Review
Fairness criteria for voting methods:
Review
Fairness criteria for voting methods:
I Condorcet criterion:
I
A candidate is the Condorcet winner if they would win in
head-to-head competition with any other candidate
Review
Fairness criteria for voting methods:
I Condorcet criterion:
I
I
A candidate is the Condorcet winner if they would win in
head-to-head competition with any other candidate
A voting method satisfies the Condorcet criterion if a
Condorcet winner will always win the election
Review
Fairness criteria for voting methods:
I Condorcet criterion:
I
I
I
A candidate is the Condorcet winner if they would win in
head-to-head competition with any other candidate
A voting method satisfies the Condorcet criterion if a
Condorcet winner will always win the election
majority criterion:
I
A voting method satisfies the majority criterion if a candidate
with a majority of first-preference votes will win the election
Review
Fairness criteria for voting methods:
I Condorcet criterion:
I
I
I
majority criterion:
I
I
A candidate is the Condorcet winner if they would win in
head-to-head competition with any other candidate
A voting method satisfies the Condorcet criterion if a
Condorcet winner will always win the election
A voting method satisfies the majority criterion if a candidate
with a majority of first-preference votes will win the election
public enemy criterion:
I
A voting method satisfies the public enemy criterion is a
candidate with a majority of last-preference votes cannot win
the election
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
B
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
B
Is there a majority winner?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
B
Is there a majority winner?
I
no
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
B
no
Is there a public enemy?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
B
no
Is there a public enemy?
I
A
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
no
Is there a public enemy?
I
I
B
A
Who is the plurality winner?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
no
Is there a public enemy?
I
I
B
A
Who is the plurality winner?
I
A
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
A
Who is the plurality winner?
I
I
no
Is there a public enemy?
I
I
B
A
Who is the instant runoff winner (10 to win)?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
A
Who is the plurality winner?
I
I
no
Is there a public enemy?
I
I
B
A
Who is the instant runoff winner (10 to win)?
I
C
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
A
Who is the instant runoff winner (10 to win)?
I
I
A
Who is the plurality winner?
I
I
no
Is there a public enemy?
I
I
B
C
Is there Borda winner?
Example
Number of Voters 7 5 6
1st choice A B C
2nd choice B C B
3rd choice C A A
I Is there a Condorcet winner?
I
I
Is there a majority winner?
I
I
A
Who is the instant runoff winner (10 to win)?
I
I
A
Who is the plurality winner?
I
I
no
Is there a public enemy?
I
I
B
C
Is there Borda winner?
I
B
Summary
Plurality
Instant Runoff
Borda
Condorcet
no
no
no
Majority
yes
yes
no
Public Enemy
no
yes
yes
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
I
Candidates are Rio de Janeiro, Madrid, and Tokyo
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
I
Candidates are Rio de Janeiro, Madrid, and Tokyo
I
Poll yields the following preferences:
Number of Voters 7
1st choice M
2nd choice R
3rd choice T
8
R
T
M
10
T
M
R
4
M
T
R
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
I
Candidates are Rio de Janeiro, Madrid, and Tokyo
I
Poll yields the following preferences:
Number of Voters 7
8 10 4
1st choice M R T M
2nd choice R T M T
3rd choice T M R R
If the election were held right now, who would win?
I
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
I
Candidates are Rio de Janeiro, Madrid, and Tokyo
I
Poll yields the following preferences:
Number of Voters 7
8 10 4
1st choice M R T M
2nd choice R T M T
3rd choice T M R R
If the election were held right now, who would win?
I
I
Rio de Janeiro is eliminated in the first round
Example
I
Instant runoffs are being used to determine the host city for
the 2016 Olympics
I
Candidates are Rio de Janeiro, Madrid, and Tokyo
I
Poll yields the following preferences:
Number of Voters 7
8 10 4
1st choice M R T M
2nd choice R T M T
3rd choice T M R R
If the election were held right now, who would win?
I
I
I
Rio de Janeiro is eliminated in the first round
Tokyo wins
Example
I
Suppose that the contingent of 4 decides to help Tokyo:
Example
I
Suppose that the contingent of
Number of Voters
1st choice
2nd choice
3rd choice
4 decides
7
8
M R
R T
T M
to help Tokyo:
10
4
T —
MT
M —
TM
R
R
Example
I
I
Suppose that the contingent of 4 decides to help Tokyo:
Number of Voters 7
8 10
4
st
1 choice M R T —
MT
2nd choice R T M —
TM
rd
3 choice T M R
R
The change looks like it only benefits Tokyo
(the presumed winner)
Example
I
I
I
Suppose that the contingent of 4 decides to help Tokyo:
Number of Voters 7
8 10
4
st
1 choice M R T —
MT
2nd choice R T M —
TM
rd
3 choice T M R
R
The change looks like it only benefits Tokyo
(the presumed winner)
Who wins?
