1. Discrete Math
Weighted Voting

2. What was the name of the company
What were the items being sold
How many people on the board
Describe a weighted voting system

3. 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

4. Over 50% is needed to pass motion 7:5, 3, 3, 2
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: 𝑤 1 , 𝑤 2 , …, 𝑤 𝑛 𝑤ℎ𝑒𝑟𝑒 𝑤 1 ≥ 𝑤 2 ≥…≥ 𝑤 𝑛

5. Venture Capitalism
Everything is fine

6. Anarchy
The quota is too low

7. Gridlock
The quota is too high

8. One Partner- One Vote
The quota is so high that the decision must be unanimous

9. Dictator
One person has enough weight to pass a motion

10. Unsuspecting Dummies
No matter how the person votes it will not help pass the motion

11. Veto Power
One person has enough votes to reject a motion but not enough to pass it by themselves

12. Your quota should be
𝑉 2 <𝑞≤𝑉 𝑤ℎ𝑒𝑟𝑒 𝑉 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑣𝑜𝑡𝑒𝑠

13. You can now do
#2, 4, 6 8, 10

14. #3 [𝑞:10, 6, 5, 4, 2] Smallest value for q Largest value for q
What is q if at least 2/3 majority is needed
What is q if over 2/3 majority is needed

15. #5 [49:4x, 2x, x, x, x] 49 is a simple majority
49 is more than 2/3 majority
49 is more than ¾ majority
[49:48, 24, 12, 12]
[49:36, 18, 9, 9]
[49:32, 16, 8, 8]

16. #7 Is there any dictators, veto holders, or dummies [15: 16, 8, 4, 1]
[18: 16, 8, 4, 1]
[24: 16, 8, 4, 1]
P1 D, all d p1 V, p4 d, p1p2 V, p3p4 d

17. #9 [q:8, 4, 2] All have veto power 𝑃 2 has veto power but 𝑃 3 does not
𝑃 3 is the only dummy.

18. 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

19. 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

20. Critical Players
The players in the winning coalition that are needed for the coalition to win
They are a critical player if:
𝑊− 𝑤 𝑝 <𝑞

21. 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

22. Who are the critical players
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
Coalition
Weight
{𝑅,𝐷}
197
{𝑅, 𝐼}
102
{𝐷, 𝐼}
101
{𝑅, 𝐷, 𝐼}
200

23. Banzhaf Power Index Make list of all winning coalitions
Find the critical players of all winning coalitions
Count the total number of times 𝑃 1 is the critical player. This is 𝐵 1 , then repeat for all players
Add all 𝐵s together and this is 𝑇
Find the ratio of each 𝐵 over 𝑇. This is now 𝛽 1 . Put in terms of a %

24. Back in Congress How many Critical Players? How many for R? D? I?6
How many for R? D? I?
2, 2, 2
What are the %?
𝑅= %, 𝐷= %, 𝐼= %
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

25. Find the Power Index of [5:3, 2, 1, 1, 1] 𝑃 1 𝑃 2 𝑃 3 𝑃 4 𝑃 5 𝑷 𝟏 𝑷 𝟐
𝑷 𝟏 𝑷 𝟐
𝑃 1 𝑃 3
𝑃 1 𝑃 4
𝑃 1 𝑃 5
𝑃 2 𝑃 3
𝑃 2 𝑃 4
𝑃 2 𝑃 5
𝑃 3 𝑃 4
𝑃 3 𝑃 5
𝑃 4 𝑃 5
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑
𝑷 𝟏 𝑷 𝟐 𝑷 𝟒
𝑷 𝟏 𝑷 𝟐 𝑷 𝟓
𝑷 𝟏 𝑷 𝟑 𝑷 𝟒
𝑷 𝟏 𝑷 𝟑 𝑷 𝟓
𝑷 𝟏 𝑷 𝟒 𝑷 𝟓
𝑃 2 𝑃 3 𝑃 4
𝑃 2 𝑃 3 𝑃 5
𝑃 2 𝑃 4 𝑃 5
𝑃 3 𝑃 4 𝑃 5
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟒
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟓
𝑷 𝟏 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓
𝑷 𝟏 𝑷 𝟐 𝑷 𝟒 𝑷 𝟓
𝑷 𝟐 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓

26. Winning Coalitions
Critical Players
𝑷 𝟏 𝑷 𝟐
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑
𝑷 𝟏 𝑷 𝟐 𝑷 𝟒
𝑷 𝟏 𝑷 𝟐 𝑷 𝟓
𝑷 𝟏 𝑷 𝟑 𝑷 𝟒
𝑷 𝟏 𝑷 𝟑 𝑷 𝟓
𝑷 𝟏 𝑷 𝟒 𝑷 𝟓
Winning Coalitions
Critical Players
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟒
𝑷 𝟏 𝑷 𝟐
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟓
𝑷 𝟏 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓
𝑷 𝟏
𝑷 𝟏 𝑷 𝟐 𝑷 𝟒 𝑷 𝟓
𝑷 𝟐 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓
𝑷 𝟏 𝑷 𝟐 𝑷 𝟑 𝑷 𝟒 𝑷 𝟓
NONE

