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class+January+25 - Math 103, Section 11, January 25, 2010...

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Math 103, Section 11, January 25, 2010 1. Go over Gateway Quiz 2. Go over homework. Exercises 33,37 and 59 3. Give out Voting Methods and Fairness Criteria filled in table. 4. Give out Instructions on Getting your Homework to Sakai sheet. 5. Homework due dates: Exercises 12,18,20,24 due on Friday, January 28. Exercises 34,38,42,60 due on Tuesday, February 1. We will be using electronic submission of homework. There are 2 methods you can use. Method #1: Type your homework into a Word document. Method#2: Write your homework on paper and then scan it. In either case, you will upload a file to Sakai, Assignments section. I will write up the details for you to follow. The good news is that there is a scanner in Tillett. I will be happy to show you how to use it before or after class. 6. Section 1.5 The Method of Pairwise Comparisons 7. Section 1.6 Rankings 8. Homework to be discussed in class on Friday, January 28. Exercises 41, 55, 56, and 57. 9. Quiz on Friday, January 28. Quiz will cover chapter 1 and will last about 15 minutes.
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Solutions to Homework a. There are 31 voters. 16 votes are needed for a majority. Plurality-with-elimination is also called instant runoff voting. It is used in San Francisco and Burlington, Vermont. A B C D Round 1 10 9 5 7
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Round 2 10 9 out 7+5=12 Round 3 10+5=15 out out 12+4=16 D wins with 16 votes, a majority. b. Find the Condorcet candidate: Compare A to B. 10+5=15 prefer A. 16 prefer B. More prefer B. Compare A to C. 10+7=17 prefer A. 14 prefer C. More prefer A. Compare A to D. 10+5=15 prefer A. 16 prefer D. More prefer D. Compare B to C. 7+5+4=16 prefer B. 15 prefer C. More prefer B. Compare B to D. 10+5+4=19 prefer B. 12 prefer D. More prefer B. Compare C to D. 10+5+5+4=24 prefer C. 7 prefer D. More prefer C. B wins head to head against A. B wins head to head against C. B wins head to head against D. So B is the Condorcet candidate. c. The Condorcet Fairness Criterion. The plurality-with-elimination method of voting violates the Condorcet Fairness Criterion. The Condorcet candidate B should win the election, but candidate D wins the election by the plurality by elimination method.
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There are a total of 5+3+5+3+2+3=21 voters. Compare A to B: 5+3+5=13 prefer A. 8 prefer B. A gets 1 point. Compare A to C: 5+3+3=11 prefer A. 10 prefer C. A gets 1 point. Compare A to D: 5+3+3=11 prefer A. 10 prefer D. A gets 1 point. Compare B to C. 5+3+3=11 prefer B. 10 prefer C. B gets 1 point. Compare B to D. 5+3=8 prefer B. 13 prefer D. D gets 1 point. Compare C to D. 5+5+3=13 prefer C. 8 prefer D. C gets 1 point. A gets 3 points, B gets 1 point. C gets 1 point. D gets 1 point. A wins the election using the method of pairwise comparisons.
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a.There are 13 voters. 7 votes are needed for a majority. A is a majority candidate. Compare A to B: 7+2=9 prefer A.
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This note was uploaded on 01/20/2012 for the course MATH 103 taught by Professor Berkowitz during the Spring '07 term at Rutgers.

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class+January+25 - Math 103, Section 11, January 25, 2010...

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