The Shapley–Shubik Power Index Another index used to measure the power of voters i...

The Shapley–Shubik Power Index

Another index used to measure the power of voters is called the Shapley–Shubik power index. It was introduced in 1954 by Lloyd Shapley and Martin Shubik. The Shapley–Shubik power index is based on the idea that voters join a coalition one by one. A coalition becomes a winning coalition when one voter’s weight first makes the coalition a winner. That voter is the pivotal voter in that particular winning coalition.

The Shapley–Shubik power index for each voter is found by considering all possible permutations, or all possible ordered coalitions, of the set of n voters (there are n! of them) and noting, in each ordered coalition, which voter is the pivotal voter. Consider three voters: P₁, P₂, and P₃. The 3! = 6 possible ways to put those three voters in order are {P₁, P₂, P₃}, {P₁, P₃, P₂}, {P₂, P₁, P₃}, {P₂, P₃, P₁}, {P₃, P₁, P₂}, {P₃, P₂, P₁}. The next figure illustrates the process of forming all possible sequential coalitions by picking the first voter, then adding the second voter to the coalition, and finally adding the third voter.

Suppose the three voters in our example are in the weighted voting system [12 | 7, 6, 5]. The next figure again shows the orders in which voters join each coalition, but the figure also includes the weights of the voters. As soon as a voter joins a coalition making the total weight greater than or equal to 12, the coalition becomes a winning coalition. The most recently added voter at that point is the pivotal voter for that coalition. The pivotal voters for each of the six ordered coalitions are circled below. Notice there is only one pivotal voter in every permutation of voters. We can find the Shapley–Shubik power index for each voter by calculating

To find the Shapley–Shubik power index for each of n voters, do the following:

STEP 1: Make a list of all possible sequential coalitions of the n voters. Remember there will be n! permutations.

STEP 2: In each coalition, consider the weights of the voters in order as they enter the coalition. Determine the first voter who, when joining the coalition, changes it from a losing coalition to a winning coalition; that is, determine who is the pivotal voter in each case.

STEP 3: Count the total number of times each voter is pivotal.

STEP 4: For each voter, find the Shapley–Shubik power index by dividing the number of times the voter is pivotal by n!.

A college committee consists of four members: a director, a coordinator, a full-time faculty member, and a part-time faculty member. The committee must vote on several proposals. The weighted voting system is [9 | 5, 4, 3, 2].

a. Calculate the Banzhaf power index for each member.

b. Calculate the Shapley–Shubik power index for each member.