1. Weighted Voting Systems Chapter 2 Objective: Calculate the Banzhaf power Index for a weighted voting system. Learn additional notation and terminology for coalitions.

2. Closing Product Exit Ticket List the winning coalitions of a weighted voting system. Identify the critical players in each winning coalition. Calculate the Banzhaf Power distribution and determine if the weighted voting system is fair.

3. Coalitions Any set of players that might join forces and vote the same way. Can be as little as one player to all players

4. Grand Coalition Consists of all players in the set. (unanimous vote)

5. Winning Coalition A coalition that has enough votes to pass the motion.

6. Blocking Coalition A coalition that has enough votes to prevent a motion from passing.

7. Critical Players A player in a winning coalition, whose votes are necessary in order to pass the motion. In other words, the one(s) that has veto power.

8. Example #1 Consider the weighted voting system [95: 65, 35, 30, 25]. Find the following: a) The total number of coalitions b) List the winning coalitions

9. Total Number of Coalitions 2 n -1 Where n=total number of players

10. Example #2 Consider the weighted voting system [22: 10, 8, 7, 2, 1] a) What is the total number of coalitions? b) List all of the coalitions

11. Banzhaf Power Index A mathematical measurement of a player’s power in a weighted voting system based on how many times that player is a critical player. β 1 = # of time Player 1 is a critical player sum of all critical players Pronounced ‘beta-one’

12. Banzhaf Power Distribution A complete list of the power indexes β 1, β 2, β 3, …… β n The sum of all the β’s is equal to 1. (or 100% if using percentages.

13. Calculation i. Make a list of all winning coalitions. ii. Determine the critical player of each coalition. iii. Count the number of times each player is critical. (B 1 for P 1, B 2 for P 2, etc.) iv. Find the total number of times all players are critical. (T = B 1 + B 2 +..B n ) v. Find the ratio for each player (β 1 = B 1 /T)

14. Consider the following ….. [49: 48, 24, 12, 12] Winning Coalitions. 1 Player2 Players3 Player4 Player NoneP 1, P 2 P 1, P 2, P 3 P 1, P 2, P 3, P 4 P 1, P 3 P 1, P 2, P 4 P 1, P 4 P 1, P 3, P 4 B 1 = 7 B 2 = 1 B 3 = 1 B 4 = 1 T = 10 β 1 = 7/10, β 2 = 7/10,β 3 = 7/10, β 4 = 7/10

15. Do the next one on your own.

16. [14: 8, 4, 2, 1] Winning Coalitions. 1 Player2 Players3 Player4 Player None P 1, P 2, P 3 P 1, P 2, P 3, P 4 B 1 = 2 B 2 = 2 B 3 = 2 B 4 = 0 T = 6 β 1 = 1/3, β 2 = 1/3,β 3 = 1/3, β 4 = 0

17. Consider the following …. What would happen if the weighted voting system was [15: 16, 8, 4, 1] ? Write down a sentence that summarizes what the Banzhaf Power Index would be for dictators and dummies.

18. Consider the following ….. Dictator – Banzhaf Power Index always 1, or 100% Dummy – Banzhaf Power Index always 0, or 0%

19. Application of Banzhaf Used to determine if a weighted voted system is set up in a fair manner. If it is fair, the relative weight of the votes should be comparable to the Banzhaf power index.

20. Nassau County Board of Supervisors Sued several times to change their weighted voting system that gave more power to bigger districts. Finally in 1991, NY Civil Liberties Union sued and won (1993). It was shown that minority districts were not receiving fair representation.

21. Nassau County Board of Supervisors Banzahf’s Power index showed for example that Hempstead had 56% of population but 70% of the power. Now districts are drawn equally, and they no longer use a weighted system.