CHAPTER 2: Weighted Voting Systems



CHAPTER 2: Weighted Voting Systems

ESSENTIAL QUESTIONS

Section 2.1: What is a Weighted Voting System?

Section 2.2: What is a Banzhaf Power Index and which players are important?

Section 2.3: What is a Banzhaf Power Index and which players are important?

Section 2.4: What is a Shapley-Shubik Power Index and which players are important?

Section 2.5: What is a Shapley-Shubik Power Index and which players are important?

WORD WALL:

WEIGHTED VOTING SYSTEM

PLAYERS

PIVOTAL PLAYER

CRITICAL PLAYER

WEIGHTS

QUOTA

BANZHAF POWER INDEX

BANZHAF POWER DISTRIBUTION

COALITION

GRAND COALITION

DUMMY

DICTATOR

VETO POWER

SHAPLEY-SHUBIK POWER INDEX

SHAPLEY-SHUBIK POWER DISTRIBUTION

CHAPTER 2 INTRODUCTION: How your grade is calculated?

1) Basic Average: Everything is worth equal number of points and at the end of the grading period you take your total number of points earned and divide by total number of possible points for grading period.

EXAMPLE: Tests: 250/ 300 Quiz: 125/150 Homework: 45/50 Participation: 22/25

What is your final grade?

2) Weighted Average: Different types of assignments are given different weights or amounts of importance to the overall grade.

Assume the following weights: Tests = 50%, Quizzes = 30%, Homework = 15%, and Participation = 5%.

Recalculate Final Grade:

Weighted Average Problems:

1) What is the highest possible grade if you never do homework?

2) What is the highest possible grade if you never score higher than 80% on a test?

FINAL GRADE WEIGHTED AVERAGE:

1st Semester = 40% 2nd Semester = 40% Final Exam = 20%

1) You earned a 92.5% in the 1st semester, an 89% in the 2nd semester and a 94% on the final what is your final grade?

2) Suppose you had a 92% 1st semester and a 95% in the 2nd semester, what is your grade before taking the exam? What score on your final is needed to get an A versus a B?

3) If you earned an 87% in the 1st semester and a 93% in the 2nd semester, what is your grade before taking the exam? What score on your final is needed to get an A versus a B?

Section 2.1 Weighted Voting Systems

• Weighted Voting System: Any formal arrangement in which voters are not necessarily equal in terms of the number of votes they control.

One Voter with _______________________ Number of Votes

** not all voters have the same number of votes**

• Motion: yes - no voting system which involves exactly _____ candidates or alternatives

• Key Elements and Notation of a Weighted Voting System

o Players:

o Weights:

o Quota, q:

o Summary of Weighted Voting System:

|Example 1: [13: 7, 4, 3, 3, 2, 1] |Example 2: [31: 12, 8, 6, 5, 5, 5, 2] |

|# of players: ______________ |# of players: ______________ |

| | |

|total # of votes: ______________ |total # of votes: ______________ |

| | |

|weight of P3: ______________ |weight of P3: ______________ |

| | |

|minimum # of votes to pass a motion: |minimum # of votes to pass a motion: |

| | |

| | |

|Which of the following are winning coalitions? |Which of the following are winning coalitions? |

|P1, P2, P3 |P1, P2, P3, P4 |

|P1, P3, P4 |P2, P3, P4, P5, P6, P7 |

|P1, P4, P5 |P1, P2, P4, P5 |

|P2, P3, P4, P5, P6 |P1, P3, P5, P7 |

• POSSIBLE SIZE of the Quota

o MINIMUM: The quota must be more than __________ the total number of votes.

o MAXIMUM: The quota less than or equal to the _____________ number of votes.

|Example 3: [31: 12, 8, 6, 5, 5, 5, 2] |Example 4: [q: 11, 9, 6, 5, 4, 3, 2] |

|3a. What percentage does the current quota represent? |4a. What is the minimum quota we can have to pass a motion? |

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|3b. A quota of 26 votes represents what percentage of votes? |4b. What quota represents needing a ¾ majority? |

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|3c. If a 2/3 majority is needed to pass a motion, what is the quota? |4c. What quota represents needing a 2/3 majority? |

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• Coalitions:

o Grand Coalition:

o Set Notation for Coalition of players:

o Winning Coalition: vote total greater than or equal to quota

o Losing Coalition: vote total less than quota

PRACTICE: For the each weighted voting systems, find all winning coalitions.