Example
I
I
I
Suppose that the contingent of 4 decides to help Tokyo:
Number of Voters 7
8 10
4
st
1 choice M R T —
MT
2nd choice R T M —
TM
rd
3 choice T M R
R
The change looks like it only benefits Tokyo
(the presumed winner)
Who wins?
I
Madrid is eliminated in the first round
Example
I
I
I
Suppose that the contingent of 4 decides to help Tokyo:
Number of Voters 7
8 10
4
st
1 choice M R T —
MT
2nd choice R T M —
TM
rd
3 choice T M R
R
The change looks like it only benefits Tokyo
(the presumed winner)
Who wins?
I
I
Madrid is eliminated in the first round
Rio de Janeiro wins!?!
Example
I
I
I
Suppose that the contingent of 4 decides to help Tokyo:
Number of Voters 7
8 10
4
st
1 choice M R T —
MT
2nd choice R T M —
TM
rd
3 choice T M R
R
The change looks like it only benefits Tokyo
(the presumed winner)
Who wins?
I
I
I
Madrid is eliminated in the first round
Rio de Janeiro wins!?!
So Tokyo getting more first round votes caused them to lose
The Monotonicity Criterion
I
The monotonicity criterion:
The Monotonicity Criterion
I
The monotonicity criterion:
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
The Monotonicity Criterion
I
The monotonicity criterion:
I
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
Instant runoffs do not satisfy the monotonicity criterion
The Monotonicity Criterion
I
The monotonicity criterion:
I
I
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
Instant runoffs do not satisfy the monotonicity criterion
Does plurality voting satisfy the monotonicity criterion?
The Monotonicity Criterion
I
The monotonicity criterion:
I
I
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
Instant runoffs do not satisfy the monotonicity criterion
Does plurality voting satisfy the monotonicity criterion?
I
Yes
The Monotonicity Criterion
I
The monotonicity criterion:
I
I
I
Instant runoffs do not satisfy the monotonicity criterion
Does plurality voting satisfy the monotonicity criterion?
I
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
Yes
Does the Borda method satisfy the monotonicity criterion?
The Monotonicity Criterion
I
The monotonicity criterion:
I
I
I
Instant runoffs do not satisfy the monotonicity criterion
Does plurality voting satisfy the monotonicity criterion?
I
I
A voting method satisfies the monotonicity criterion if the
winner of an election cannot lose a repeat election if
preferences are altered only to the benefit of the original
winner
Yes
Does the Borda method satisfy the monotonicity criterion?
I
Yes
Summary
Plurality
Instant Runoff
Borda
Condorcet
no
no
no
Majority
yes
yes
no
Public Enemy
no
yes
yes
Monotonicity
yes
no
yes
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
3
D
B
C
A
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
I
A
3
D
B
C
A
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
I
I
3
D
B
C
A
A
Who is the instant runoff winner (11 to win)?
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
I
I
3
D
B
C
A
A
Who is the instant runoff winner (11 to win)?
I
A
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
I
I
A
Who is the instant runoff winner (11 to win)?
I
I
3
D
B
C
A
A
Who is the Borda winner?
One Last Criterion
Number of Voters 8 6 4
1st choice A B C
2nd choice D C D
3rd choice B A A
4th choice C D B
I Who is the plurality winner?
I
I
A
Who is the instant runoff winner (11 to win)?
I
I
3
D
B
C
A
A
Who is the Borda winner?
I
A
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
4
C
D
A
B
X
3
D
BC
X
CA
X
A
X
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
I
C
4
C
D
A
B
X
3
D
BC
X
CA
X
A
X
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
I
I
4
C
D
A
B
X
3
D
BC
X
CA
X
A
X
C
Who is the instant runoff winner (11 to win)?
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
I
I
4
C
D
A
B
X
3
D
BC
X
CA
X
A
X
C
Who is the instant runoff winner (11 to win)?
I
C
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
I
I
3
D
BC
X
CA
X
A
X
C
Who is the instant runoff winner (11 to win)?
I
I
4
C
D
A
B
X
C
Who is the Borda winner?
One Last Criterion
Now suppose
Number of
1st
2nd
3rd
I
that B drops out:
Voters
8
6
choice
A
BC
X
choice D
CA
X
choice X
BC X
AD
C
X
D
X
Who is the plurality winner?
I
I
3
D
BC
X
CA
X
A
X
C
Who is the instant runoff winner (11 to win)?
I
I
4
C
D
A
B
X
C
Who is the Borda winner?
