**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 v*oting 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,

__ Probability
of Being a Critical Voter__.
Suppose

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.