I Everybody has individual preferences I Want to transform individual preferences to a single societal preference I Want to do this fairly

Voting Theory

I Big question: what is the best way to hold an election? I Want to transform individual preferences to a single societal preference I Want to do this fairly

Voting Theory

I Big question: what is the best way to hold an election?

I Everybody has individual preferences I Want to do this fairly

Voting Theory

I Big question: what is the best way to hold an election?

I Everybody has individual preferences I Want to transform individual preferences to a single societal preference Voting Theory

I Big question: what is the best way to hold an election?

I Everybody has individual preferences I Want to transform individual preferences to a single societal preference I Want to do this fairly I Don’t require majority to win

I A gets 32% I B gets 40% I C gets 28%

I B

I Candidate with most votes wins

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I Who wins?

Plurality Voting

Plurality Voting

I Everyone gets one vote I A gets 32% I B gets 40% I C gets 28%

I B

I Don’t require majority to win

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I Who wins?

Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins I A gets 32% I B gets 40% I C gets 28%

I B

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I Who wins?

Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins

I Don’t require majority to win I A gets 32% I B gets 40% I C gets 28%

I B

I Example:

I Who wins?

Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins

I Don’t require majority to win

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I B

I Who wins?

Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins

I Don’t require majority to win

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I A gets 32% I B gets 40% I C gets 28% I B

Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins

I Don’t require majority to win

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I A gets 32% I B gets 40% I C gets 28% I Who wins? Plurality Voting

Plurality Voting

I Everyone gets one vote I Candidate with most votes wins

I Don’t require majority to win

I Method for voting Governors, Congressmen, President (ignoring electoral colleges) I Example:

I A gets 32% I B gets 40% I C gets 28% I Who wins?

I B I B should win

I The least preferred candidate wins! I Example of vote splitting

I 1: Supporters of both A and C have B as their second choice

I 2: Supporters of both A and C have B as their last choice

I Possible scenarios:

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I B should win

I The least preferred candidate wins! I Example of vote splitting

I 1: Supporters of both A and C have B as their second choice

I 2: Supporters of both A and C have B as their last choice

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios: I The least preferred candidate wins! I Example of vote splitting

I B should win

I 2: Supporters of both A and C have B as their last choice

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios:

I 1: Supporters of both A and C have B as their second choice I The least preferred candidate wins! I Example of vote splitting

I 2: Supporters of both A and C have B as their last choice

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios:

I 1: Supporters of both A and C have B as their second choice

I B should win I The least preferred candidate wins! I Example of vote splitting

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios:

I 1: Supporters of both A and C have B as their second choice

I B should win

I 2: Supporters of both A and C have B as their last choice I Example of vote splitting

Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios:

I 1: Supporters of both A and C have B as their second choice

I B should win

I 2: Supporters of both A and C have B as their last choice

I The least preferred candidate wins! Plurality Voting

I Example:

I A gets 32% I B gets 40% I C gets 28% I Possible scenarios:

I 1: Supporters of both A and C have B as their second choice

I B should win

I 2: Supporters of both A and C have B as their last choice

I The least preferred candidate wins! I Example of vote splitting I Bush: 48.38% I Gore: 47.87% I Nader: 2.74%

I Probable that many preferred Nader, but did not want to “throw away their vote”

I From game theory: it is rarely a dominant strategy to enter the race I 2000 Presidential election:

I Do these numbers truly reflect first preference?

I Makes new third candidates unviable

Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Bush: 48.38% I Gore: 47.87% I Nader: 2.74%

I Probable that many preferred Nader, but did not want to “throw away their vote”

I From game theory: it is rarely a dominant strategy to enter the race I 2000 Presidential election:

I Do these numbers truly reflect first preference?

Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Makes new third candidates unviable I Bush: 48.38% I Gore: 47.87% I Nader: 2.74%

I Probable that many preferred Nader, but did not want to “throw away their vote”

I 2000 Presidential election:

I Do these numbers truly reflect first preference?

Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enter the race I Probable that many preferred Nader, but did not want to “throw away their vote”

I Do these numbers truly reflect first preference?

Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enter the race I 2000 Presidential election:

I Bush: 48.38% I Gore: 47.87% I Nader: 2.74% I Probable that many preferred Nader, but did not want to “throw away their vote”

Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enter the race I 2000 Presidential election:

I Bush: 48.38% I Gore: 47.87% I Nader: 2.74%

I Do these numbers truly reflect first preference? Plurality Voting

I If there’s only two candidates, the most preferred candidate wins I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enter the race I 2000 Presidential election:

I Bush: 48.38% I Gore: 47.87% I Nader: 2.74%

I Do these numbers truly reflect first preference?

