Honors Discrete: Quiz 2.1 – 2.3 Review Guide
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1) Identify and define the essential components.
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a) QUOTA
b) WEIGHTS
c) PLAYERS
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Examples: Identify the quota, # of players, and weight of the 4th player.
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a. [35: 12, 7, 3, 8, 9, 6, 1]
b. [20: 8, 7, 4, 3, 2, 1]
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2) Quota - Relationship to voters
a) Determine if coalitions have met the quota (winning or losing).
b) MINIMUM:
c) MAXIMUM:
d) Change the quota to meet new criterion.
EXAMPLE: For the given weighted voting system, [20: 8, 7, 4, 3, 2, 1]
a. What is the least number of votes for a quota?
b. What is the most number of votes for a quota?
c. What is the quota to have at least a 3/5 majority?
d. What is the quota to have more than 3/5 majority?
e. What is the quota to have a ¾ majority?
f. What WHOLE percentage does the current quota represent in the system?
3) Dictator, Dummies, Veto Power
a) Provide a definition for each
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DICTATOR:
DUMMIES:
VETO POWER:
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b) Identify each in a weighted voting system
Example:
a. Determine in the weighted voting system [27: 11, 9, 8, 5]
Who is a dummy, dictator, or has veto power?
b. Determine in the weighted voting system [12: 9, 6, 3]
Who is a dummy, dictator, or has veto power?
4) COALITION
a) Definition
b) Find specific coalitions (winning, losing, GRAND)
c) Determine the total number of different coalitions
Example: [35: 12, 7, 3, 8, 9, 6, 1]
a. Write out 3 winning coalitions.
b. how many total possible different coalitions exist?
5) Banzhaf Power Distribution
a) Definition Critical Player
b) Calculate the Banzhaf Power Index
EXAMPLE:
a. Perform the Banzhaf power distribution on [10: 6, 5, 4]
b. Perform the Banzhaf power distribution on [19: 9, 8, 5, 3]
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Honors Discrete: Quiz 2.1 – 2.3 Review Guide Solutions
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1) Identify and define the essential components.
a) Quota: minimum number of votes required to pass a motion
b) Weights: number of votes that a player controls
c) Players: voters
EXAMPLE: Identify the quota, # of players, and weight of the 4th player.
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a. [35: 12, 7, 3, 8, 9, 6, 1]
Quota = 35
Players = 7
P4 = 8
b. [20: 8, 7, 4, 3, 2, 1]
Quota = 20
Players = 6
P4 = 3
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2) Quota - Relationship to voters
a) Determine if coalitions have met the quota.
b) Minimum: majority of votes (MORE THAN HALF)
c) Maximum: total of the votes
d) Change the quota to meet new criterion: multiply the total by the requirement (round up)
EXAMPLE: For the given weighted voting system, [20: 8, 7, 4, 3, 2, 1]
a. what is the least number of votes for a quota
= (8 + 7 + 4 + 3 + 2 +1 )/2↑ = 13
b. what is the most number of votes for a quota
= 8 + 7 + 4 + 3 + 2 + 1 = 25
c. What is the quota to have at least a 3/5 majority?
= 25*3/5 = 15
d. What is the quota to have more than 3/5 majority?
= 25*3/5 = 15↑ = 16
e. What is the quota to have a 3/4 majority?
= 25*3/4 = 18.75 ↑ = 19
f. What percentage does the current quota represent in the system?
20/25 = 60%
3) Dictator, Dummies, Veto Power
a) Provide a definition for each
· Dictator:
o Player’s weight is greater than or equal to quota.
o Winning Coalition by itself
· Dummies:
o Any player’s weight that won’t affect the outcome
o Never is a critical player
· Veto Power:
o Quota total cannot be met in a coalition unless this player votes with them
o Player(s) that is in EVERY winning coalition AND in each coalition is ALWAYS critical
b) Identify each in a weighted voting system
EXAMPLE:
a. Determine in the weighted voting system [27: 11, 9, 8, 5]
Who is a dummy, dictator, or has veto power?
{11, 9, 8} {11, 9, 8, 5}
Dummy = P4 or 5 Dictator = None Veto Power = P1, P2, P3 or 11, 9, 8
b. Determine in the weighted voting system [12: 9, 6, 3]
Who is a dummy, dictator, or has veto power?
{9, 6} {9, 6, 3} {9, 3}
Dummy = NONE Dictator = NONE Veto Power = P1 or 9
4) Coalitions
a) Definition: group of players that vote the same way
b) Find specific coalitions
winning = total votes greater than or equal to quota
losing = total votes less than quota
grand = all players together)
c) Determine the total number of different coalitions = 2N - 1
EXAMPLE: ) [35: 12, 7, 3, 8, 9, 6, 1]
a. write out 3 winning coalitions.
Examples: {12+9+8+6}, {12+9+8+7}, {12+8+7+6+3}
b. how many total possible different coalitions exist?
27 – 1 = 127 coalitions
5) Banzhaf Power Distribution
a) Definition Critical Player: player that a coalition needs to be winning
b) Calculate the Banzhaf Power Index: number of times a player is critical
number of all players are critical
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EXAMPLE:
a. Perform the Banzhaf power distribution on [10: 6, 5, 4]
Winning: {6, 5} {6, 4} {6, 5, 4}
Power Indexes: P1 = 3/5 P2 = 1/5 P3 = 1/5
b. Perform the Banzhaf power distribution on [19: 9, 8, 5, 3]
Winning: {9, 8, 3} {9, 8, 5} {9, 8, 5, 3}
Power Indexes: P1 = 3/ 8 P2 = 3/8 P3 = 1/8 P4 = 1/8
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