1) [15: 9, 8, 7]

2) [16: 10, 6, 2]

3) [12: 8, 6, 4]

4) [8: 4, 3, 2, 1]

5) [10: 4, 3, 2, 1]

6) [18: 20, 8, 4 ]

Section 2.1 Weighted Voting Systems Part 2

• Critical Player: player that a winning coalition MUST have to win.

Consider the following winning coalitions: Who is critical?

1) [14: 10, 8, 6, 3]

{10, 8}

{10, 6}

{8, 6}

{10, 8, 3}

{10, 6, 3}

{8, 6, 3}

{10, 8, 6, 3}

2) [8: 4, 3, 2, 1]

{4, 3, 1}

{4, 3, 2}

{4, 3, 2, 1}

3) [10: 8, 6, 4, 1]

{8, 6}

{8, 4}

{6, 4}

{8, 6, 1}

{8, 6, 4}

{8, 4, 1}

{6, 4, 1}

{8, 6, 4, 1}

• POWER Definitions

o DICTATOR: Player’s weight is _____________________________ to the quota

All the power, so there can only ever be one dictator in any weighted voting system

▪ [18: 20, 10, 5]

o DUMMY: Any player’s weight that won’t affect the outcome, (powerless to stop anything)

*** Special Case: When a system has a dictator, all other players are dummies***

▪ [8: 6, 3, 1]

o VETO POWER: Quota total cannot be met unless the player votes in favor

***All other players can’t meet quota with their votes***

▪ [10: 5, 4, 3, 2]

o Example 1: [11: 12, 5, 4]

• Dictator:

• Dummies:

• Veto power:

o Example 2: [15: 14, 8, 7]

▪ Dictator:

▪ Dummies:

▪ Veto power:

o Example 3: [6: 4, 2, 1]

▪ Dictator:

▪ Dummies:

▪ Veto power:

o Example 4: [6: 5, 5, 1]

▪ Dictator:

▪ Dummies:

▪ Veto power:

o Example 5: [6: 4, 1, 1]

▪ Dictator:

▪ Dummies:

▪ Veto power:

o Example 6: [11: 8, 4, 3, 1]

▪ Dictator:

▪ Dummies:

▪ Veto power:

HOMEWORK: p. 72 #1 – 9 (odd)

Section 2.2 and 2.3 Banzhaf Power Index Applications

Find all winning coalitions and then identify which players are critical.

• EXAMPLE #1: [10: 8, 4, 1]

•

• EXAMPLE #2: [101: 99, 98, 3]

• EXAMPLE #3: [15: 8, 6, 4, 2]

• EXAMPLE #4: [6: 5, 3, 3]

• EXAMPLE #5: [20: 12, 10, 8, 2]

• EXAMPLE #6: [12: 7, 6, 5, 4]

Banzhaf’s Power Interpretation: Player’s Power Should be measured by how often player is CRITICAL

• Calculating the Banzhaf Power Index:

1) Find all winning coalitions.

2) Identify critical players in each winning coalition.

3) Count the total number of times that each player, Pi, is critical, Bi.

4) Find the total number of times all players are critical, T = [pic]

5) Banzhaf Power Index, β (beta): ratio of how often a player is critical to all players being critical; βi = [pic]is the Banzhaff Power Index for player Pi

Find the Banzhaf Power Index for each player:

• EXAMPLE #1: [10: 8, 4, 1]

| |P1 = 8 |P2 = 4 |P3 = 1 |

|# times player is critical, Bi| | | |

|Banzhaf Power Index, βi | | | |

• EXAMPLE#2: [101: 99, 98, 3]

| |P1 = 99 |P2 = 98 |P3 = 3 |

|# times player is critical, Bi| | | |

|Banzhaf Power Index, βi | | | |

Find the Banzhaf Power Index for each player in the following:

• EXAMPLE #3: [4: 3, 2, 1]

•

• EXAMPLE #4: [8: 5, 4, 3, 1]

Banzhaf Power Distribution: complete list of all Banzhaf Power Indexes β1 , β2, β3,…, βN

• The sum of all Banzhaf Power Indexes = 1 = β1 + β2 + β3 + … + βN

BE CAREFUL OF REPEATING WEIGHTS: [8: 6, 4, 2, 2] (Use subscripts to help)

What is the total number of possible coalitions (winning and losing) depending on the number of players?

