Voting and Elections

VI. Coalitions and the Banzhaf Power Index

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:
Here are a collection of definitions we will need in order to discuss weighted voting systems.
We'll use the notation [q:w1, w2, w3, ...,wn-1, wn] as a shorthand way of describing a weighted voting system with weights w1, w2, w3, ...,wn-1, wn 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. 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.