Voting Theory

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

I Everybody has individual preferencesI Want to transform individual preferences to a single societal

preferenceI 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 societalpreference

I Want to do this fairly

Voting Theory

I Big question: what is the best way to hold an election?I Everybody has individual preferencesI 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 preferencesI Want to transform individual preferences to a single societal

preferenceI Want to do this fairly

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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

Plurality Voting

Plurality Voting

I Everyone gets one voteI 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

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

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

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

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

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

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

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enterthe 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”

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviable

I From game theory: it is rarely a dominant strategy to enterthe 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”

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviableI 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”

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviableI From game theory: it is rarely a dominant strategy to enter

the raceI 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”

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviableI From game theory: it is rarely a dominant strategy to enter

the raceI 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”

Plurality Voting

I If there’s only two candidates, the most preferred candidatewins

I Makes new third candidates unviableI From game theory: it is rarely a dominant strategy to enter

the raceI 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”

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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I One possible solution: hold runoff electionsI 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

Runoff Elections

I Perks:

I Voters will more likely vote their preferenceI Least preferred candidate can’t win

I Problems:

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:

I Inefficient; need to hold election over multiple days

Runoff Elections

I Perks:I Voters will more likely vote their preferenceI Least preferred candidate can’t win

I Problems:

I Inefficient; need to hold election over multiple days

Runoff Elections

I Perks:I Voters will more likely vote their preferenceI Least preferred candidate can’t win

I Problems:

I Inefficient; need to hold election over multiple days

Runoff Elections

I Perks:I Voters will more likely vote their preferenceI Least preferred candidate can’t win

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

Approval Voting

I One possible solution: approval voting

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

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve of

I Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve ofI Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve ofI Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve ofI Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve ofI Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

Approval Voting

I One possible solution: approval votingI Each voter checks off candidates that they approve ofI Candidate with most votes wins

I Example:

A B C

Voter 1 XVoter 2 X XVoter 3 X XVoter 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 theU.N. Secretary-General

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 fornobody

I Third party candidates are more legitimateI 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 fornobody

I Third party candidates are more legitimateI 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 fornobody

I Third party candidates are more legitimateI 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 fornobody

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 fornobody

I Third party candidates are more legitimateI 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 fornobody

I Third party candidates are more legitimateI Easy to understand

I Problems will be covered later

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?

Ranked Voting

I Another method: ranked votingI 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 votingI 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 votingI 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?

Instant Runoffs

I Instant Runoffs:

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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choice

I If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they win

I Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Instant Runoffs

I Instant Runoffs:I Look at everyone’s first choiceI If one candidate has > 50%, they winI Otherwise, eliminate candidate with fewest first choice votesI 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

Borda Count

I Another method for tallying ranked votes: the Borda method

I Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

Borda Count

I Another method for tallying ranked votes: the Borda methodI Candidate gets n points if a first preferenceI 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

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 inhead-to-head competition with any other candidate

I A voting method satisfies the Condorcet criterion if aCondorcet 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 inhead-to-head competition with any other candidate

I A voting method satisfies the Condorcet criterion if aCondorcet 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 inhead-to-head competition with any other candidate

I A voting method satisfies the Condorcet criterion if aCondorcet 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 inhead-to-head competition with any other candidate

I A voting method satisfies the Condorcet criterion if aCondorcet winner will always win

Plurality Voting and the Condorcet Criterion

I Suppose that:

I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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

Plurality Voting and the Condorcet Criterion

I Suppose that:I 32% prefer A then B then CI 28% prefer B then A then CI 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