Banzhaf Power Index and Shapley-Shubik Power Indices
Brief Introduction (For a more complete explanation, see For All Practical Purposes, 10th Edition, New York, WH Freeman 2015, Chapter 11)
A weighted voting system is a decision-making device with participants, called voters, who are asked to decide upon questions by “yea” or “nay” votes. Each voter is assigned a voting weight. The question is decided in the affirmative if the sum of the voting weights is greater than a specified quota. For a simple example consider three voters, A, B, and C. Their voting weights are 2, 1, and 1, respectively. The quota must be more than 2 and not more than 4: if more than 4, no question would be affirmed; if 2 or less, contradictory questions might be affirmed.
A winning coalition is a set of voters whose voting weights add up to more than the quota. In the example just mentioned, with quota of 3, the winning coalitions are {A,B}, {A,C}, and the grand coalition {A,B,C}, consisting of all three voters. If the quota is 4, then the grand coalition is the only winning coalition—to be affirmed, a question must be supported unanimously.
Banzhaf Power Index
Consider a winning coalition S, and a voter X who belongs to S. Let q be the quota, w(S) be the total weight of S, and w(X) be the voting weight of X. Since S is a winning coalition, w(S) is at least q. The voter X is a critical voter in S if w(S) – w(X) < q. In this case X is essential to the coalition: if X decides to vote “nay” the total weight of what remains of S will be less that the quota, so the question will fail.
The Banzhaf Power Index of a voter X is the number of winning coalitions that X belongs to and in which X is critical. In our example, A is critical in all three winning coalitions, so the Banzhaf Power Index of A is 3. B and C are not critical voters in the grand coalition, which has total weight 4; with voting weight of only 1, if one of them deserts, the remaining two voters would still form a winning coalition. Thus B is only critical in the winning coalition {A,B}, and hence the Banzhaf Power Index of B is 1. C has the same Banzhaf power index.
Banzhaf Share of Power. Consider the sum of the Banzhaf Power Indices of all the voters. (In our example, the sum is 5.) If we divide each voter's Banzhaf Power Index by the sum, we get that voter's share of the voting power, according to the Banzhaf model. In our example, A has 60% of the power; B and C each have 20%. The sum of the Banzhaf Shares of Power is 100%.
Probability of Being a Critical Voter. Suppose X wants to know her chances of being a critical voter when a question Q is under consideration. (She plans to vote “yea.”) In a real-world situation, she would consider what's going on in the minds of her fellow voters, but we can't do that. The goal is to measure voting power, as it is built into the system. For example, if X finds out that her Banzhaf Power Index is 0, how the other voters feel about Q makes no difference to X: she will never be a critical voter in any winning coalition, and might as well not bother to vote. We therefore make an unrealistic assumption, from the point of view of real-world politics, that the other voters toss coins to determine their votes. If there are n voters (including X), there are 2n-1 coalitions of voters that include X, and vote “yea.” Some of these coalitions are losing coalitions, others are winning. Under the coin-tossing assumption, all are equally likely, and the number of these that are winning, and in which X is a critical voter, is the Banzhaf Power Index of X. Therefore, the probability that X will be a critical voter in a winning coalition is her Banzhaf Power Index, divided by 2n-1. In our example, 2n-1 = 4, so the probability that A will cast a critical vote is 75%, while B and C each have as 25% chance of being a critical voter. These probabilities will not add up to 100%.
Imagine a real-life question, like a tax cut, that is to be decided by a weighted voting system. Some would like to eliminate taxes altogether, some would like the tax rate to be negative, others would like to raise taxes to eliminate a deficit. We could line the voters up in order by their opinions: the amount that taxes they would favor would determine their place in line. Then the tax bill could be crafted so as to capture just enough votes to pass, by moving down the line until a voter P is reached, whose voting weight, when combined with the weights of those behind him, is just enough to bring the total in favor to cover the quota. The line of voters is called a permutation, and the voter P is called the pivot of the permutation.
The Shapley-Shubik Power Index of a voter is the number of permutations of the set of voters in which he or she is the pivot, divided by the number of permutations that are possible. If there are n voters, there are
n! = 1 x 2 x 3 x … x n
permutations. To see how this works with our example, consider the voter A, with voting weight 2. She is pivot if she is second or third in a permutation. There are 4 such permutations: BAC, CAB, BCA, and CBA, and since 3! = 6, the Shapley-Shubik Power Index of A is 4/6 = 2/3. B and C share the remaining two permutations, so each has Shapley-Shubik power index equal to 1/6.
Counting Problems
To calculate these power indices is a counting problem. Such problems can be trivial when the number are small, but how about the U.S. Electoral College, in which the states vote for the president of the United States, using weighted voting? Each state's weight is equal to the number of members of the Senate and the House of Representatives representing it; in addition the District of Columbia has 3 votes. This brings the total voting weight to 538, and the quota is 270. There are 56 participants (including five congressional districts that vote independently) with weights ranging from 1 for a congressional district to 55. Counting coalitions or permutations one at a time is not feasible. A computer algorithm is available, and you can find it by following the link shown below. The program allows entry of up to 24 different voting weights. In our example there were only two: 1, which we count as having multiplicity 2—because two voters have that weight—and 2, which has multiplicity 1. If you enter a big system, you can expect the computer to take some time. For the Electoral College, enter the 21 different weights with their multiplicities, put the quota at 270, click “Submit,”and go get a coffee. When you return, the results of a lot of counting will be on display.
Before you take off and use the calculator please consider this: if a person needs a calculator to find 8% tax on a $2.00 purchase, perhaps they don't understand the principles. This calculator will determine the Power Indices for the simple example that was mentioned above, but it should not be necessary. When doing textbook problems, do not use the Power Index Calculator unless the text says you should do so. Of course, it is all right to use the calculator to confirm a result that you have obtained with pencil and paper.