In many voting procedures the rules are one person, one vote, and a simple majority is required to elect a candidate or to pass a motion. But it is not unusual to have a procedure where individual voters have more than one vote or where something other than a simple majority is required. An example of such a situation is when not all shareholders in a company own the same number of shares, and each shareholder has as many votes as shares. Does a shareholder with twice as many shares as another have twice as much control, or power, over the company? In this experiment you will investigate this question and some related ones. First, we begin with some definitions. The number of votes that a voter has is called the voter's weight. Here only counting numbers can be weights. The total number of votes needed to elect a candidate or to pass a motion is the quota. The collection of the quota and the individual weights for all voters is called a weighted voting system. If the voters are designated v1, v2, ..., vk with corresponding weights w1, w2, ..., wk and q is the quota, then the weighted voting system may be conveniently represented by [q: w1, w2, ..., wk]. For ease of computations, the weights are usually listed from largest to smallest.

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 v1's power relative to the other voters.
More interesting cases arise when the power of each voter is not so obvious as in these first examples. One way to measure a voter's power was developed by John Banzhaf in 1965. A coalition is a subset of the voters in a weighted voting system. If the total number of votes controlled by the members of the coalition equals or exceeds the quota, we call the coalition a winning coalition. If not, this is a losing coalition.

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].
Banzhaf's idea is to measure a voter's 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 voter's Banzhaf power index.

6. Here is a test for Banzhaf's 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.