider the second scenario from the previous homework, in which the officers and first enlisted class acted as a single voter and so the weighted voting system became [97 : 98,30,20,20,20,5]. Find both the Banzhaf indices and Shapley-Shubik indices for all voters. Hint: What kind of voter is the officer/first enlisted class in this system?

Q3. Consider a weighted voting system containing three voters, in which the circled voters are pivotal in each

arrangement: (v₁,v₂,v₃)

(v₁, v₃,v₂)

Assume the usual organization of voters (so that v₁ has the largest, v₂ has the second-largest, and v₃ the smallest).

a. How many additional missing (i.e. unknown) arrangements are there in this voting system?

b. Which voter is pivotal in the arrangement (v₃, v₁, v₂)? How do you know?

c. Explain why we know that v₁ is pivotal in the arrangement (v₂, v_3,, v₁), even without more information about the system.

d. Finish the necessary computations to find the Shapley-Shubik index for each of the three voters.