• The order of the players in the coalition doesn’t matter

• The size or number of players in the coalition can change.

|Players |Coalitions |What do all the coalitions look like? |

|1 | | |

|2 | | |

|3 | | |

|4 | | |

Total Number of Possible Coalitions from N Players:

HOMEWORK: p. 73 #11, 12, 15, 20 and p.75# 41, 42, 45

Section 2.4 and 2.5 Shapley-Shubik Power Index and Applications Part 1

• Banzhaf Coalitions

•

• Shapley-Shubik Coalitions

• Sequential Coalitions: an ordered list of all players voting.

• Find all sequential coalitions for each of the following weighted voting systems?

o [12: 8, 6, 4]

o [16: 10, 5, 5]

• How many possible sequential coalitions with N players are there?

(ordered sequence containing N items)

o Multiplication Rule: If there _______ ways to do X and ______ ways to do Y, then there is __________ ways to do X and Y.

o Factorial, N!: The _______________________ of the first N positive integers

3! =

5! =

[pic]

Number of Sequential Coalitions N players is __ ___________

Practice with Factorials: Evaluate each expressions [MATH] – [PRB] – [4:!]

1) 7!

2) 8!

3) [pic]

4) 6!

5) 4!

6) [pic]

7) [pic]

8) [pic]

9) [pic]

10) If 10! = 3628800

a. 9!

b. 11!

• Pivotal Player: The player in a sequential coalition who’s votes makes the coalition winning based on the order of voting from Left to Right

EXAMPLE: Find the pivotal player in each of the following sequential coalitions.

1) Quota = 15

a. < 3, 4, 8, 2 >

b. < 2, 8, 4, 3 >

c. < 2, 3, 4, 8 >

2) Quota = 27

a. < 10, 12, 4, 3 >

b. < 12, 3, 10, 4 >

c. < 3, 10, 4, 12 >

3) Quota = 18

a. < 20, 8, 5, 3 >

b. < 3, 5, 8, 20 >

c. < 5, 3, 20, 8 >

HOMEWORK: p.75 #37, 43 – 44

Section 2.4 and 2.5 Shapley-Shubik Power Index and Applications Part 2

For the following weighted voting system: Find all sequential coalitions and identify who is pivotal.

Example 1: [8: 6, 3, 2] Example 2: [11: 7, 4, 3, 1]

Shapley – Shubik Interpretation of Power:

Player’s Power Should be measured by how often player is PIVOTAL

• Calculating the Shapley – Shubik Power

1. Find all possible sequential coalitions for the N players

2. Find pivotal player in each sequential coalition

3. Count number of times each player, Pi, is pivotal = SSi

4. Shapley-Shubik Power Index, σ, (sigma): Ratio of how often a player is pivotal to the number of sequential coalitions

[pic], where T = total number of sequential coalitions

• Shapley- Shubik Power Distribution: Complete list of σ for each player

Find the Shapley – Shubik Power Distribution in each of the following examples:

• Example 1: [5: 3, 2, 1]

|Sequential Coalitions |Pivotal Totals |Shapley-Shubik Power Index |

| | | |

| | | |

| | | |

• Example 2: [16: 9, 8, 7]

• Example 3: [20: 15, 5, 5] (Be careful of repeated weights)

HOMEWORK: p. 74# 24 –25, 27 (parts a, d, e)

24) Consider the weighted voting system [8: 7, 6, 2]

(a) Write down all the sequential coalitions, and in each sequential coalition underline the pivotal player.

(b) Find the Shapley-Shubik power distribution of this weighted voting system.

25) Find the Shapley-Shubik power distribution of the weighted voting system [5: 3, 2, 1, 1]

27) Find the Shapley-Shubik power distribution of each of the following weighted voting system

(a) [8: 8, 5, 1]

(d) [8: 6, 5, 1]

(e) [8: 6, 5, 3]

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