math 180 - voting theorychhays/lecture30.pdfplurality voting i if there’s only two candidates, the...
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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