The document discusses different methods for analyzing weighted voting systems, including determining winning and losing coalitions, critical players, Banzhaf power indices, possible coalitions, and Shapley-Shubik power indices. Examples are provided to illustrate calculating these values for weighted voting systems with different numbers of players and vote distributions.

1. Weighted Voting Systems

6. · Make a list of all possible coalitions. ·Determine whether they are winning or losing coalitions ·Of the winning, determine which Players are critical players ·Count the total number of times player P is critical (B) ·Count the total number of times all players are critical (T) The Banzhaf power index of player P is then given by the fraction B/T Finding the Banzhaf Power Index of Player P

9. Homework: p. 61 # 1-4 all

10. NBA Draft: Ma ny of the teams use a weighted voting system in determining which college player to draft. In one system, the head coach (HC) has 4 votes, the general manager (GM) has 3 votes, the director of scouting operations (DS) has 2 votes and the team psychiatrist (TP) has 1 vote. A simple majority of 6 votes is required for a yes vote on a player. Describe the weighted voting system using common notation: Determine the Banzhaf Power Distribution:

12. How many coalitions are there? That's alot of coalitions. Is there a faster way to determine the number of coaltions?

13. Quiz ( #1-4: 2 points each; #5: 10 points) Co nsider the following weighted voting system: [10: 7, 5, 4, 2] 1. How many votes are needed to carry a motion? 2. How many players are there? 3. How many total votes are there? 4. How many possible coalitions are there? 5. Determine the Banzhaf Power Distribution of this weighted voting system.

15. The TSU Promotion and Tenure committee consists of 5 members: the dean (D) and four other faculty members of equal standing (F1, F2, F3, F4). In this committee motions are carried by strict majority, but the dean never votes except to break a tie. How is power distributed in this voting system? Another Example:

16. Consider the following weighted voting system: [13: 8, 5, 5, 4, 2] 1. How many players are there? 2. How many votes are needed to pass a motion? 3. How many total votes are there? 4. How many coalitions are there? 5. Determine the Banzhaf Power Distribution for the above voting system. 6. Create a weighted voting system that has a dictator. 7. Create a weighted voting system where one player has veto power. 8. Create a weighted voting system that needs an unanimous vote in order to pass a motion.

17. The Shapely-Shubik Power Index Key component: sequential coalitions . Based on the idea that every coalition starts with a first player, who may be joined by others. Which brings in the question of order, i.e. permutations and factorials. [12:6, 4, 4, 3, 2, 1] how many sequential coalitions? In each of the coalitions, there is one player that tips the scales and moves the coalition from a losing one to a winning one, this player is considered to be the pivotal player. The number of sequential coalitions with N players is N!

18. · Make a list of all sequential coalitions. There are N! of them. ·Determine the pivotal player. There is one in each coalition. ·Count the number of times player P is pivotal (S) The Shapley-Shubik Power Index is then given by the fraction S/N! Finding the Shapley-Shubik Power Index of Player P

19. Example: Consider the following Weighted Voting System [6:4, 3, 2, 1] Determine the Shapley-Shubik Power Index. Determine the Shapley-Shubik Power Index.

21. Homework: p. 62 # 7, 8, 12-16 all Test on Chapter 2, April 1st

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