Weighted Voting 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
`
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
`
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
`
<4, 1, 3, 2>
<1, 4, 2, 3>
<2, 1, 4, 3>
<3, 4, 2, 1>