Discrete Math
Weighted Voting
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What was the name of the company
What were the items being sold
How many people on the board
Describe a weighted voting system
Words to know
• Players: The voters in the weighted system
• Weights: The number of votes each player
holds
• Quota: The minimum number of votes
required to pass a motion
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James has 5 votes
Morgan has 3 votes
Kyle has 3 votes
Clarke has 2 votes
Over 50% is needed to pass motion
• 7: 5, 3, 3, 2
• It will be written as follows:
– 𝑞: 𝑤1 , 𝑤2 , … , 𝑤𝑛 𝑤ℎ𝑒𝑟𝑒 𝑤1 ≥ 𝑤2 ≥ ⋯ ≥ 𝑤𝑛
Venture Capitalism
• Everything is fine
Anarchy
• The quota is too low
Gridlock
• The quota is too high
One Partner- One Vote
• The quota is so high that the decision must be
unanimous
Dictator
• One person has enough weight to pass a
motion
Unsuspecting Dummies
• No matter how the person votes it will not
help pass the motion
Veto Power
• One person has enough votes to reject a
motion but not enough to pass it by
themselves
Your quota should be
𝑉
<𝑞≤𝑉
2
𝑤ℎ𝑒𝑟𝑒 𝑉 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑣𝑜𝑡𝑒𝑠
You can now do
#2, 4, 6
8, 10
#3
• [𝑞: 10, 6, 5, 4, 2]
(a) Smallest value for q
(b) Largest value for q
(c) What is q if at least 2/3 majority is needed
(d) What is q if over 2/3 majority is needed
#5
• [49:4x, 2x, x, x, x]
(a) 49 is a simple majority
(b) 49 is more than 2/3 majority
(c) 49 is more than ¾ majority
[49:48, 24, 12, 12]
[49:36, 18, 9, 9]
[49:32, 16, 8, 8]
#7
• Is there any dictators, veto holders, or
dummies
(a) [15: 16, 8, 4, 1]
(b) [18: 16, 8, 4, 1]
(c) [24: 16, 8, 4, 1]
P1 D, all d
p1 V, p4 d,
p1p2 V, p3p4 d
#9
• [q:8, 4, 2]
(a) All have veto power
(b) 𝑃2 has veto power but 𝑃3 does not
(c) 𝑃3 is the only dummy.
Banzhaf Power Index
• Who is the most important voter
• In congress everyone votes along party lines
• There are 99 Republicans, 98 Democrats, and
3 Independents.
• They are all important
More Words
• Coalitions: A group of players that will join
forces and vote the same
• Grand Coalitions: All players vote the same
• Winning/Losing Coalition: The group that wins
or loses
Critical Players
• The players in the winning coalition that are
needed for the coalition to win
• They are a critical player if:
𝑊 − 𝑤𝑝 < 𝑞
Lets look at Congress
• 99 Republicans
• 98 Democrats
• 3 Independents
• List the winning coalitions:
Coalition
Weight
{𝑅, 𝐷}
197
{𝑅, 𝐼}
102
{𝐷, 𝐼}
101
{𝑅, 𝐷, 𝐼}
200
Who are the critical players
Coalition
Coalition
Weight Weight
Critical Players
{𝑅, 𝐷} {𝑅, 𝐷}
197
197
𝑅 𝑎𝑛𝑑 𝐷
{𝑅, 𝐼} {𝑅, 𝐼}
102
102
𝑅 𝑎𝑛𝑑 𝐼
{𝐷, 𝐼} {𝐷, 𝐼}
101
101
𝐷 𝑎𝑛𝑑 𝐼
{𝑅, 𝐷, 𝐼}{𝑅, 𝐷, 𝐼}
200
200
𝑁𝑜𝑛𝑒
Banzhaf Power Index
1. Make list of all winning coalitions
2. Find the critical players of all winning
coalitions
3. Count the total number of times 𝑃1 is the
critical player. This is 𝐵1 , then repeat for all
players
4. Add all 𝐵s together and this is 𝑇
5. Find the ratio of each 𝐵 over 𝑇. This is now
𝛽1 . Put in terms of a %
Back in Congress
• How many Critical Players?
•6
• How many for R? D? I?
• 2, 2, 2
• What are the %?
