2 The Mathematics of Power

2. 2 The Mathematics of Power 2.1 An Introduction to Weighted Voting 2.2 The Banzhaf Power Index 2.3 Applications of the Banzhaf Power Index 2.4 The Shapley-Shubik Power Index 2.5 Applications of the Shapley-Shubik Power Index

3. The Electoral College Calculating the Shapley-Shubik power index of the states inthe Electoral College is no easy task. There are 51! sequential coalitions, anumber 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 sophisticatedmathematical shortcuts that, when coupled with the right kind of software, allowthe calculations to be done by an ordinary computer in a matter of seconds (seereference 16 for details).

4. The Electoral College 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 ofColumbia. Comparing the Banzhaf and the Shapley-Shubik power indexes showsthat there is a very small difference between the two. This example shows that insome situations the Banzhaf and Shapley-Shubik power indexes give essentiallythe same answer. The United Nations Security Council example next illustratesa very different situation.

5. The United Nations Security Council The UnitedNations Security Council consists of 15 member nations – 5 are permanentmembers and 10 are nonpermanent members appointed on a rotating basis. Fora motion to pass it must have a Yes vote from each of the 5 permanent membersplus at least 4 of the 10 nonpermanent members. It can be shown that thisarrangement is equivalent to giving the permanent members 7 votes each, thenonpermanent members 1 vote each, and making the quota equal to 39 votes.

6. The United Nations Security Council We will sketch a rough outline of how the Shapley-Shubik power distributionof the Security Council can be calculated. The details, while not terribly difficult,go beyond the scope of this book. 1. There are 15! sequential coalitions of 15 players (roughly about 1.3trillion).

7. The United Nations Security Council 2. A nonpermanent member can be pivotal only if it is the 9th player in thecoalition, preceded by all five of the permanent members and three nonpermanent members. (There are approximately 2.44 billion sequential coalitionsof this type.)

8. The United Nations Security Council 3. From steps 1 and 2 we can conclude that the Shapley-Shubik power index of anonpermanent member is approximately 2.44 billion/1.3 trillion ≈ 0.0019 = 0.19%.(For the purposes of comparison it is worth noting that there is a bigdifference between this Shapley-Shubik power index and the correspondingBanzhaf power index of 1.65% obtained in Section 2.3.)

9. The United Nations Security Council 4. The 10 nonpermanent members (each with a Shapley-Shubik power index of0.19%) have together 1.9% of the power pie, leaving the remaining 98.2% tobe divided equally among the 5 permanent members. Thus, the Shapley-Shubikpower index of each permanent member is approximately 98.2/5 =19.64%.

10. The United Nations Security Council This analysis shows the enormous difference between the Shapley-Shubikpower of the permanent and nonpermanent members of the Security Council–permanent members have roughly 100 times the Shapley-Shubik power of non-permanent members!

11. The European Union We introduced the European Union Council of Ministers inSection 2.3 and observed that the Banzhaf power index of each country isreasonably close to that country’s weight when the weight is expressed as apercent of the total number of votes. We will now do a similar analysis for theShapley-Shubik power distribution.

12. The European Union 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 theShapley-Shubik power distribution in the EU Council of Ministers.The calculations took just a couple of seconds using an ordinary desktop computer and somefancy mathematics software.

13. The European Union

14. The European Union When we compare the Banzhaf and Shapley-Shubik power indexes of thevarious 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 casesthere 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 theyshould and the smaller countries have a little less power than they should; withthe Banzhaf power distribution this situation is exactly reversed.