Discrete Math Weighted Voting

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

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

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 2 <𝑞≤𝑉 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] 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

#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]

#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

#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.

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− 𝑤 𝑝 <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 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

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 %

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=33 1 3 %, 𝐷=33 1 3 %, 𝐼=33 1 3 % 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] 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

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

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 :

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

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

#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

#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]

#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%

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

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

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

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!= 5!= 4∗3∗2∗1=24 5∗4∗3∗2∗1=120

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

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%

You can do #26, 28, 30

#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

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

#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]

#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