Math and Elections A Lesson in the Math

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

About This Presentation 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 2003 -04 and 2004 -05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami Nov. 2004 Edition Released First Nov. 2004 Revised Math and Elections Revised 2

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 What type of drink should the cafeteria serve with school lunches this year? Apple juice Nov. 2004 Math and Elections Orange juice Ralph Nader Grape juice David Cobb Milk 3

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

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

True, When We Have Only 2 Choices Apple juice George Bush John Kerry Orange juice Proposition 71: □ Yes □ No Blank or doubly marked votes do not count Only one possible complication: tie votes (no winner, prop fails) 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 Nov. 2004 A A Math and Elections O O O M M M MM 6

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

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

Activity 1: Polling 1. On small pieces of paper, vote for: A O M A A O O O A A Apple juice Orange juice Milk Nov. 2004 M M M M A 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? M O Math and Elections M 9

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

Indicating First and Second Choices A A O O O A A M M M Second choices: M A’s prefer O over M O’s prefer A over M M’s half are A > O, the other half O > A M A A Number of A kids who prefer O over M 4 5 O Nov. 2004 Number of A kids who prefer M over O 4 0 5 8 M O Math and Elections 4 0 4 M 11

Vote Tallying in Rounds A A O O O M M M M A 4 0 95 Collect the ordered choices of voters Remove the lowest vote getter (A) O 4 0 4 M 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 Nov. 2004 Math and Elections 12

Borda Voting A A O O O M M M M 2 points for 1 st choice 1 point for 2 nd choice 0 point for 3 rd choice M A Number of A kids who prefer O over M A points: 4 × 2 + 9 × 1 = 17 O points: 5 × 2 + 8 × 1 = 18 M points: 8 × 2 + 0 × 1 = 16 4 0 Is this outcome fair? 5 No, M has the most first-place votes Yes, O would win against A or M Nov. 2004 Number of A kids who prefer M over O O Math and Elections 4 0 4 M 13

Activity 2: Ordered Preferences 1. On small pieces of paper, vote for your first and second choices among A, O, M A A O O O A A M M M M A 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? Nov. 2004 O Math and Elections M 14

Activity 3: A Variant of Borda Voting A A O O O M M M M What happens if we change the points to: 3 for 1 st choice 2 for 2 nd choice 1 for 3 rd choice A 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. 2004 4 0 5 O Math and Elections 4 0 4 M 15

Activity 4: Borda Voting A A O O O M M M M A A points: __ × 2 + __ × 1 = __ O points: __ × 2 + __ × 1 = __ M points: __ × 2 + __ × 1 = __ Number of A kids who prefer O over M Number of A kids who prefer M over O 4 0 Is the outcome fair? ____________________________ Nov. 2004 Show that if one of the M voters changes his/her 2 nd choice, A can win 5 O Math and Elections 4 0 4 M 16

Borda Voting: Conspiracy A A O O O M M M M Number of A kids who prefer O over M A: 4 × 2 + 10 × 1 = 18 points O: 5 × 2 + 6 × 1 = 16 points M: 8 × 2 + 1 × 1 = 17 points Number of A kids who prefer M over O 4 01 Is this outcome fair? No, M would not win against A or O 3 5 Yes, M has the most 1 st place votes Nov. 2004 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) A O Math and Elections 0 45 43 M 17

Approval Voting Each voter lists all the choices that are acceptable to him/her A A AG AG A A AO AO AO O O GO GG G A, G A Votes are tallied and the total for each choice is found G Approval voting makes majority vote more likely 2 4 A = 9, G = 8, O = 11 (wins) Nov. 2004 GO GO 3 0 A, O Math and Elections 3 3 G, O 5 O 18

Activity 5: Approval Voting 1. On small pieces of paper, vote for all your approved juice choices among A, O, G A A AG AG A A AO AO AO GO GG G A, G A 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? Nov. 2004 O O O GO GO Math and Elections G A, O G, O O 19

Conclusions A A O O O M M M M 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. Run-off election among the top two vote getters solves some, but not all, of the problems. Ordering all the candidates, and not just voting for the top one, combined with Borda voting (point system) is usually the best. Nov. 2004 Math and Elections 20

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