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!.

Sometimes the candidate receiving the plurality of the popular vote for U.S. President does not become the President. This is due to the way votes are cast by the electoral college. The number of electors allotted to each state is the same as the combined number of seats in the House of Representatives and the Senate for that state. There are a total of 538 electoral college votes. The quota is a majority of the electors, which is 270 votes. The following table contains the numbers of electoral college votes allocated

to each state and the District of Columbia for the decade after the 1990 census and the decade after the 2000 census.

Calculating the Banzhaf power index or the Shapley–Shubik power index for each state is impossible to do by hand. The elector distribution in 1990 had over 51 trillion winning coalitions! Fortunately, power index calculators are available on the Internet. One such calculator can be found at www.math.temple.edu/~cow/bpi.html. This site will calculate the Banzhaf power index values and is supported by the National Science Foundation and Temple University. Another calculator will give Shapley–Shubik power index values and can be found at http://www.misojiro.t.u-tokyo.ac.jp/~tomomi/cgi-bin/vpower/index-e.cgi. This site is supported by Tomomi Matsui from the University of Tokyo, Japan.

a. Calculate the Banzhaf power index for each state’s electors based on the 1990 census. Use an Internet power index calculator.

b. Calculate the Banzhaf power index for each state’s electors based on the 2000 census. Use an Internet power index calculator. Compare the results from parts (a) and (b).

c. Calculate the Shapley–Shubik power index for each state’s electors based on the 1990 census. Use an Internet power index calculator.

d. Calculate the Shapley–Shubik power index for each state’s electors based on the 2000 census. Use an Internet power index calculator. Compare the results from parts (c) and (d).