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.

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.

q q+A-1

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: A 1 1 1 1 1 - here A is not pivotal• Case 2: 1 A 1 1 1 1 - again, A not pivotal• Case 3: 1 1 A 1 1 1 - now A is pivotal • Case 4: 1 1 1 A 1 1 - A is pivotal • Case 5: 1 1 1 1 A 1 - A is pivotal • Case 6: 1 1 1 1 1 A - A is not pivotal (5 votes reached before A joins)

• Conclusion: A is pivotal 3 out of 6 times, therefore has a Shapley-Shubik 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%).