The Mathematics of Elections Kim Klinger-Logan St. Olaf College - - PowerPoint PPT Presentation

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

In the news . . .

Bangor Daily News

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 2nd: D 3rd: B 4th: A 1st: D 2nd: C 3rd: B 4th: A 1st: C 2nd: B 3rd: D 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: C 2nd: B 3rd: D 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: D 2nd: C 3rd: B 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: B 2nd: D 3rd: C 4th: A 1st: A 2nd: B 3rd: C 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 2nd: D 3rd: B 4th: A 1st: D 2nd: C 3rd: B 4th: A 1st: C 2nd: B 3rd: D 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: C 2nd: B 3rd: D 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: D 2nd: C 3rd: B 4th: A 1st: A 2nd: B 3rd: C 4th: D 1st: B 2nd: D 3rd: C 4th: A 1st: A 2nd: B 3rd: C 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?

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes.

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes. No winner so eliminate B.

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes. No winner so eliminate B.

Round 2: B’s votes go to D. First place votes for . . . A=14, C=11, and D=12.

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes. No winner so eliminate B.

Round 2: B’s votes go to D. First place votes for . . . A=14, C=11, and D=12. No winner so eliminate C.

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes. No winner so eliminate B.

Round 2: B’s votes go to D. First place votes for . . . A=14, C=11, and D=12. No winner so eliminate C. Round 3: C’s votes go to D. First place votes for . . . A=14 and D=23. D is the winner (then A, C and B).

Plurality-with-Elimination Method

What happens in the Math Club Election using the Plurality-with-Elimination 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 Round 1: First place votes for . . . A=14, B =4, C=11, and D=8

• ut of 37 votes. No winner so eliminate B.

Round 2: B’s votes go to D. First place votes for . . . A=14, C=11, and D=12. No winner so eliminate C. Round 3: C’s votes go to D. First place votes for . . . A=14 and D=23. D is the winner (then A, C and B). Trump v. Clinton 2016

• IV. Method Pairwise Comparisons

Given any two candidates, we can count how many voters votes

• ne above the other and vice versa, the one with the highest

number wins the pairwise comparison. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner.

• IV. Method Pairwise Comparisons

Given any two candidates, we can count how many voters votes

• ne above the other and vice versa, the one with the highest

number wins the pairwise comparison. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner. What happens in the Math Club Election using the Method of Pairwise Comparisons?

• IV. Method Pairwise Comparisons

Given any two candidates, we can count how many voters votes

• ne above the other and vice versa, the one with the highest

number wins the pairwise comparison. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner. What happens in the Math Club Election using the Method of Pairwise Comparisons? 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

• IV. Method Pairwise Comparisons

Given any two candidates, we can count how many voters votes

• ne above the other and vice versa, the one with the highest

number wins the pairwise comparison. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner. What happens in the Math Club Election using the Method of Pairwise Comparisons? 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 Pairwise Comparison: Votes: Winner: A OR B A (14); B (23) B

• IV. Method of Pairwise Comparisons

Given any two candidates, we can count how many voters votes

• ne above the other and vice versa, the one with the highest

number wins the pairwise comparison. For each possible pairwise comparison between candidates, give 1 point to the winner and 0 points to the loser (if a tie, give each 1/2 a point). The candidate with the most points is the winner. What happens in the Math Club Election using the Method of Pairwise Comparisons? Pairwise Comparison: Votes: Winner: A OR B A (14); B (23) B A OR C A (14); C (23) C A OR D A (14); D (23) D B OR C B (18); C (19) C B OR D B (28); D (9) B C OR D C (25); D (12) C C=3, B=2, D=1, A=0. Thus C is the winner!

Summary of Results

For the Math Club Election we had the following outcomes

• I. Plurality Method: A, C, D, B
• II. Borda Count Method: B, C, D, A
• III. Plurality-with-Elimination Method: D, A, C, B
• IV. Method of Pairwise Comparisons: C, B, D, A

Summary of Results

For the Math Club Election we had the following outcomes

• I. Plurality Method: A, C, D, B
• II. Borda Count Method: B, C, D, A
• III. Plurality-with-Elimination Method: D, A, C, B
• IV. Method of Pairwise Comparisons: C, B, D, A

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 Of all of the methods discussed above which one is the best?

Fairness Criterion

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible.

Majority Criterion: A majority candidate should always be the winner.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible.

Majority Criterion: A majority candidate should always be the winner. Monotonicity Criterion: If candidate X is the winner, then X would still be the winner had a voter ranked X higher in his preference ballot.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible.

Majority Criterion: A majority candidate should always be the winner. Monotonicity Criterion: If candidate X is the winner, then X would still be the winner had a voter ranked X higher in his preference ballot.

◮ Unanimity: If every voter prefers one candidate over another,

the outcome will reflect that.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible. ◮ Unanimity: If every voter prefers one candidate over another,

the outcome will reflect that.

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: If

candidate X is the winner then X would still be the winner had one or more of the losing candidates not been in the race.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible. ◮ Unanimity: If every voter prefers one candidate over another,

the outcome will reflect that.

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: If

candidate X is the winner then X would still be the winner had one or more of the losing candidates not been in the race. Condorcet Criterion: A Condorcet candidate (i.e. a candidate that beats each of the other candidate in a pairwise comparison) should always be the winner.

Fairness Criterion

Arrow set a minimum set of requirements that a method should have:

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible. ◮ Unanimity: If every voter prefers one candidate over another,

the outcome will reflect that.

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: If

candidate X is the winner then X would still be the winner had one or more of the losing candidates not been in the race.

