Discrete Math Weighted Voting What was the name

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

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 •

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

#5 • [49: 4 x, 2 x, 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] [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] P 1 D, all d p 3 p 4 d p 1 V, p 4 d, p 1 p 2 V,

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 •

Lets look at Congress • 99 Republicans • 98 Democrats • 3 Independents • List the winning coalitions: Coalition Weight

Who are the critical players Coalition Weight Critical Players

Banzhaf Power Index •

Back in Congress • Coalition Weight Critical Players

Find the Power Index of •

Winning Coalitions Critical Players NONE

Find the % •

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

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 •

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!!! •

• Step 1: list the Sequential Coalitions • Step 2: find the pivotal players

You can do #26, 28, 30

#27 • Sequential Coalitions

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