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 . Simple Majority Supermajority Quota S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 2 / 11
Coalitions Definition A coalition is a nonempty set of voters. Winning Coalition Losing Coalition S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 3 / 11
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. S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 4 / 11
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% S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 5 / 11
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? { P 1 , P 4 , P 6 } { P 2 , P 3 , P 6 } { P 2 , P 3 , P 4 } { P 3 , P 4 , P 5 , P 6 } S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 6 / 11
Number of Coalitions 2 n − 1 Example In a weighted voting system with 15 voters, how many coalitions are possible? S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 7 / 11
“Special” Voters Dictators Dummies Veto Power Critical Voter S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 8 / 11
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] S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 9 / 11
Banzhaf Power Index Banzhaf Power Total Banzhaf Power Banzhaf Power Index S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 10 / 11
Exercise For [6 | 5 , 3 , 1], find the Banzhaf power index for each voter. S Butler (Michigan Tech) Chapter 3 Section 3 September 22, 2006 11 / 11
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