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible.

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates L(C) = the set of all full linear orderings of candidates

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates L(C) = the set of all full linear orderings of candidates P(C) = set of all preference schedules

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates L(C) = the set of all full linear orderings of candidates P(C) = set of all preference schedules = L(C) × L(C) × · · · × L(C)

• N times

= L(C)N for N = number of voters

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates L(C) = the set of all full linear orderings of candidates P(C) = set of all preference schedules = L(C) × L(C) × · · · × L(C)

• N times

= L(C)N for N = number of voters social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

Arrow’s Impossibility Theorem

For elections involving three or more candidates, a method for determining election results that is always fair is mathematically impossible. C = set of all candidates L(C) = the set of all full linear orderings of candidates P(C) = set of all preference schedules = L(C) × L(C) × · · · × L(C)

• N times

= L(C)N for N = number of voters social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

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 → 1st: C 2nd: D 3rd: B 4th: A

Proof of Arrow’s Impossibility Theorem

social welfare function: F : P(A) → L(A) (b1, b2, . . . , bN) → bW

◮ Non-dictatorship: There is no one voter who determines the

• utcome of every election.

◮ Universality: Each election has a unique outcome. ◮ Non-imposition: Every election outcome is possible. ◮ Unanimity: If every voter prefers one candidate over another,

the outcome will reflect that.

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: If

candidate X is the winner then X would still be the winner had one or more of the losing candidates not been in the race.

Proof of Arrow’s Impossibility Theorem

social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

◮ Non-dictatorship: F is not a projection. ◮ Universality: F is a well-defined function. ◮ Non-imposition: F is surjective. ◮ Unanimity: If A is ranked higher than B for all ballots bi for

i ∈ {1, . . . , N} then A should be ranked higher than B in F(b1, . . . , bN).

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: For

two preference schedules (b1, . . . , bN) and (c1, . . . , cN) if the candidates A and B have the same order in bi as in ci such that for all i, then A and B have the same order in F(b1, . . . , bN) and F(c1, . . . , cN).

Proof of Arrow’s Impossibility Theorem

social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

◮ Non-dictatorship: F is not a projection. ◮ Universality: F is a well-defined function. ◮ Non-imposition: F is surjective. ◮ Unanimity: If A is ranked higher than B for all ballots bi for

i ∈ {1, . . . , N} then A should be ranked higher than B in F(b1, . . . , bN).

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: For

two preference schedules (b1, . . . , bN) and (c1, . . . , cN) if the candidates A and B have the same order in bi as in ci such that for all i, then A and B have the same order in F(b1, . . . , bN) and F(c1, . . . , cN).

Proof of Arrow’s Impossibility Theorem

social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

◮ Non-dictatorship: F is not a projection. ◮ Universality: F is a well-defined function. ◮ Non-imposition: F is surjective. ◮ Unanimity: If A is ranked higher than B for all ballots bi for

i ∈ {1, . . . , N} then A should be ranked higher than B in F(b1, . . . , bN).

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: For

two preference schedules (b1, . . . , bN) and (c1, . . . , cN) if the candidates A and B have the same order in bi as in ci such that for all i, then A and B have the same order in F(b1, . . . , bN) and F(c1, . . . , cN).

Proof of Arrow’s Impossibility Theorem

social welfare function: F : P(C) → L(C) (b1, b2, . . . , bN) → bW

◮ Non-dictatorship: F is not a projection. ◮ Universality: F is a well-defined function. ◮ Non-imposition: F is surjective. ◮ Unanimity: If A is ranked higher than B for all ballots bi for

i ∈ {1, . . . , N} then A should be ranked higher than B in F(b1, . . . , bN).

◮ Independence-of-irrelevant-alternatives (IIA) Criterion: For

two preference schedules (b1, . . . , bN) and (c1, . . . , cN) if the candidates A and B have the same order in bi as in ci such that for all i, then A and B have the same order in F(b1, . . . , bN) and F(c1, . . . , cN). Then show that a social welfare function F cannot consistently share all these properties!

Hope?

Arrow’s Impossibility Theorem For elections involving three or more candidates, a method for determining election results that is always meets all of Arrow’s fairness criteria is mathematically impossible. That this does not mean that every election is unfair or that every voting method is equally bad, nor does it mean that we should stop trying to improve the quality of our voting experience!

Hope?

Arrow’s Impossibility Theorem For elections involving three or more candidates, a method for determining election results that is always meets all of Arrow’s fairness criteria is mathematically impossible. That this does not mean that every election is unfair or that every voting method is equally bad, nor does it mean that we should stop trying to improve the quality of our voting experience! Do any of these criteria seem fishy?

Hope?

Arrow’s Impossibility Theorem For elections involving three or more candidates, a method for determining election results that is always meets all of Arrow’s fairness criteria is mathematically impossible. That this does not mean that every election is unfair or that every voting method is equally bad, nor does it mean that we should stop trying to improve the quality of our voting experience! Do any of these criteria seem fishy? People have considered weakening IIA and investigating

• ther rules.

Thank You! math.umn.edu/∼kling202

Bangor Daily News, “Poliquin sues Dunlap to stop ranked-choice count in Maine’s 2nd District”: http://bangordailynews.com/2018/11/13/politics/poliquin-sues- dunlap-to-stop-ranked-choice-count-in-maines-2nd- district/? ga%253D2.257568523.1649929019.1542033437- 1922745084.1494327202 CNN 2016 Election Results: https://www.cnn.com/election/2016/results Radio Lab “Tweak the Vote”: https://www.wnycstudios.org/shows/radiolab