Nov. 2004Math and ElectionsSlide 1 Math and Elections A Lesson in the “Math + Fun!” Series

Nov. 2004Math and ElectionsSlide 2 About This Presentation EditionReleasedRevised FirstNov This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during the and school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami

Nov. 2004Math and ElectionsSlide 3 We Vote to Choose Our Leaders or Indicate Our Preferences Who would you like to be our president (senator, school board member) for the next few years? George Bush John Kerry David Cobb Ralph Nader Apple juice Orange juice Grape juice Milk What type of drink should the cafeteria serve with school lunches this year?

Nov. 2004Math and ElectionsSlide 4 We Use Different Voting Methods Marked ballot with optical reading Write-in ballot with manual counting Computerized touch-screen voting Punched-card or punched-paper ballot processed by special reader devices

Nov. 2004Math and ElectionsSlide 5 Isn’t Counting All There Is to Voting? 20 kids voting O O O O O MM M M M M M M G G G A A A A 4 prefer apple juice 5 prefer orange juice 3 prefer grape juice 8 prefer milk

Nov. 2004Math and ElectionsSlide 6 True, When We Have Only 2 Choices George Bush John Kerry Apple juice Orange juice Blank or doubly marked votes do not count Only one possible complication: tie votes (no winner, prop fails) Proposition 71: □ Yes □ No Things get tricky as soon as we go to three or more choices In 1952, mathematical economist Kenneth Arrow proved that there is no consistent method of making a fair choice among 3 or more candidates All examples to follow will assume three choices; you can imagine that problems can only get worse if there are more than three choices O O O O O MM M M M M M M A A A A

Nov. 2004Math and ElectionsSlide 7 Majority and Plurality Voting 17 kids voting O O O O O MM M M M M M M A A A A Juice or Milk? Juice gets a majority of votes (majority means more than half) AJ, OJ, or Milk? Milk gets a plurality of votes (plurality means more than others)

Nov. 2004Math and ElectionsSlide 8 Meaning of Fairness in Voting Results of 2-way races: O or A?9 to 8 O or M?9 to 8 A or M?9 to 8 O should win! A solution: Run-off between the top two vote getters A MO 4 85 In a 3-way race: A gets 4 votes O gets 5 votes M gets 8 votes So, M wins! O O O O O MM M M M M M M A A A A Juice people always prefer juice to milk; milk people are equally divided among A and O as second choice

Nov. 2004Math and ElectionsSlide 9 Activity 1: Polling 1. On small pieces of paper, vote for: A Apple juice O Orange juice M Milk 2. Collect and tally the votes; enter results in this triangle. 3. Suppose after the vote has been tallied, you are informed that the top choice is no longer available. Can you make a fair choice without voting again? A MO O O O O O MM M M M M M M A A A A

Nov. 2004Math and ElectionsSlide 10 Indicating Two-Way Preferences No it does not! if 5 > 3 and 3 > 2 then 5 > 2 Alice’s preferences: A over O O over M M over A Does this make sense? A voter who prefers A > O, O > M, and M > A is “confused” 523 Nonconfused voters can order their choices from most to least desirable: e.g., A > O > M Apple juice Orange juice Milk

Nov. 2004Math and ElectionsSlide 11 Indicating First and Second Choices O O O O O MM M M M M M M A A A A A MO Second choices: A’s prefer O over M O’s prefer A over M M’s half are A > O, the other half O > A A M 4 85 O Number of A kids who prefer O over M Number of A kids who prefer M over O

Nov. 2004Math and ElectionsSlide 12 Vote Tallying in Rounds O O O O O MM M M M M M M A A A A A MO Remove the lowest vote getter (A) Adjust the voter choices to account for the removed candidate Repeat the process with the remaining choices until only two candidates remain; then tally the votes as usual Collect the ordered choices of voters 9

Nov. 2004Math and ElectionsSlide 13 Borda Voting O O O O O MM M M M M M M A A A A A MO 2 points for 1 st choice 1 point for 2 nd choice 0 point for 3 rd choice Number of A kids who prefer O over M Number of A kids who prefer M over O A points: 4 × × 1 = 17 O points: 5 × × 1 = 18 M points: 8 × × 1 = 16 Is this outcome fair? No, M has the most first-place votes Yes, O would win against A or M

Nov. 2004Math and ElectionsSlide 14 Activity 2: Ordered Preferences 1. On small pieces of paper, vote for your first and second choices among A, O, M 2. Collect and tally the votes; enter results in this triangle. 3. Tally the votes in rounds 4. Tally the votes according to Borda voting rules 5. Are the results fair? Why? O O O O O MM M M M M M M A A A A A MO

Nov. 2004Math and ElectionsSlide 15 Activity 3: A Variant of Borda Voting O O O O O MM M M M M M M A A A A A MO What happens if we change the points to: 3 for 1 st choice 2 for 2 nd choice 1 for 3 rd choice A points: __ × 3 + __ × 2 + __ × 1 = ___ (was 17) O points: __ × 3 + __ × 2 + __ × 1 = ___ (was 18) M points: __ × 3 + __ × 2 + __ × 1 = ___ (was 16) Is the outcome fair? ____________________________

Nov. 2004Math and ElectionsSlide 16 Activity 4: Borda Voting O O O O O MM M M M M M M A A A A A MO Show that if one of the M voters changes his/her 2 nd choice, A can win Number of A kids who prefer O over M Number of A kids who prefer M over O A points: __ × 2 + __ × 1 = __ O points: __ × 2 + __ × 1 = __ M points: __ × 2 + __ × 1 = __ Is the outcome fair? ____________________________

Nov. 2004Math and ElectionsSlide 17 Borda Voting: Conspiracy O O O O O MM M M M M M M A A A A A MO Suppose one A > O > M and one M > O > A voter conspire to change their votes to A > M > O and M > A > O (i.e., each tries to help the other) Number of A kids who prefer O over M Number of A kids who prefer M over O A: 4 × × 1 = 18 points O: 5 × × 1 = 16 points M: 8 × × 1 = 17 points Is this outcome fair? Yes, M has the most 1 st place votes No, M would not win against A or O

Nov. 2004Math and ElectionsSlide 18 Approval Voting Each voter lists all the choices that are acceptable to him/her Votes are tallied and the total for each choice is found A = 9, G = 8, O = 11 (wins) Approval voting makes majority vote more likely A G O A,OG,O A,G AG O O O O O A A A A G G G GO AO

Nov. 2004Math and ElectionsSlide 19 Activity 5: Approval Voting 1. On small pieces of paper, vote for all your approved juice choices among A, O, G 2. Collect and tally the votes; enter results in this diagram. 3. Tally the approval votes and choose a winner. 4. Are the results fair? Why? A G O A,OG,O A,G AG O O O O O A A A A G G G GO AO

Nov. 2004Math and ElectionsSlide 20 Conclusions O O O O O MM M M M M M M A A A A When there are three or more choices, no voting method guarantees a fair outcome in all cases. Choosing the candidate or option with the most votes (plurality) is not a good idea, unless he/she/it has a majority of the votes. Ordering all the candidates, and not just voting for the top one, combined with Borda voting (point system) is usually the best. Run-off election among the top two vote getters solves some, but not all, of the problems.

Nov. 2004Math and ElectionsSlide 21 Next Lesson Thursday, December 2, 2004