Weighted Voting Systems

·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

Homework: p. 61 # 1-4 all

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:

That's alot of coalitions.Is there a faster way todetermine the number ofcoaltions?

How many coalitions are there?

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.

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:

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.

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?

·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

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

Homework:

p. 62 # 7, 8, 12-16 all

Test on Chapter 2, April 1st

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