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. Weighted voting in New York dates back to the 1800s Weighted voting systems used in some New York counties were seriously flawed (and unconstitutional) John Banzhaf introduced the Banzhaf Power Index in an article entitled Weighted Voting Doesn’t Work key point was that in weighted voting votes do not necessarily imply power The Nassau County (N.Y.) Board of Supervisors

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. 1960s, Nassau County was divided into 6 uneven districts, 4 with high populations and 2 rural districts with low populations Table shows names of the 6 districts and their weights based on their respective populations The Nassau County (N.Y.) Board of Supervisors

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Quota was a simple majority of 58 (out of 115) votes A district’s power should be in proportion to its population, thus ensuring to all citizens the “equal protection” guaranteed by the Constitution. Banzhaf: Instead of looking at the weights, one should focus on which districts are critical players in the many winning coalitions that can be formed on the Board. The Nassau County (N.Y.) Board of Supervisors

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Banzhaf argued that only the 3 largest The Nassau County (N.Y.) Board of Supervisors districts–Hempstead #1, Hempstead #2, and Oyster Bay–could ever be critical players, and consequently, the other 3 districts had no power whatsoever. Notice that North Hempstead had no power, never mind its 21 votes on the Board!

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Banzhaf’s mathematical analysis of the unfair power distribution in the Nassau County Board set the stage for a series of lawsuits against Nassau and other New York state counties based on the argument that weighted voting violated the “equal protection” guarantees of the Fourteenth Amendment (which was the very reason that weighted voting was instituted to begin with). The Nassau County (N.Y.) Board of Supervisors

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The final result was a federal court decision in 1993 abolishing weighted voting in New York state. In 1996, after a protracted fight, the Nassau County Board of Supervisors became a 19- member legislature, each member having just one vote and representing districts of roughly equal population. The Nassau County (N.Y.) Board of Supervisors

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The United Nations Security Council is an international body responsible for maintaining world peace and security. As currently constituted, the Security Council consists of 15 voting nations – 5 of them are the permanent members (Britain, China, France, Russia, and the United States); the other ten nations are nonpermanent members appointed for a two-year period on a rotating basis. The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. To pass a motion in the Security Council requires a yes vote from each of the five permanent members (in effect giving each permanent member veto power) plus at least 4 additional yes votes from the 10 nonpermanent members. In other words, a winning coalition in the Security Council must include all 5 of the permanent members plus 4 or more nonpermanent members. The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. All in all, there are 848 possible winning coalitions in the Security Council– too many to list one by one. However, it’s not too difficult to figure out the critical player story: In the winning coalitions with 9 players (5 permanent members plus 4 nonpermanent members) every member of the coalition is a critical player; in all the other winning coalitions (10 or more players) only the permanent members are critical players. The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Using a few simple (but not elementary) calculations, one can find that of the 848 winning coalitions, there are 210 with 9 members and the rest have 10 or more members. Carefully piecing together this information leads to the following surprising conclusion: the Banzhaf power index of each permanent member is 848/5080 (roughly 16.7%), while the Banzhaf power index of each nonpermanent member is 84/5080 (roughly 1.65%). The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. Note the discrepancy in power between the permanent and non-permanent members – a permanent member has more than 10 times as much power as a nonpermanent member. Was this really the intent of the United Nations charter or the result of a lack of understanding of the mathematics behind weighted voting? The United Nations Security Council

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The European Union (EU) is a political and an economic confederation of European nations, a sort of United States of Europe. As of the writing of this edition, the EU consists of 27 member nations (Table 2-8), with three more countries (Turkey, Croatia, and Macedonia) expected to join the EU in the near future. 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. The legislative body for the EU (called the EU Council of Ministers) operates as a weighted voting system where the different member nations have weights that are roughly proportional to their respective populations (but with some tweaks that favor the small countries). The second column shows each nation’s weight in the Council of Ministers. The total number of votes is V = 345, with the quota set at q = 255 votes. The European Union

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. The third column gives the relative weights (weight/345) expressed as a percent. The last column shows the Banzhaf power index of each member nation, also expressed as a percent. We can see from the last two columns that there is a very close match between Banzhaf power and weights (when we express the weights as a percentage of the total number of votes). The European Union

Excursions in Modern Mathematics, 7e: Copyright © 2010 Pearson Education, Inc. This is an indication that, unlike in the Nassau County Board of Supervisors, in the EU votes and power go hand in hand and that this weighted voting system works pretty much the way it was intended to work. The European Union