I Probable that many preferred Nader, but did not want to “throw away their vote” I A gets 32% I B gets 40% I C gets 28%

I After election, eliminate weakest candidate(s) I Hold another election I Round 1:

I C gets eliminated

I Round 2: people who voted for C get their second preference

I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections I A gets 32% I B gets 40% I C gets 28%

I Hold another election I Round 1:

I C gets eliminated

I Round 2: people who voted for C get their second preference

I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I A gets 32% I B gets 40% I C gets 28%

I Round 1:

I C gets eliminated

I Round 2: people who voted for C get their second preference

I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I Hold another election I C gets eliminated

I Round 2: people who voted for C get their second preference

I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I Hold another election I Round 1:

I A gets 32% I B gets 40% I C gets 28% I Round 2: people who voted for C get their second preference

I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I Hold another election I Round 1:

I A gets 32% I B gets 40% I C gets 28%

I C gets eliminated I Used in French presidential elections

Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I Hold another election I Round 1:

I A gets 32% I B gets 40% I C gets 28%

I C gets eliminated

I Round 2: people who voted for C get their second preference Runoff Elections

I One possible solution: hold runoff elections

I After election, eliminate weakest candidate(s) I Hold another election I Round 1:

I A gets 32% I B gets 40% I C gets 28%

I C gets eliminated

I Round 2: people who voted for C get their second preference

I Used in French presidential elections I Inefficient; need to hold election over multiple days

I Voters will more likely vote their preference I Least preferred candidate can’t win I Problems:

Runoff Elections

I Perks: I Inefficient; need to hold election over multiple days

I Least preferred candidate can’t win I Problems:

Runoff Elections

I Perks:

I Voters will more likely vote their preference I Inefficient; need to hold election over multiple days

I Problems:

Runoff Elections

I Perks:

I Voters will more likely vote their preference I Least preferred candidate can’t win I Inefficient; need to hold election over multiple days

Runoff Elections

I Perks:

I Voters will more likely vote their preference I Least preferred candidate can’t win I Problems: Runoff Elections

I Perks:

I Voters will more likely vote their preference I Least preferred candidate can’t win I Problems:

I Inefficient; need to hold election over multiple days I Each voter checks off candidates that they approve of I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Candidate with most votes wins I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes I Used in many professional societies, and the election for the U.N. Secretary-General

Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins Approval Voting

I One possible solution: approval voting

I Each voter checks off candidates that they approve of I Candidate with most votes wins

I Example: A B C Voter 1 X Voter 2 X X Voter 3 X X Voter 4 X X I A gets 3 votes, B gets 2 votes, C gets 2 votes

I A wins

I Used in many professional societies, and the election for the U.N. Secretary-General I Note that voting for everybody is equivalent to voting for nobody

I Voters get to choose to vote for or against a candidate

I Third party candidates are more legitimate I Easy to understand

I Problems will be covered later

Approval Voting

I Perks: I Note that voting for everybody is equivalent to voting for nobody

I Third party candidates are more legitimate I Easy to understand

I Problems will be covered later

Approval Voting

I Perks:

I Voters get to choose to vote for or against a candidate I Third party candidates are more legitimate I Easy to understand

I Problems will be covered later

Approval Voting

I Perks:

I Voters get to choose to vote for or against a candidate

I Note that voting for everybody is equivalent to voting for nobody I Easy to understand

I Problems will be covered later

Approval Voting

I Perks:

I Voters get to choose to vote for or against a candidate

I Note that voting for everybody is equivalent to voting for nobody

I Third party candidates are more legitimate I Problems will be covered later

Approval Voting

I Perks:

I Voters get to choose to vote for or against a candidate

I Note that voting for everybody is equivalent to voting for nobody

I Third party candidates are more legitimate I Easy to understand Approval Voting

I Perks:

I Voters get to choose to vote for or against a candidate

I Note that voting for everybody is equivalent to voting for nobody

I Third party candidates are more legitimate I Easy to understand

I Problems will be covered later I Voters rank candidates from most preferred to least preferred

I Example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I Question: how do we tally the votes?

Ranked Voting

I Another method: ranked voting I Example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I Question: how do we tally the votes?

Ranked Voting

I Another method: ranked voting

I Voters rank candidates from most preferred to least preferred I Question: how do we tally the votes?