1
3
1
3
1
3
• 𝑅 = 33 %, 𝐷 = 33 %, 𝐼 = 33 %
Coalition
Weight
Critical Players
{𝑅, 𝐷}
197
𝑅 𝑎𝑛𝑑 𝐷
{𝑅, 𝐼}
102
𝑅 𝑎𝑛𝑑 𝐼
{𝐷, 𝐼}
101
𝐷 𝑎𝑛𝑑 𝐼
{𝑅, 𝐷, 𝐼}
200
𝑁𝑜𝑛𝑒
Find the Power Index of
[5: 3, 2, 1, 1, 1]
𝑃1
𝑃2
𝑃3
𝑃4
𝑃5
𝑷𝟏 𝑷𝟐
𝑃1 𝑃3
𝑃1 𝑃4
𝑃1 𝑃5
𝑃2 𝑃3
𝑃2 𝑃4
𝑃2 𝑃5
𝑃3 𝑃4
𝑃3 𝑃5
𝑃4 𝑃5
𝑷𝟏 𝑷𝟐 𝑷𝟑
𝑷𝟏 𝑷𝟐 𝑷𝟒
𝑷𝟏 𝑷𝟐 𝑷𝟓
𝑷𝟏 𝑷𝟑 𝑷𝟒
𝑷𝟏 𝑷𝟑 𝑷𝟓
𝑷𝟏 𝑷𝟒 𝑷𝟓
𝑃2 𝑃3 𝑃4
𝑃2 𝑃3 𝑃5
𝑃2 𝑃4 𝑃5
𝑃3 𝑃4 𝑃5
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟒
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟓
𝑷𝟏 𝑷𝟑 𝑷𝟒 𝑷𝟓
𝑷𝟏 𝑷𝟐 𝑷𝟒 𝑷𝟓
𝑷𝟐 𝑷𝟑 𝑷𝟒 𝑷𝟓
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟒 𝑷𝟓
Winning Coalitions
Critical Players
Winning Coalitions
Critical Players
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟒
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐 𝑷𝟑
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟓
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐 𝑷𝟒
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟑 𝑷𝟒 𝑷𝟓
𝑷𝟏
𝑷𝟏 𝑷𝟐 𝑷𝟓
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟐 𝑷𝟒 𝑷𝟓
𝑷𝟏 𝑷𝟐
𝑷𝟏 𝑷𝟑 𝑷𝟒
𝑷𝟏 𝑷𝟑 𝑷𝟒
𝑷𝟐 𝑷𝟑 𝑷𝟒 𝑷𝟓
𝑷𝟐 𝑷𝟑 𝑷𝟒 𝑷𝟓
𝑷𝟏 𝑷𝟑 𝑷𝟓
𝑷𝟏 𝑷𝟑 𝑷𝟓
𝑷𝟏 𝑷𝟐 𝑷𝟑 𝑷𝟒 𝑷𝟓
NONE
𝑷𝟏 𝑷𝟒 𝑷𝟓
𝑷𝟏 𝑷𝟒 𝑷𝟓
• Total Number of Critical Player:
– 28
• Critical Players for 𝑷𝟏 :
– 11
• Critical Players for 𝑷𝟐 :
–8
• Critical Players for 𝑷𝟑 :
–3
• Critical Players for 𝑷𝟒 :
–3
• Critical Players for 𝑷𝟓 :
–3
Find the %
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11
𝛽1 :
= 39.29%
28
8
𝛽2 : = 28.57%
28
3
𝛽3 : = 10.71%
28
3
𝛽4 : = 10.71%
28
3
𝛽5 : = 10.71%
28
You can do
# 12, 14, 18, 20, 22
#11
• 10: 6, 5, 4, 2
(a) What is the weight of the coalition formed by
𝑃1 𝑎𝑛𝑑 𝑃3
(b) What are all the winning coalitions
(c) Who is the critical players in {𝑃1 , 𝑃2 , 𝑃3 }
(d) Find the Banzhaf Power Index
#13
(a) Find the Banzhaf Power Index of 6: 5, 2, 1
𝛽1 = 60%
𝛽2 = 20%
𝛽3 = 20%
(b) Find the Banzhaf Power Index of [3: 2, 1, 1]
𝛽1 = 60%
𝛽2 = 20%
𝛽3 = 20%
#19
• A weighted voting system has 3 players. The
only winning coalitions are the following:
𝑃1 , 𝑃2 , 𝑃1 , 𝑃3 , 𝑃1 , 𝑃2 , 𝑃3
(a) Find the Critical Players of each
𝑃1 , 𝑃2
𝑃1 , 𝑃3
𝑃1
(a) Find the Banzhaf Power Index
60%
20%
20%
Where did Banzhaf come from
• Nassau County
District
Weight
Hempstead #1
31
Hempstead #2
31
Oyster Bay
28
North Hempstead
21
Long Beach
2
Glen Cove
2
• Find the Banzhaf Power Index of Nassau County
Shapley- Shubik
• Sequential Coalition: a coalition that the order
matters.
