Weighted voting system most naturally arise in voting among shareholders, were each voter controls a number of votes (the number of shares they control). These voters use this system to make a decision on a yes/no question, or motion. Weighted voting systems are characterized by the following:

• a collection of voters-These n voters we label v₁, v₂, v₃, ...,v_n-1, v_n.
• a collection of weights-We associate with each voter a positive number called the voter's weight, which is understood to be the number of votes held by that voter. We will use the notations w_i to represent the weight of voter v_i.
• a quota-The quota for a weighted voting system is a positive number q such that a motion will pass if the sum of te weights of all the voters who vote "yes" on the motion equals of exceeds q, and will fail other wise.

Here are a collection of definitions we will need in order to discuss weighted voting systems.

• a coalition is a colletion of voters (possibly empty) in a weighted voting system, with any number of members ranging from no voters to all the voters in the system.
• a weight of a coalition is the sum of the weights of all the voters in the coalition.
• a winning coalition is a coalition that can single handedly force a motion to pass: that is, if every member of the coaltion were to vote in favor of passing a motion, then the motion would pass, regardless of how any of the other voters (those not belonging to the coalition) voted.
• a losing coalitions is a coalition that cannot single handedly force a motion to pass; that is, even f every member of the coalition were to vote in favor of passing a motion, the motion could still fail.
• a minimal winning coaltion is a winning coalition that would become a losing coalition if any single voter were removed from it.

We'll use the notation [q:w₁, w₂, w₃, ...,w_n-1, w_n] as a shorthand way of describing a weighted voting system with weights w₁, w₂, w₃, ...,w_n-1, w_n and quota q.

Problem Set 11 :Suppose voter v₁ (say Doug) proposed a motion in the weighted voting system [101:101, 97,2], which combinations of voters could force Doug's motion to pass by voting in favor of it? With this quota, what distinguishes Doug from the other voters?

If the quote was changed to 103 (so we would have [103:101,97,2]), which combinations of voters could force Doug's motion to pass by voting in favor of it? What distinguishes Doug from the other voters in this case?

If the quote was changed to 105 (so we would have [105:101,97,2]), which combinations of voters could force Doug's motion to pass by voting in favor of it? What distinguishes Doug from other voters in the system? What distinguishes v₃ (say Elisabeth) from the other voters? This series of questions shows that in some cases, two different weighted voting systems can produce the exact same winning coalitions. When this occurs, it is natural to say that the two systems are essentially the same, or as mathematician like to say, we call the two systems isomorphic. For each of the following weighted voting systems, list all of the winning coalitions. Then decide which of the systems are isomorphic to each other.

1. [4:2,2,1]
2. [4:3,2,1]
3. [5:3,2,1]
4. [5:3,2,2]
5. [5:3,3,2]

End of Problem Set 11

In the first three questions in the last section we made some important observations leading us to some new concepts. Namely, when the quota was 101 Doug was a dictator, meaning that he was present in every winning coalition and absent from every losing coalitions. When the quota was 103, Doug was only present in every winning coalition, we say he had veto power. When the quota was 105, we saw that Elisabeth was absent from every minimal winning coalition, we could call her a dummy, meaning that she plays no real role in the election.

The United Nations Security Council consist of fifteen different representative countries. Five representatives (from China, France, Great Britain, Russia, and the United States) are considered permanent members, and the remaining ten change from year to year, with nonpermanent members serving two-year terms. Passage of a motion requires votes in favor from all five permanent members and at least four of the nonpermanent members. This can be thought of as the weighted voting system [39:7,7,7,7,7,1,1,1,1,1,1,1,1,1,1] Each of the permanent members by definition have veto power, and each of the nonpermanent members.

Clearly the permanent members of the UNSC have much more power than the nonpermanent members (it was after all designed this way), but how can we measure quantitatively how much more power. Well, there are methods of doing so called power indices. We will study one such power index (to learn another, one can refer to [Hodge, 2005] in chapter seven).
John F. Banzhaf III believed that in a (yes/no) voting system a particular voter is more powerful that another if that voter's membership in winnin coalitions is more frequently essential, or critical, to keep the coalitions from being losing coalitions. To make this more exact, the following are defined for all yes/no voting systems.

• A voter in a winning coalition is said to be critical if the voter's withdrawal from the coalition would cause it to become a losing coalition.
• The Banzhaf power of a voter is the number of winning coalitions in which the voter is critical
• The total Banzhaf power of a system is the sum of the Banzhaf powers of all the voters in the system.
• The Banzhaf index of a voter is the Banzhaf power of the voter divided by the total Banzhaf power of the system.

Let us work though a simple example with [103:101,97,2]. Now the second voter has nearly 48 times higher weight that the third, but do they yield more power? The second voter is in only one winning coaltion in which they are critical (namely the first and second voter). The third voter is also in only one winning coalition in which they are critical (namely the first and third voters). Then, the first voter is in three winning coalitions in which they are critical (write them out). So the Banzhaf index of voter one is 3/5, voter two's Banzhaf index is 1/5, and voter three's Banzhaf index is 1/5. Meaning voter two and three yield the same amount of power in this weighted voting system, makes since right?

Problem Set 12 :The final problem is perhaps the most challenging (you may need to learn or recall some things that can be found in the first lesson on probability, here's a quick link). Calculate the Banzhaf power index of a permanent member and a nonpermanent member. How much more power does a permanent member have?

End of Problem Set 12

References and Further Reading:
[1]Jonathan K. Hodge and Richard E. Klima.The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, R.I., 2005.