Ranked Voting

I Another method: ranked voting

I Voters rank candidates from most preferred to least preferred

I Example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A Ranked Voting

I Another method: ranked voting

I Voters rank candidates from most preferred to least preferred

I Example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I Question: how do we tally the votes? I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

So B wins I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice

So B wins I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win

So B wins I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes

So B wins I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

So B wins I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

So B wins I A has 2 votes

I B has 4 votes

I C has 3 votes

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

So B wins Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A has 2 votes

I B has 4 votes

I C has 3 votes So B wins I B has 5 votes

I C has 4 votes So B wins

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred —A B B B C C —A C B B C C C —A —A C B C Least Preferred C —A —A —A B B B —A —A

I ——————A has 2 votes So B wins

Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred —A B B B C C —A C B B C C C —A —A C B C Least Preferred C —A —A —A B B B —A —A

I ——————A has 2 votes

I B has 5 votes

I C has 4 votes Instant Runoffs

I Instant Runoffs:

I Look at everyone’s first choice I If one candidate has > 50%, they win I Otherwise, eliminate candidate with fewest first choice votes I Repeat as necessary

I Used in Australian and Irish national elections

I Back to the example: Voters Most Preferred —A B B B C C —A C B B C C C —A —A C B C Least Preferred C —A —A —A B B B —A —A

I ——————A has 2 votes

I B has 5 votes

I C has 4 votes So B wins . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I Candidate gets n − 1 points if a second preference

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

. .

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I A gets

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

3 · 2 + 2 · 2 + 1 · 5 = 15 points

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I B gets

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

3 · 4 + 2 · 2 + 1 · 3 = 19 points

so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points

I C gets so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points 3 · 3 + 2 · 5 + 1 · 1 = 20 points so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets so C wins

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preference I Candidate gets n − 1 points if a second preference . . I Candidate gets 1 point if a last preference

I Back to the example: Voters Most Preferred A B B B C C A C B B C C C A A C B C Least Preferred C A A A B B B A A

I A gets 3 · 2 + 2 · 2 + 1 · 5 = 15 points

I B gets 3 · 4 + 2 · 2 + 1 · 3 = 19 points

I C gets 3 · 3 + 2 · 5 + 1 · 1 = 20 points so C wins I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win

I One good idea is the Condorcet criterion:

Fair Voting

I Want to determine if the outcome of the election is “fair” I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win

Fair Voting

I Want to determine if the outcome of the election is “fair” I One good idea is the Condorcet criterion: I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win

Fair Voting

I Want to determine if the outcome of the election is “fair” I One good idea is the Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate Fair Voting

I Want to determine if the outcome of the election is “fair” I One good idea is the Condorcet criterion:

I A candidate is the Condorcet winner if they would win in head-to-head competition with any other candidate I A voting method satisfies the Condorcet criterion if a Condorcet winner will always win I A would get 72%; B would get 36%

I A would get 60%; C would get 40%

I B would get 60%; C would get 40%

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I Who would win A vs. C?

I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that: I A would get 72%; B would get 36%

I A would get 60%; C would get 40%

I B would get 60%; C would get 40%

I Who would win A vs. B?

I Who would win A vs. C?

I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I A would get 60%; C would get 40%

I B would get 60%; C would get 40%

I A would get 72%; B would get 36% I Who would win A vs. C?

I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B? I A would get 60%; C would get 40%

I B would get 60%; C would get 40%

I Who would win A vs. C?

I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I B would get 60%; C would get 40%

I A would get 60%; C would get 40% I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C? I B would get 60%; C would get 40%

I Who would win B vs. C?

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I B would get 60%; C would get 40%

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I Who would win B vs. C? I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I Who would win B vs. C?

I B would get 60%; C would get 40% I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I Who would win B vs. C?

I B would get 60%; C would get 40%

I A is the Condorcet winner I Plurality voting does not satisfy the Condorcet criterion

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I Who would win B vs. C?

I B would get 60%; C would get 40%

I A is the Condorcet winner

I In a plurality election, C wins the election! Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then C I 28% prefer B then A then C I 40% prefer C then A then B I Who would win A vs. B?

I A would get 72%; B would get 36% I Who would win A vs. C?

I A would get 60%; C would get 40% I Who would win B vs. C?

I B would get 60%; C would get 40%

I A is the Condorcet winner

I In a plurality election, C wins the election!

I Plurality voting does not satisfy the Condorcet criterion