1. Homework Read Pages 6 – 23 Page 30: 18, 19, 24, 28, 29, 30, 33, 59, 60, 71

2. Office hours Wednesday 1/2410:00 – 11:00 12:00 – 1:00 1:00 – 2:30 Thursday 1/2510:00 – 11:00 11:00 – 12:00 http://staff.kings.edu/mrevans

3. Core 120 Math lab The mathematics help lab provides students the opportunity to receive assistance toward understanding the topics presented in their Core 120 or class. The lab will give students the chance to ask questions from their class notes or text, and then reinforce their understanding of these topics by working on related problems from the text. The lab will also provide assistance in comprehending and working homework exercises that have been previously attempted by the student. Any student who has questions in these classes is invited to attend for any part of the times listed below. All help will be provided in small group sessions on a walk-in basis. Topics will be covered at a first come first serve basis, with the depth of each topic depending on the number of questions asked. Instruction and assistance in the lab will be provided by Marc Evans, King’s College Mathematics Specialist. Core 120 – Quantitative Reasoning Reinforcement Lab –Mondays6:00 – 7:30 pmAdministration 426 –Thursdays4:00 – 5:30 pmAdministration 426

4. Put problems that we will look at up on board Which problems did the group have difficulty with Pick groups to put up 19a, 29 Discuss 19, 24, 29, 59

5. Which methods satisfy which criterion? YN N N N N N Y

6. The Condorcet Criterion (page 8) - If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election.

7. IV.THE METHOD OF PAIRWISE COMPARISONS TABLE 1-11

8. TABLE 1-12

9. IV.THE METHOD OF PAIRWISE COMPARISONS (page 17) Each candidate is matched one-on-one with with every other candidate. Each of these one-on-one matchups is called a pairwise comparison. In a pairwise comparison between candidates X and Y each vote is assigned to either X or Y, the vote going to whichever of the candidates is higher on the ballot. The winner of the pairwise comparison is the one with the most votes and a win is worth one point. In case of a tie each candidate gets 1/2 point. The winner of the election is the candidate with the most points after all of the pairwise comparisons are tabulated. In the case of a tie we can have more than one winner or use a predetermined tie-breaking procedure.

10. IV.THE METHOD OF PAIRWISE COMPARISONS TABLE 1-11

11. Objective 4: Synthesizing results from previous discoveries to create new ideas

12. Winner MASVoting Method AlishaPlurality BorisBorda Count DavePlurality-with-elimination CarmenPairwise comparisons MATHEMATICS APPRECIATION SOCIETY FINAL RESULTS What’s wrong with the method of pairwise comparisons?

13. A vs B: 7 votes to 15 votes. B gets 1 point A vs C: 16 votes to 6 votes. A gets 1 point A vs D: 13 votes to 9 votes. A gets 1 point A vs E: 18 votes to 4 votes. A gets 1 point B vs C: 10 votes to 12 votes. C gets 1 point B vs D: 11 votes to 11 votes. B gets.5 point, D gets.5 point B vs E: 14 votes to 8 votes. B gets 1 point C vs D: 12 votes to 10 votes. C gets 1 point C vs E: 10 votes to 12 votes. E gets 1 point D vs E: 18 votes to 14 votes. D gets 1 point FINAL TALLY A-3 B-2.5 C-2 D-1.5 E-1 Winner is A. Example 1.12. LA Laxers Draft Choices (page 19) TABLE 1-13

14. A vs B: 7 votes to 15 votes. B gets 1 point A vs D: 13 votes to 9 votes. A gets 1 point A vs E: 18 votes to 4 votes. A gets 1 point B vs D: 11 votes to 11 votes. B gets.5 point, D gets.5 point B vs E: 14 votes to 8 votes. B gets 1 point D vs E: 18 votes to 14 votes. D gets 1 point FINAL TALLY A-2 B-2.5 D-1.5 E-0 Winner is B. TABLE 1-14

15. The Independence-of-Irrelevant-Alternatives Criterion. If choice X is a winner of an election and one (or more) of the other choices is removed and the ballots recounted, then X should still be a winner of the election. The method of pairwise comparisons violates the independence-of-alternatives criterion. 4 th criteria for a fair election:

16. Which methods satisfy which criterion? YN N N N N N YN

17. The method of pairwise comparisons may end in a tie. What else is wrong with the method of pairwise comparisons? H vs P: 9 votes to 2 votes. H gets 1 point P vs D: 6 votes to 5 votes. P gets 1 point D vs H: 7 votes to 4 votes. D gets 1 point FINAL TALLY H-1 P-1 D-1 Tie. Winner is ? Example 1.13. Hockey Team Dinner Preferences TABLE 1-15

18. NFL tiebreakers

19. With N candidates there are : 1+2+3+…+(N-3)+(N-2)+(N-1)= How many pairwise comparisons?

20. Homework Read Pages 19 - 26, 42 – 46 Page 35: 35, 38, 63, 64