Weighted Voting
Examples with larger systems
• Banzhaf power index
• Shapley-Shubik power index
Important Assumption
• Because these weighted voting systems are larger (5, 6
or maybe 7 voters) we can not easily calculate the power
index by the definition. This is because the total number
of coalitions and permutations becomes too large to
handle without a computer.
• For example, with 5 voters, to calculate the ShapleyShubik index from the definition would require a table
with 120 rows. To handle this, we would need a
computer with a program like Excel.
Important Assumption
• In systems with larger numbers of voters, we will calculate the
Banzhaf and Shapley-Shubik power index using counting techniques
rather than listing every coalition or every permutation. Essentially,
this amounts to short-cuts to listing out every single coalition or
permutation.
• There are two goals: (1) Simply to be able to calculate the index for
larger systems and (2) Develop skill in logic, problem solving and
using advanced counting techniques.
• The short-cut counting technique we’ll use depends on the following
assumption:
All of the voters will have the same weight except one voter.
– for example, the system [5: 3, 1, 1, 1, 1]
• Therefore, even though the systems are big, they are not too complex
and we can reason through the possible combinations and count all of
the possible coalitions and permutations using counting techniques.
Important Assumption
• The following is a fact that we could prove to be
true deductively…
If voters have the same weight, then they will have
the same power (in either the Banzhaf or
Shapley-Shubik measure).
• Don’t forget, the converse is not true: If voters
have the same power that does not mean they will
have the same weight.
Example #1
• Consider the system: [ 7: 4, 2, 2, 2, 2 ]
• Let’s calculate the Banzhaf power index first…
Label the voters A, B, C, D, E, from left to right, so A has weight 4.
We will consider A’s power first…
A will be critical when a member of a coalition of weight w that satisfies
the inequality 7 < w < 10. (Here we can always use the quota as the
left endpoint and the value q+wi – 1, where wi is the weight of voter i,
as the right endpoint.)
We consider each case:
If w = 7: This can not happen given the weights of the voters.
If w = 8: This can happen when A is joined by any two of the others,
thus 4C2 = 6 ways.
If w = 9: This can not happen – no combination sums to 9.
If w = 10: This can happen when A is joined by any three of the others,
thus 4C3 = 4 ways.
Therefore, A can be critical 6 + 4 = 10 ways, and has a Banzhaf power
of 10x2 = 20.
Example #1
•
Continuing with the same system: [ 7: 4, 2, 2, 2, 2 ]
•
Now, let’s calculate the power for B, C, D and E, which will all be the same,
since they all have the same weight…
•
So let’s just do the power for B.
•
B will be critical when belonging to a coalition of weight w that satisfies 7 < w
< 8. (Again, we can use the quota as the left endpoint and the value
q + wi – 1, where wi is the weight of voter i, as the right endpoint.)
•
•
•
So check how many ways each case can occur…
W = 7: this can not occur because no combination of voters will add to 7.
W = 8: this can happen when B is joined by the other 3 voters with weight 2,
which can happen 3C3 = 1 way.
W = 8: or it can happen when B is joined by A and one of the voters C, D, or
E which can happen 3C1= 3 ways.
Thus, B can be critical in 1 + 3 = 4 ways, so has power 4x2 = 8.
•
•
Example #1
• Conclusion – we were given the system: [ 7: 4, 2, 2, 2, 2 ]
• Now we have what we need to write the Banzhaf power index.
• The Banzhaf power for this system is… (20,8,8,8,8).
• To write in terms of percentages, we would get about
(38.5%, 15.4%, 15.4%, 15.4%, 15.4% )
(notice how the percentages don’t add to exactly 100% because of
rounding – different ways to handle this are considered in another
chapter, when we study apportionment)
Example #2
• Suppose we were given the system: [ 7: 4, 2, 2, 2, 2 ]
• Let’s calculate the Shapley-Shubik power index…
• Again, label the voters A, B, C, D, E. We’ll do the power for A first…
• Shapley-Shubik requires counting all permutations of voters and the
power for a voter is the fraction of times that voter is pivotal out of the
total number of permutations.
• We have 5 voters and therefore 5! = 120 total permutations.
• In how many of those 120 permutations is A pivotal?
• To answer that, determine in which of the 5 possible positions would
A be pivotal…
Example #2
• We were given the system: [ 7: 4, 2, 2, 2, 2 ]
• We’ll now determine in which of the 5 possible positions would A be
pivotal…
• In which is A pivotal? Notice each case below represents 16
different possibilities because the 2’s can be permutated in each.
• 4
2
2
2
2
no – not enough to reach 7 when A is first
• 2
4
2
2
2
no – still not enough – only 6 when A joins
• 2
2
4
2
2
yes – now A will be the pivotal voter, making 8
• 2
2
2
4
2
yes – again, A is the pivotal voter
• 2
2
2
2
4
no – A is not pivotal, total is reached at 4th voter
Example #2
• Each line below represents 24 possibilities because each of the 4
voters with weight-2 can be permutated while leaving the weight-4
voter in place. Because there are 5 cases and each has 24
possibilities, this makes for a total of 5x24=120 total permutations
• But we’ve shortened the list to only 5 cases. In how many of these
cases is A pivotal? The answer is 2 out of 5. So A’s power will be
2/5 as measured by the Shapley-Shubik method.
• 4
2
2
2
2
no – not enough to reach 7 when first
• 2
4
2
2
2
no – still not enough – only 6 when A joins
• 2
2
4
2
2
yes – now A will be the pivotal voter, making 8
• 2
2
2
4
2
yes – again, A is the pivotal voter
• 2
2
2
2
4
no – A is not pivotal, total is reach at 4th voter
Example #2
• So far we’ve concluded A’s power will be 2/5 as
measured by the Shapley-Shubik method.
• Therefore all of the remaining 3/5 of the power will
be divided equally among the remaining 4 voters
since each of them has equal weight. Thus each
will have a power index of
3 1 3
 3
 4   
5 4 20
5
Example #2
• Conclusion: The Shapley-Shubik power index for
the system [ 7: 4, 2, 2, 2, 2 ] will be …
(2/5, 3/20, 3/20, 3/20, 3/20)
Or, if writing as percentages…
(40%, 15%, 15%, 15%, 15%)
(Notice how close the answer is to the Banzhaf power
index in this case…)
Example #3
• Don’t forget how to do the nominal measure of power…
• For the system [ 7: 4, 2, 2, 2, 2 ] it will be …
(4/12, 2/12, 2/12, 2/12, 2/12) which reduces to …
(1/3, 1/6, 1/6, 1/6, 1/6) and in percentage form we get
( 33%, 16.7%, 16.7%, 16.7%, 16.7%)
To Summarize
• For the system [ 7: 4, 2, 2, 2, 2 ], when we measure the power of
voters in this system, depending on the method, we get…
Nominal: (33%, 16.7%, 16.7%, 16.7%, 16.7%)
Banzhaf: (38.5%, 15.4%, 15.4%, 15.4%, 15.4% )
Shapley-Shubik: (40%, 15%, 15%, 15%, 15%)
These answers are all very close, but that is partly because these
are simple systems (they are big, but still simple). Remember, we
can have systems that are big and the measure of power could
be very different depending on the method used. An important
point to keep in mind is that the nominal power method is very
limited – it doesn’t show, for example, when a voter has no power
at all in a system – you will see that in some other examples in
the notes.