ToC | Next Labs: Voting and Social Choice. Part 2. Math Alive

## Weighted Voting Systems

A weighted voting system is one in which the participants have varying numbers of votes. One of the most common examples of a weighted voting system is the U.S. Electoral College. Under the Electoral College system, the number of votes for each state is based upon that state's population. California, one of the most populous states, can cast 54 electoral votes while Alaska may cast only 3 votes.

The "power'' of a participant in such a weighted voting system can be roughly defined as the ability of that participant to influence a decision. As our measurement of this power we will use the Banzhaf power index, which is a much more accurate measure of a participant's power than the number of votes that the participant can cast.

A participant's Banzhaf power index is the number of distinct coalitions in which the participant is a swing vote. A voter is a swing vote whenever the outcome of the election can change if the participant switches votes. Let's look at the following example:

[3: 2, 1, 1]

The first number represents the number of votes needed to win. The second, third and fourth numbers represent the number of votes that candidates one, two and three can cast, respectively. In order to calculate the Banzhaf power index for each voter, let's look at each coalition up close.

y(s)yy4Win
y(s)y(s)n3Win
y(s)ny(s)3Win
yn(s)n(s)2Lose
n(s)yy2Lose
n(s)yn1Lose
n(s)ny1Lose
nnn0Lose

In this table, y stands for a "Yes" vote, "n" stands for "No", "Win" means that the proposition wins, and "Lose" that it loses. Swing votes are marked by placing (s) after "y" or "n" vote. Recall that the proposition needs a majority (i.e. 3 votes) to win. Let's look at the first voting coalition. All the participants have voted Yes, giving a total of 4 votes and a winning outcome. If we change the first participant's vote from Yes to No, then we are left with only 2 votes and a losing outcome. Since participant one has the power to change the outcome by switching his vote, he is a swing vote in this coalition, and his Banzhaf power index increments by 1. If we change the vote of the second participant, we will still have 3 votes and a winning coalition. The same thing occurs if we change the third participant's vote. Therefore, neither participants 2 nor 3 are swing votes, and their Banzhaf power index does not change.

Now let's look at the second coalition. The first two participants vote Yes, while the third participant voted No. This gives a total of 3 votes and a winning outcome. Once again, participant one is a swing vote, because the loss of his two votes leaves only 1 vote and a losing coalition. If we switch participant two's vote, we have only 2 votes and a losing coalition, so participant two is a swing vote. If we switch participant three's vote, we have 4 votes and a winning coalition; the outcome has not changed. Therefore, participant three is not a swing vote.

If we continue this process, we find that participant one is a swing vote in six coalitions, while participants two and three are each swing votes in two coalitions. Our Banzhaf power index is therefore (6, 2, 2). It is important to notice that even though participant one has only 1 more vote than participants two and three, he has three times as much power in Banzhaf's sense of power.

(Note that in this lab we just count the number of coalitions in which each voter is a swing voter. Unlike the standard Banzhaff power index, as defined in class, we do NOT divide by the total number of coalitions. Our main reason is that this gives cleaner numbers - if we don't divide, we always have integers. If you'd like to find the decimals given in class, you can just divide by the total number of coalitions, which is the power of 2 corresponding to the number of voters. But in any case, the important aspect of these power indices is not really their value per se, but how they compare with each other. For instance, whether we write [3:2,1,1]=(6,2,2) as we do here, or [3:2,1,1]=(.75,.25,.25) as in class (where we divided by 8, the total number of coalitions), the important thing is that the first voter has three times as great a power index as compared to the other two, something that is clear in both conventions.)

You are now ready to calculate the Banzhaf power index of several weighted voting systems.