Weighted voting
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Weighted Voting Systems
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·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 fractionB/T
Finding the Banzhaf Power Index of Player P
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Homework: p. 61 # 1-4 all
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NBA Draft:
Many 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. Asimple 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:
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That's alot of coalitions.Is there a faster way todetermine the number ofcoaltions?
How many coalitions are there?
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Quiz (#1-4: 2 points each; #5: 10 points)
Consider 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.
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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, butthe dean never votes except to break a tie. How is power distributedin this voting system?
Another Example:
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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.
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In each of the coalitions, there is one player that tips the scales andmoves the coalition from a losing one to a winning one, this player isconsidered to be the pivotal player.
The number of sequential coalitions with N players is N!
The Shapely-Shubik Power Index
Key component: sequential coalitions. Based on the idea that everycoalition starts with a first player, who may be joined by others. Whichbrings in the question of order, i.e. permutations and factorials.
[12:6, 4, 4, 3, 2, 1]
how many sequential coalitions?
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·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 fractionS/N!
Finding the Shapley-Shubik Power Index of Player P
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Example: Consider the following Weighted Voting System [6:4, 3, 2, 1]Determine the Shapley-Shubik Power Index.Determine the Shapley-Shubik Power Index.
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Homework:
p. 62 # 7, 8, 12-16 all
Test on Chapter 2, April 1st
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