I
C
Independence of Irrelevant Alternatives
I
The independence of irrelevant alternatives criterion:
Independence of Irrelevant Alternatives
I
The independence of irrelevant alternatives criterion:
I
A voting method satisfies the I.I.A. criterion if the winner of
an election would still win if other candidates were disqualified
Independence of Irrelevant Alternatives
I
The independence of irrelevant alternatives criterion:
I
I
A voting method satisfies the I.I.A. criterion if the winner of
an election would still win if other candidates were disqualified
Plurality, instant runoffs, and the Borda method do not satisfy
I.I.A.
Summary
Plurality
Instant Runoff
Borda
Cond.
no
no
no
Maj.
yes
yes
no
P.E.
no
yes
yes
Mono.
yes
no
yes
I.I.A.
no
no
no
Method of Pairwise Comparisons (Condorcet Method)
Method of pairwise comparisons
I
Compare each pair of candidates
Method of Pairwise Comparisons (Condorcet Method)
Method of pairwise comparisons
I
Compare each pair of candidates
I
Candidate earns one point for each candidate that they beat
Method of Pairwise Comparisons (Condorcet Method)
Method of pairwise comparisons
I
Compare each pair of candidates
I
Candidate earns one point for each candidate that they beat
I
Candidate with the most points wins
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
7 5
A B
B C
C A
method
6
C
B
A
of pairwise comparisons?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
B
7 5
A B
B C
C A
method
6
C
B
A
of pairwise comparisons?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
7 5
A B
B C
C A
method
6
C
B
A
of pairwise comparisons?
B
The method of pairwise comparisons satisfies the Condorcet
criterion
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
7 5
A B
B C
C A
method
6
C
B
A
of pairwise comparisons?
B
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
7 5
A B
B C
C A
method
6
C
B
A
of pairwise comparisons?
B
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
yes
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
6
C
B
A
of pairwise comparisons?
B
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the public enemy criterion?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
6
C
B
A
of pairwise comparisons?
B
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the public enemy criterion?
I
yes
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
B
yes
Does it satisfy the public enemy criterion?
I
I
6
C
B
A
of pairwise comparisons?
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the monotonicity criterion?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
B
yes
Does it satisfy the public enemy criterion?
I
I
6
C
B
A
of pairwise comparisons?
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the monotonicity criterion?
I
yes
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
yes
yes
Does it satisfy the monotonicity criterion?
I
I
B
Does it satisfy the public enemy criterion?
I
I
6
C
B
A
of pairwise comparisons?
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the I.I.A. criterion?
Example
Number of Voters
1st choice
2nd choice
3rd choice
I Who wins via the
I
I
I
yes
yes
Does it satisfy the monotonicity criterion?
I
I
B
Does it satisfy the public enemy criterion?
I
I
6
C
B
A
of pairwise comparisons?
The method of pairwise comparisons satisfies the Condorcet
criterion
Does it satisfy the majority criterion?
I
I
7 5
A B
B C
C A
method
yes
Does it satisfy the I.I.A. criterion?
I
no
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
I
If there are n candidates, you have to do
n2 −n
2
comparisons
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
2
I
I
If there are n candidates, you have to do n 2−n comparisons
For other methods, you have to do ∼ n computations
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
2
I
I
I
If there are n candidates, you have to do n 2−n comparisons
For other methods, you have to do ∼ n computations
There might not be winner:
Number of Voters 7 5
1st choice A B
2nd choice B C
3rd choice C A
6
C
A
B
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
2
I
I
I
If there are n candidates, you have to do n 2−n comparisons
For other methods, you have to do ∼ n computations
There might not be winner:
Number of Voters 7 5
1st choice A B
2nd choice B C
3rd choice C A
I
6
C
A
B
Who wins via the method of pairwise comparisons?
Method of Pairwise Comparisons
Problems with the method of pairwise comparisons?
I Complicated/inefficient:
2
I
I
I
If there are n candidates, you have to do n 2−n comparisons
For other methods, you have to do ∼ n computations
There might not be winner:
Number of Voters 7 5
1st choice A B
2nd choice B C
3rd choice C A
I
6
C
A
B
Who wins via the method of pairwise comparisons?
I
Nobody (rock, paper, scissors)
Summary
Plurality
Instant Runoff
Borda
Pairwise Comparisons
Cond.
no
no
no
yes
Maj.
yes
yes
no
yes
P.E.
no
yes
yes
yes
Mono.
yes
no
yes
yes
I.I.A.
no
no
no
no
Another Method?
I
Is there a method that satisfies all of these criteria?
Another Method?
I
Is there a method that satisfies all of these criteria?
Arrow’s Impossibility Theorem
There is no voting method that satisfies:
I
Condorcet criterion
I
majority criterion
I
monotonicity criterion
I
I.I.A. criterion