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Chapter 3 Section 3 MA1020 Quantitative Literacy Sidney Butler Michigan Technological University September 22, 2006 S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 1 / 11 Weighted Voting Systems Notation: P n and W n .

SLIDE 1

Chapter 3 Section 3

MA1020 Quantitative Literacy Sidney Butler

Michigan Technological University

September 22, 2006

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SLIDE 2

Weighted Voting Systems

Notation: Pn and Wn. Simple Majority Supermajority Quota

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SLIDE 3

Coalitions

Definition A coalition is a nonempty set of voters. Winning Coalition Losing Coalition

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Exercise

Suppose representatives for five zones have voting weights of 4, 6, 2, 8, and 10, respectively.

1 If passing a motion requires a simple majority of yes votes, then what

is the smallest weight required to pass a motion?

2 If passing a motion requires a two-thirds supermajority of yes votes,

then what is the smallest weight required to pass a motion?

3 If the quota is 25, then give the notation for this weighted voting

system.

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Exercise

The following set of numbers represents the weights assigned to voters in a weighted voting system. Following the weight is the percentage required for measures to pass. Determine the quota and express the weighted voting system using the proper notation. [8, 5, 5, 3, 3, 2]; 60%

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Exercises

Example Consider the weighted voting system [14|5, 4, 3, 2]. What must happen in order to pass a motion? Example For the weighted voting system [16|9, 7, 6, 4, 3, 2], determine if the following coalitions of voters are winning or losing coalitions? {P1, P4, P6} {P2, P3, P6} {P2, P3, P4} {P3, P4, P5, P6}

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Number of Coalitions

2n − 1 Example In a weighted voting system with 15 voters, how many coalitions are possible?

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“Special” Voters

Dictators Dummies Veto Power Critical Voter

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Exercise

Identify voters who are dictators, dummies or have veto power. [8|5, 4, 3] [25|14, 13, 12, 8] [7|7, 2, 2, 2]

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SLIDE 10

Banzhaf Power Index

Banzhaf Power Total Banzhaf Power Banzhaf Power Index

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Exercise

For [6|5, 3, 1], find the Banzhaf power index for each voter.

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