Excursions in Modern Mathematics Sixth Edition

1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

2. Chapter 2Weighted Voting Systems The Power Game

3. Weighted Voting SystemsOutline/learning Objectives • Represent a weighted voting system using a mathematical model. • Use the Banzhaf and Shapley-Shubik indices to calculate the distribution of power in a weighted voting system.

4. Weighted Voting Systems 2.1 Weighted Voting Systems

5. Weighted Voting Systems • The Players The voters in a weighted voting system. • The Weights That each player controls a certain number of votes. • The Quota The minimum number of votes needed to pass a motion (yes-no votes)

6. Weighted Voting Systems • Dictator The player’s weight is bigger than or equal to the quota. Consider [11:12, 5, 4] owns enough votes to carry a motion single handedly.

7. Weighted Voting Systems • Dummy A player with no power. Consider [30: 10, 10, 10, 9] turns out to be a dummy! There is never going to be a time when is going to make a difference in the outcome of the voting.

8. Weighted Voting Systems • Veto Power If a motion cannot pass unless player votes in favor of the motion. Consider [12: 9, 5, 4, 2] has the power to obstruct by preventing any motion from passing.

9. Weighted Voting Systems 2.2 The Banzhaf Power Index

10. Weighted Voting Systems • Coalitions Any set of players that might join forces and vote the same way. The coalition consisting of all the players is called a grand coalition.

11. Weighted Voting Systems • Winning Coalitions Some coalitions have enough votes to win and some don’t. We call the former winning coalitions and the latter losing coalitions.

12. Weighted Voting Systems • Critical players In a winning coalition, a player is said to be a critical player for the coalition if the coalition must have that player’s votes to win. If and only if

13. Weighted Voting Systems • Computing a Banzhaf Power Distribution • Step 1. Make a list of all possible winning coalitions.

14. Weighted Voting Systems • Computing a Banzhaf Power Distribution • Step 2. Within each winning coalition determine which are the critical players. (To determine if a given player is critical or not in a given winning coalition, we subtract the player’s weight from the total number of votes in the coalition- if the difference drops below the quota q, then that player is critical. Otherwise, that player is not critical.

15. Weighted Voting Systems • Computing a Banzhaf Power Distribution • Step 3.Count the number of times that is critical. Call this number Repeat for each of the other players to find

16. Weighted Voting Systems • Computing a Banzhaf Power Distribution • Step 4. Find the total number of times all players are critical. This total is given by

17. Weighted Voting Systems • Computing a Banzhaf Power Distribution • Step 5. Find the ratio 1 . This gives the Banzhaf power index of . Repeat for each of the other players to find 2, 3, …, N . The complete list of ’sgives the Banzhaf power distribution of the weighted voting system.

18. Weighted Voting Systems 2.3 Applications of Banzhaf Power

19. Weighted Voting Systems Applications of Banzhaf Power • The Nassau County Board of Supervisors John Banzhaf first introduced the concept • The United Nations Security Council Classic example of a weighted voting system • The European Union (EU) Relative Weight vs Banzhaf Power Index

20. Weighted Voting Systems 2.4 The Shapley-Shubik Power Index

21. Weighted Voting Systems Three-Player Sequential Coalitions

22. Weighted Voting Systems Shapley-Shubik- Pivotal Player The player that contributes the votes that turn what was a losing coalition into a winning coalition.

23. Weighted Voting Systems • Computing a Shapley-Shubik Power Distribution • Step 1. Make a list of all possible sequential coalitions of the N players. Let T be the number of such coalitions.

24. Weighted Voting Systems • Computing a Shapley-Shubik Power Distribution • Step 2. In each sequential coalition determine the pivotal player.

25. Weighted Voting Systems • Computing a Shapley-Shubik Power Distribution • Step 3.Count the number of times that is pivotal. Call this number . Repeat for each of the other players to find

26. Weighted Voting Systems • Computing a Shapley-Shubik Power Distribution • Step 4. Find the ratio 1 This gives the Shapley Shubik power index of . Repeat for each of the other players to find 2, 3, …, N . The complete list of  ’sgives the Shapley-Shubik power distribution of the weighted voting system.

27. Weighted Voting Systems The Multiplication Rule If there are m different ways to do X, and n different ways to do Y, then X and Y together can be done in m x n different ways.

28. Weighted Voting Systems The Number of Sequential Coalitions The number of sequential coalitions with N players is N! = N x (N-1) x…x 3 x 2 x 1.

29. Weighted Voting Systems Applications of Shapley-Shubik Power • The Electoral College There are 51! Sequential coalitions • The United Nations Security Council Enormous difference between permanent and nonpermanent members • The European Union (EU) Relative Weight vs Shapley-Shubik Power Index

30. Weighted Voting Systems Conclusion • The notion of power as it applies to weighted voting systems • How mathematical methods allow us to measure the power of an individual or group by means of an index. • We looked at two different kinds of power indexes– Banzhaf and Shapley-Shubik