1. For the weighted voting system [9 : 9, 4, 2, 1], what is the quota? How many voters are there? What is the total number of votes available? 2. In a weighted voting system [q : w1, w2, , wk], what are the restrictions on the possible values of q? Explain each restriction. 3. For the weighted voting system [9 : 9, 4, 2, 1], describe how much power voter v1 has. Such a voter is called a dictator. Why is this appropriate? Could a system have two dictators? Explain why or why not. 4. For [8 : 5, 3, 2, 1], is v1 a dictator? Describe v1s power relative to the other voters. 5. (a) List all the coalitions for [9 : 9, 4, 2, 1]. Which of these are winning coalitions? (b) List all the winning coalitions for [8 : 5, 3, 2, 1].

Banzhafs idea is to measure a voters power by examining how many times removal of that voter from a coalition would change the coalition from winning to losing. Consider the system [7 : 5, 4, 3]. The winning coalitions are {v1, v2}, {v1, v3}, {v2, v3}, and {v1, v2, v3}. Each member of the first three coalitions has the power to change it from winning to losing, but none have this power in the last coalition. All together there are six opportunities for change. Each of v1, v2, v3 has two of these opportunities. We record this information as the Banzhaf power distribution for the system: v1: 2 over 6 , v2: 2 over 6 , v3: 2 over 6 . According to this analysis, all three voters have the same amount of power despite having different weights. The fraction of power assigned to a voter is the voters Banzhaf power index.

6. Here is a test for Banzhafs definition of power. Calculate the Banzhaf power distribution for [9 : 9, 4, 2, 1]. Explain how the results are consistent with the designation of v1 as a dictator.