• Pivotal Player: The person that cast the
winning vote.
List all the sequential coalitions
• 𝑃1 , 𝑃2 , 𝑃3
𝑃1 , 𝑃2 , 𝑃3
𝑃3 , 𝑃1 , 𝑃2
𝑃1 , 𝑃3 , 𝑃2
𝑃2 , 𝑃3 , 𝑃1
𝑃2 , 𝑃1 , 𝑃3
𝑃3 , 𝑃2 , 𝑃1
How many Sequential Coalitions
will we have?
• The multiplication rule: If there is X choices
and Y choices we have X * Y total choices
Factorial!!!
• 𝑁! = 𝑁 ∗ 𝑁 − 1 ∗ 𝑁 − 2 ∗ … ∗ 3 ∗ 2 ∗ 1
• 4! =
4 ∗ 3 ∗ 2 ∗ 1 = 24
• 5! =
5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
4: 3, 2, 1
• Step 1: list the Sequential Coalitions
𝑷𝟏 , 𝑷𝟐 , 𝑷𝟑
𝑷𝟐 , 𝑷𝟑 , 𝑷𝟏
𝑷𝟏 , 𝑷𝟑 , 𝑷𝟐
𝑷𝟑 , 𝑷𝟏 , 𝑷𝟐
𝑷𝟐 , 𝑷𝟏 , 𝑷𝟑
𝑷𝟑 , 𝑷𝟐 , 𝑷𝟏
• Step 2: find the pivotal players
𝑷𝟐
𝑷𝟏
𝑷𝟑
𝑷𝟏
𝑷𝟏
𝑷𝟏
• Step 3: Count the pivotal player for each
player.
𝑆𝑆1 = 4, 𝑆𝑆2 = 1, 𝑆𝑆3 = 1
• Step 4: Shapley-Shubik power Distribution
4
𝜎1 = = 66.67%
6
1
𝜎2 = = 16.67%
6
1
𝜎3 = = 16.67%
6
You can do
#26, 28, 30
#25
• 16: 9,8,7
(a) List all the sequential coalitions, and ID the
pivotal players.
9, 𝟖, 7
8, 𝟗, 7
7, 𝟗, 8
9, 𝟕, 8
8,7, 𝟗
7,8, 𝟗
#25
• 16: 9,8,7
(b) Find the Shapley-Shubik distribution
𝑇=6
𝑆𝑆1 = 4
𝑆𝑆2 = 1
𝑆𝑆3 = 1
4
1
1
𝜎1 =
𝜎2 =
𝜎3 =
6
6
6
#27
• Find the Shapley-Shubik power Distribution of
each
(a) 15: 16,8,4,1
(b) 18: 16,8,4,1
(c) 24: 16,8,4,1
(d)[28: 16,8,4,1]
#27
• Sequential Coalitions
16,8,4,1
16,8,1,4
16,1,8,4
16,1,4,8
16,4,1,8
16,4,8,1
8,16,4,1
8,16,1,4
8,4,16,1
8,4,1,16
8,1,4,16
8,1,16,4
4,16,8,1
4,16,1,8
4,8,16,1
4,8,1,16
4,1,16,8
4,1,8,16
1,16,8,4
1,16,4,8
1,8,16,4
1,8,4,16
1,4,16,8
1,4,8,16
Problems
#2, 4, 6
8, 10, 12,
14, 18, 20,
22, 26, 28,
30, 38, 40