Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 2 The Mathematics of Power 2.1An Introduction to Weighted Voting 2.2The Banzhaf Power Index 2.3 Applications of the Banzhaf Power Index 2.4The Shapley-Shubik Power Index 2.5Applications of the Shapley-Shubik Power Index

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Calculating the Shapley-Shubik power index of the states in the Electoral College is no easy task. There are 51! sequential coalitions, a number so large (67 digits long) that we don’t even have a name for it. Individually checking all possible sequential coalitions is out of the question, even for the world’s fastest computer. There are, however, some sophisticated mathematical shortcuts that, when coupled with the right kind of software, allow the calculations to be done by an ordinary computer in a matter of seconds (see reference 16 for details). The Electoral College

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Appendix A at the end of this book shows both the Banzhaf and the Shapley-Shubik power indexes for each of the 50 states and the District of Columbia. Comparing the Banzhaf and the Shapley-Shubik power indexes shows that there is a very small difference between the two. This example shows that in some situations the Banzhaf and Shapley-Shubik power indexes give essentially the same answer. The United Nations Security Council example next illustrates a very different situation. The Electoral College

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The United Nations Security Council consists of 15 member nations – 5 are permanent members and 10 are nonpermanent members appointed on a rotating basis. For a motion to pass it must have a Yes vote from each of the 5 permanent members plus at least 4 of the 10 nonpermanent members. It can be shown that this arrangement is equivalent to giving the permanent members 7 votes each, the nonpermanent members 1 vote each, and making the quota equal to 39 votes. The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We will sketch a rough outline of how the Shapley-Shubik power distribution of the Security Council can be calculated. The details, while not terribly difficult, go beyond the scope of this book. The United Nations Security Council 1. There are 15! sequential coalitions of 15 players (roughly about 1.3 trillion).

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 2.A nonpermanent member can be pivotal only if it is the 9th player in the coalition, preceded by all five of the permanent members and three nonpermanent members. (There are approximately 2.44 billion sequential coalitions of this type.) The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 3. From steps 1 and 2 we can conclude that the Shapley-Shubik power index of a nonpermanent member is approximately 2.44 billion/1.3 trillion ≈ = 0.19%. (For the purposes of comparison it is worth noting that there is a big difference between this Shapley-Shubik power index and the corresponding Banzhaf power index of 1.65% obtained in Section 2.3.) The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 4. The 10 nonpermanent members (each with a Shapley-Shubik power index of 0.19%) have together 1.9% of the power pie, leaving the remaining 98.2% to be divided equally among the 5 permanent members. Thus, the Shapley-Shubik power index of each permanent member is approximately 98.2/5 =19.64%. The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This analysis shows the enormous difference between the Shapley-Shubik power of the permanent and nonpermanent members of the Security Council– permanent members have roughly 100 times the Shapley-Shubik power of non- permanent members! The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. We introduced the European Union Council of Ministers in Section 2.3 and observed that the Banzhaf power index of each country is reasonably close to that country’s weight when the weight is expressed as a percent of the total number of votes. We will now do a similar analysis for the Shapley- Shubik power distribution. The European Union

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Computing Shapley-Shubik power in a weighted voting system with 27 players cannot be done using the direct approach we introduced in this chapter. The last column of Table 2-11 shows the Shapley- Shubik power distribution in the EU Council of Ministers.The calculations took just a couple of seconds using an ordinary desktop computer and some fancy mathematics software. The European Union

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The European Union

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. When we compare the Banzhaf and Shapley-Shubik power indexes of the various nations in the EU (Tables 2-8 and 2- 11), we can see that there are differences, but the differences are small (less than 1% in all cases). In both cases there is a close match between weights and power but with a twist: In the Shapley-Shubik power distribution the larger countries have a tad more power than they should and the smaller countries have a little less power than they should; with the Banzhaf power distribution this situation is exactly reversed. The European Union