Honors Discrete: CHAPTER 2 TEST REVIEW GUIDE
1) Identify and define the essential components.
a) QUOTA
b) WEIGHTS
c) PLAYERS
Examples: Identify the quota, # of players, and weight of the 4th player.
a. [35: 12, 7, 3, 8, 9, 6, 1]
b. [20: 8, 7, 4, 3, 2, 1]
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 or majority statement.
e) Find the percentage that the quota represents
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?
d. What is the quota to have more than 3/5
majority?
b. What is the most number of votes for a quota?
e. What is the quota to have a ¾ majority?
c. What is the quota to have at least a 3/5
majority?
f. What WHOLE percentage does the current
quota represent in the system?
3) Dictator, Dummies, Veto Power
a) Provide a definition for each
DICTATOR:
DUMMIES:
b) Identify dummy, dictator, or veto power?
a. [27: 11, 9, 8, 5]
c. [19: 8, 7, 5, 3, 2]
b. [12: 9, 6, 3]
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d. [15: 16, 7, 3, 2]
VETO POWER:
e. [17: 12, 5, 2, 2]
4) COALITION
a) Definition
b) Find specific coalitions (winning, losing, grand, sequential)
c) Determine the total number of possible coalitions
d) Determine the total number of sequential coalitions
Example:
#1) [35: 12, 7, 3, 8, 9, 6, 1]
a. Write out 3 winning coalitions.
b. how many total possible different coalitions exist?
c. how many sequential coalitions exist?
#2) Consider a weighted voting system of FIVE players (P1 to P5)
a. How many total sequential coalitions exist?
b. How many total possible coalitions exist?
c. How many possible coalitions in this system do not include P3?
d. How many possible coalitions in this system do not include P2 and P3?
e. How many possible coalitions in this system do include P2 and P3?
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]
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b. Perform the Banzhaf power distribution on [19: 9, 8, 5, 3]
c. Perform the Banzhaf power distribution on [14: 5, 5, 4, 4]
d. Perform the Banzhaf power distribution on [16:10, 6, 2]
6) Perform Operations with Factorials: (SHOW YOUR WORK)
8!
5!
9!
12!
13!
9!
3!
5!
7) Shapley-Shubik Power Distribution
a) Definition of Pivotal Player
b) Calculate Shapley-Shubik Power Index
a. Perform the Shapley-Shubik Power Distribution on [17:10, 5, 2]
b. Perform the Shapley-Shubik Power Distribution on [9: 4, 3, 2, 1]
c. Perform the Shapley-Shubik Power Distribution on [10: 4, 3, 2, 1]
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Honors Discrete: CHAPTER 2 TEST Review Guide Solutions
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.
a. [35: 12, 7, 3, 8, 9, 6, 1]
b. [20: 8, 7, 4, 3, 2, 1]
Quota = 35
Quota = 20
Players = 7
Players = 6
P4 = 8
P4 = 3
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 dummy, dictator, or veto power?
a. [27: 11, 9, 8, 5]
{11, 9, 8}
{11, 9, 8, 5}
Dummy = P4 or 5
Dictator = None
Veto Power = P1, P2, P3 or 11, 9, 8
b. [12: 9, 6, 3]
{9, 6} {9, 6, 3} {9, 3}
Dummy = NONE
Dictator = NONE
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Veto Power = P1 or 9
c. [19: 8, 7, 5, 3, 2]
8+7+5+3+2
8+7+3+2
8+7+5+3
8+7+5+2
8+7+5
Dummy = NONE
Dictator = NONE
Veto Power = P1 or P2
d. [15: 16, 7, 3, 2]
16
16+7
16+3
16+7+3
16+7+2
16+3+2
Dummy = P2, P3, P4 Dictator = P1 Veto Power = P1
e. [17: 12, 5, 2, 2]
12 + 5
12 + 5 + 21
Dummy = P3, P4
12 + 5 + 22
Dictator = None
16+2
16+7+3+2
12 + 5 + 21 + 22
Veto Power = P1 and P2
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)
sequential = ordered coalition of ALL players
c) Determine the total number of different coalitions = 2N - 1
d) Determine the total number of sequential coalitions = N!
EXAMPLE:
#1) [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 possible coalitions
c. how many sequential coalitions exist?
7! Sequential coalitions
#2) Consider a weighted voting system of FIVE players (P1 to P5)
a. how many total sequential coalitions exist?
5! SEQUENTIAL
b. how many total possible coalitions exist?
25 – 1 = 31 possible
c. How many possible coalitions in this weighted voting system do not include P3?
24 – 1 = 15 possible
d. How many possible coalitions in this weighted voting system do not include P2 and P3?
23 – 1 = 7 possible
e. How many possible coalitions in this weighted voting system do include P2 and P3?
31 – 7 = 24 possible
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
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
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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
c. Perform the Banzhaf power distribution on [14: 5, 5, 4, 4]
Winning Coalitions: 5 + 5 + 41
5 + 5 + 42
5 + 5 + 41 + 42
2
2
1
1
P1  , P2  , P3  , P4 
6
6
6
6
d. Perform the Banzhaf power distribution on [16:10,6,2]
Winning Coalitions: 10 + 5 + 2
1
1
1
P1  , P2  , P3 
3
3
3
6) Perform Operations with Factorials: (SHOW YOUR WORK)
8!
9!
1
1
 6 * 7 * 8  336


5!
12! 10 * 11 * 12 1320
13!
3!
1
1
 10 * 11 * 12 * 13  17160


5! 4 * 5 20
9!
7) Shapley-Shubik Power Distribution
a) Definition of Pivotal Player: Player in the ordered addition of a sequential coalition that makes
the total votes win
b) Calculate Shapley-Shubik Power Index
a. Perform the Shapley-Shubik Power Distribution on [17:10, 5, 2]
<10, 6, 2>
<10, 2, 6> <6, 10, 2>
<6, 2, 10>
<2, 10, 6>
<2,6, 10>
1
1
1
P1  , P2  , P3 
3
3
3
b. Perform the Shapley-Shubik Power Distribution on [9: 4, 3, 2, 1]
<4, 3, 2, 1> <4, 3, 1, 2> <4, 2, 1, 3>
<4, 2, 3, 1> <4, 1, 2, 3>
<1, 3, 2, 4> <1, 3, 4, 2>
<1, 2, 4, 3> <1, 2, 3, 4> <1, 4, 3, 2>
<2, 3, 4, 1> <2, 3, 1, 4>
<2, 4, 1, 3> <2, 4, 3, 1>
<2, 1, 3, 4>
<3, 1, 2, 4>
<3, 1, 4, 2> <3, 2, 4, 1>
<3, 2, 1, 4> <3, 4, 1, 2>
1
1
1
P1  , P2  , P3  , P4  0
3
3
3
c. Perform the Shapley-Shubik Power Distribution on [10: 4, 3, 2, 1]
Quota= 10 means the last player in sequential coalition is pivotal
1
1
1
1
P1  , P2  , P3  , P4 
4
4
4
4
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<4, 1, 3, 2>
<1, 4, 2, 3>
<2, 1, 4, 3>
<3, 4, 2, 1>