The Mathematics of Elections Kim Klinger-Logan St. Olaf College February 2019 It’s not the voting that’s democracy; it’s the counting. – Tom Stoppard, Jumpers
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Ranked Choice Voting Consider the following preference ballot in a Math Club Election. 1st: Carmen 2nd: Ben 3rd: Darius 4th: Alice
Ranked Choice Voting Consider the following preference ballot in a Math Club Election. 1st: Carmen 2nd: Ben 3rd: Darius 4th: Alice What does this ballot tell us about the voter’s preferences?
Ranked Choice Voting Consider the following preference ballot in a Math Club Election. 1st: Carmen 2nd: Ben 3rd: Darius 4th: Alice What does this ballot tell us about the voter’s preferences? Well it obviously tells us the order of the voter’s preferences. It also tells us unequivocally which candidate the voter would choose between any two candidates (i.e. between A and B the voter would choose B). Finally, the relative preferences of the ballot would not change if one of the candidates withdraws or is eliminated.
Ranked Choice Voting Suppose we have the following small batch of ballots 1st: C 1st: D 1st: C 1st: A 1st: C 2nd: D 2nd: C 2nd: B 2nd: B 2nd: B 3rd: B 3rd: B 3rd: D 3rd: C 3rd: D 4th: A 4th: A 4th: A 4th: D 4th: A 1st: A 1st: D 1st: A 1st: B 1st: A 2nd: B 2nd: C 2nd: B 2nd: D 2nd: B 3rd: C 3rd: B 3rd: C 3rd: C 3rd: C 4th: D 4th: A 4th: D 4th: A 4th: D A: Alice, B: Ben, C: Carmen, and D: Darius How do we count these ballots? Number of Voters: 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A
Ranked Choice Voting Suppose we have the following small batch of ballots 1st: C 1st: D 1st: C 1st: A 1st: C 2nd: D 2nd: C 2nd: B 2nd: B 2nd: B 3rd: B 3rd: B 3rd: D 3rd: C 3rd: D 4th: A 4th: A 4th: A 4th: D 4th: A 1st: A 1st: D 1st: A 1st: B 1st: A 2nd: B 2nd: C 2nd: B 2nd: D 2nd: B 3rd: C 3rd: B 3rd: C 3rd: C 3rd: C 4th: D 4th: A 4th: D 4th: A 4th: D A: Alice, B: Ben, C: Carmen, and D: Darius How do we count these ballots? Number of Voters: 4 2 2 1 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A
Math Club Election Suppose we count the rest of the ballots and we end up with the following preference schedule: Number of Voters: 14 10 8 4 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A A: Alice, B: Ben, C: Carmen, and D: Darius
Math Club Election Suppose we count the rest of the ballots and we end up with the following preference schedule: Number of Voters: 14 10 8 4 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A A: Alice, B: Ben, C: Carmen, and D: Darius Who is the winner of the election?
I. Plurality Method The candidate with the most first place votes is winner, the candidate with the second most votes is second, and so on.
I. Plurality Method The candidate with the most first place votes is winner, the candidate with the second most votes is second, and so on. What happens in the Math Club Election using the Plurality Method?
I. Plurality Method The candidate with the most first place votes is winner, the candidate with the second most votes is second, and so on. What happens in the Math Club Election using the Plurality Method? Number of Voters: 14 10 8 4 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A
I. Plurality Method The candidate with the most first place votes is winner, the candidate with the second most votes is second, and so on. What happens in the Math Club Election using the Plurality Method? Number of Voters: 14 10 8 4 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A Alice is 1st, Carmen is 2nd, Darius is 3rd and Ben is 4th.
Plurality Method What is the appeal of this method?
Plurality Method What is the appeal of this method? simplicity
Plurality Method What is the appeal of this method? simplicity What are the potential problems with this method?
Plurality Method What is the appeal of this method? simplicity What are the potential problems with this method? 1. When there are more than two candidates we can end up with a winner that does not have more than 50% of the votes.
Plurality Method What is the appeal of this method? simplicity What are the potential problems with this method? 1. When there are more than two candidates we can end up with a winner that does not have more than 50% of the votes. 2. The closeness of the election: This causes it to be the most easily manipulated by insincere voters.
Plurality Method What is the appeal of this method? simplicity What are the potential problems with this method? 1. When there are more than two candidates we can end up with a winner that does not have more than 50% of the votes. 2. The closeness of the election: This causes it to be the most easily manipulated by insincere voters. 3. A candidate may be preferred by voters over all other candidates yet not win.
Plurality Method What is the appeal of this method? simplicity What are the potential problems with this method? 1. When there are more than two candidates we can end up with a winner that does not have more than 50% of the votes. 2. The closeness of the election: This causes it to be the most easily manipulated by insincere voters. 3. A candidate may be preferred by voters over all other candidates yet not win. Number of Voters: 14 10 8 4 1 1st: A C D B C 2nd: B B C D D 3rd: C D B C B 4th: D A A A A
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points.
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points.The candidate with the most points wins the candidate with the second most points gets second place, and so on.
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points.The candidate with the most points wins the candidate with the second most points gets second place, and so on. What happens in the Math Club Election using the Borda Count Method?
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points.The candidate with the most points wins the candidate with the second most points gets second place, and so on. What happens in the Math Club Election using the Borda Count Method? Number of Voters: 14 10 8 4 1 1st (4pts): A C D B C 2nd (3pts): B B C D D 3rd (2pts): C D B C B 4th (1pts): D A A A A
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points. The candidate with the most points wins the candidate with the second most points gets second place, and so on. What happens in the Math Club Election using the Borda Count Method? Number of Voters: 14 10 8 4 1 1st (4pts): A (56) C D B C 2nd (3pts): B B C D D 3rd (2pts): C D B C B 4th (1pts): D A (10) A (8) A (4) A (1) A: 56+10+8+4+1 = 79 pts
II. Borda Count Method In an election with n candidates, we give 1 point for last place, 2 points for second from the last place and so on so that the first place candidate gets n points. The candidate with the most points wins the candidate with the second most points gets second place, and so on. What happens in the Math Club Election using the Borda Count Method? Number of Voters: 14 10 8 4 1 1st (4pts): A (56) C (40) D (32) B (16) C (4) 2nd (3pts): B (42) B (30) C (24) D (12) D (3) 3rd (2pts): C (28) D (20) B (16) C (8) B (2) 4th (1pts): D (14) A (10) A (8) A (4) A (1) A: 56+10+8+4+1 = 79 pts B: 42+30+16+16+2 = 106 pts Ben wins (then C: 28+40+24+8+4 = 104 pts Carmen, Darius D: 14+20+32+12+3 = 81 pts and Alice).
III. Plurality-with-Elimination Method ◮ Round 1: Count the 1st place votes for each candidate, just as you would in the plurality method. If a candidate has a majority of 1st place votes, then that candidate is the winner. Otherwise, eliminate the candidate(s) with the fewest 1st place votes. ◮ Round 2: Cross out the names of the candidates eliminated from the preference schedule and transfer those votes to the next eligible candidates on those ballots. Recount the votes. If a candidate has a majority then declare that candidate the winner. Otherwise eliminate the candidate with the fewest votes. ◮ Round 3, 4, . . . : Repeat the process, each time eliminating the candidate with the fewest votes and transferring those votes to the next eligible candidate. Continue until there is a candidate with the majority. That candidate is the winner of the election.
Plurality-with-Elimination Method What happens in the Math Club Election using the Plurality-with-Elimination Method?
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