Calculating Power in Larger Systems
Nominal, Banzhaf and
Shapley-Shubik
A Special Case
• We will consider only one special case here:
• All voters will have the same weight except one
with a larger weight.
• This technique is intended to be applied in the
special case when there are 4 or more voters
and listing every case would be more lengthy.
Given [ 5: 3,1,1,1,1,1]
• Here we have 6 voters and a quota of q = 5.
• Calculate the nominal power, write as a percent.
• (3/8, 1/8, 1/8, 1/8, 1/8, 1/8)
= ( 38%, 12%, 12%, 12%, 12%, 12%).
Given [ 5: 3,1,1,1,1,1]
• Calculate Banzhaf measure of power, write in percent.
• Name the voters A, B, C, D, E, F , and do power for A first.
• A is critical when weight of coalition to which A belongs is 5 to 7.
q
q+A-1
w = 5: A + 1 + 1 can happen 5 choose 2 ways = 10 ways
w = 6: A + 1 + 1 + 1 can happen 5 choose 3 ways = 10 ways
w = 7: A + 1 + 1 + 1 + 1 can happen 5 choose 4 ways = 5 ways
• There are 25 ways that A could be critical to a winning coalition, and
therefore 25 ways that A could be critical to a blocking coalition.
• So power Banzhaf power index for A is 25+25 = 50.
Given [ 5: 3,1,1,1,1,1]
• Next we calculate power for B only. The result will be the same for
voters C, D, E and F…
• B will be critical when belonging to a coalition with weight 5 only.
w = 5: B + 1 + 1 + 1 + 1 can happen 4 choose 4 which is only 1 way.
w = 5: B + A + 1 can happen 4 choose 1 = 4 ways.
• Therefore B can be critical to win in 5 ways (and critical to block in 5
ways) and therefore has Banzhaf index 5 + 5 = 10.
• So the Banzhaf index for this system is (50, 10, 10, 10, 10, 10) and
writing the answer in percent form, we get
(50%, 10%, 10%, 10%, 10%, 10%).
Given [ 5: 3,1,1,1,1,1]
• Calculate the Shapley-Shubik power index, write answer in percent
form.
• For Shapley-Shubik, the calculation for each voter is the number of
times that voter is pivotal out of the total number of permutations.
Therefore, when we have a large system with the special case that
all voters are the same except one voter, we can take an additional
shortcut in the calculations.
• We only need to do one voter. For example, if we get the power of A
(the voter with 3 votes), the remaining power is shared among the
others and so we just divide the remaining power among the others –
rather than compute the number of times the others are pivotal. It’s a
little faster than doing the Banzhaf index for the large system – in this
special case.
Given [ 5: 3,1,1,1,1,1]
• Calculate the Shapley-Shubik power index, write answer in percent
form.
• We calculate the power index for A – the voter with 3 votes.
• Another shortcut is that for each of the 6 positions (because there are
6 voters total) in the list of voters that A could occupy as we list all
permutations, we have the same number of permutations (it would be
5! = 120 in this case) of all the other identical voters. So we don’t
really have to count that because it will remain the same throughout
the calculation.
• We just count in which of the 6 positions would A be pivotal?
Given [ 5: 3,1,1,1,1,1]
• We just count - in which of the 6 positions would A be pivotal?
•
•
•
•
•
•
Case 1:
Case 2:
Case 3:
Case 4:
Case 5:
Case 6:
A11111
1A1111
11A111
111A11
1111A1
11111A
- here A is not pivotal
- again, A not pivotal
- now A is pivotal 
- A is pivotal 
- A is pivotal 
- A is not pivotal (5 votes reached before A joins)
• Conclusion: A is pivotal 3 out of 6 times, therefore has a ShapleyShubik power index of 3/6 which reduces to ½ and is equal 50%.
• Next, we conclude that 50% of the power remains for the other
voters, and because there are 5 (with equal votes) who will share it
equally we get each has power 50%/5 = 10%.
Given [ 5: 3,1,1,1,1,1]
• So the Shapley-Shubik power index is (50%, 10%, 10%, 10%, 10%).
• Now we can summarize:
• Given the above weighted voting system, the measure of power
among the voters is (depending on the method used)
• Nominal power:
( 38%, 12%, 12%, 12%, 12%, 12%).
• Banzhaf power:
(50%, 10%, 10%, 10%, 10%, 10%).
• Shapley-Shubik power:
(50%, 10%, 10%, 10%, 10%, 10%).