27. Total Number of Critical Player: Critical Players for 𝑷 𝟏 : 28
Critical Players for 𝑷 𝟏 : 11
Critical Players for 𝑷 𝟐 : 8
Critical Players for 𝑷 𝟑 : 3
Critical Players for 𝑷 𝟒 :
Critical Players for 𝑷 𝟓 :

28. Find the % 𝛽 1 : 11 28 =39.29% 𝛽 2 : 8 28 =28.57% 𝛽 3 : 3 28 =10.71%
𝛽 1 : =39.29%
𝛽 2 : 8 28 =28.57%
𝛽 3 : 3 28 =10.71%
𝛽 4 : 3 28 =10.71%
𝛽 5 : 3 28 =10.71%

29. You can do
# 12, 14, 18, 20, 22

30. #11
10:6, 5, 4, 2
What is the weight of the coalition formed by 𝑃 1 𝑎𝑛𝑑 𝑃 3
What are all the winning coalitions
Who is the critical players in { 𝑃 1 , 𝑃 2 , 𝑃 3 }
Find the Banzhaf Power Index

31. #13 Find the Banzhaf Power Index of 6:5, 2, 1
𝛽 1 =60%
𝛽 2 =20%
𝛽 3 =20%
Find the Banzhaf Power Index of [3:2, 1, 1]

32. #19
A weighted voting system has 3 players. The only winning coalitions are the following: 𝑃 1 , 𝑃 2 , 𝑃 1 , 𝑃 3 , 𝑃 1 , 𝑃 2 , 𝑃 3
Find the Critical Players of each
𝑃 1 , 𝑃 2 𝑃 1 , 𝑃 3 𝑃 1
Find the Banzhaf Power Index
60% 20% 20%

33. Where did Banzhaf come from
Nassau County
Find the Banzhaf Power Index of Nassau County
District
Weight
Hempstead #1 31
Hempstead #2
Oyster Bay 28
North Hempstead 21
Long Beach 2
Glen Cove

34. Shapley- Shubik
Sequential Coalition: a coalition that the order matters.
Pivotal Player: The person that cast the winning vote.

35. List all the sequential coalitions
𝑃 1 , 𝑃 2 , 𝑃 3
𝑃 1 , 𝑃 2 , 𝑃 3
𝑃 1 , 𝑃 3 , 𝑃 2
𝑃 2 , 𝑃 1 , 𝑃 3
𝑃 3 , 𝑃 1 , 𝑃 2
𝑃 2 , 𝑃 3 , 𝑃 1
𝑃 3 , 𝑃 2 , 𝑃 1

36. How many Sequential Coalitions will we have?
The multiplication rule: If there is X choices and Y choices we have X * Y total choices

37. Factorial!!! 𝑁!=𝑁 ∗ 𝑁−1 ∗ 𝑁−2 ∗ …∗3∗2∗1 4!= 5!= 4∗3∗2∗1=24
5∗4∗3∗2∗1=120

38. 4:3, 2, 1 Step 1: list the Sequential Coalitions
Step 2: find the pivotal players
𝑷 𝟏 , 𝑷 𝟐 , 𝑷 𝟑
𝑷 𝟏 , 𝑷 𝟑 , 𝑷 𝟐
𝑷 𝟐 , 𝑷 𝟏 , 𝑷 𝟑
𝑷 𝟐 , 𝑷 𝟑 , 𝑷 𝟏
𝑷 𝟑 , 𝑷 𝟏 , 𝑷 𝟐
𝑷 𝟑 , 𝑷 𝟐 , 𝑷 𝟏
𝑷 𝟐
𝑷 𝟑
𝑷 𝟏

39. Step 3: Count the pivotal player for each player.
𝑆𝑆 1 =4, 𝑆𝑆 2 =1, 𝑆𝑆 3 =1
Step 4: Shapley-Shubik power Distribution
𝜎 1 = 4 6 =66.67%
𝜎 2 = 1 6 =16.67%
𝜎 3 = 1 6 =16.67%

40. You can do
#26, 28, 30

41. #25
16:9,8,7
List all the sequential coalitions, and ID the pivotal players.
9,𝟖,7
9,𝟕,8
8,𝟗,7
8,7,𝟗
7,𝟗,8
7,8,𝟗

42. #25 16:9,8,7 (b) Find the Shapley-Shubik distribution 𝑇=6 𝑆𝑆 1 =4
𝑆𝑆 2 =1
𝑆𝑆 3 =1
𝜎 1 = 𝜎 2 = 𝜎 3 = 1 6

43. #27 Find the Shapley-Shubik power Distribution of each 15:16,8,4,1
18:16,8,4,1
24:16,8,4,1
[28:16,8,4,1]

44. #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

45. Problems
#2, 4, 6 8, 10, 12, 14, 18, 20, 22, 26, 28, 30, 38, 40