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 can happen 5 choose 2 ways = 10 ways w = 6: A can happen 5 choose 3 ways = 10 ways w = 7: A 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 = 50. qq+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 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 = 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 here A is not pivotal Case 2: 1 A again, A not pivotal Case 3: 1 1 A now A is pivotal  Case 4: A A is pivotal  Case 5: A 1 - A is pivotal  Case 6: 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%).