Fairness
Criteria
The plurality,
plurality-with-elimination, and pairwise comparisons methods all satisfy this
criterion (think about why).
However,
look at the following example with 3 candidates and 5 voters:
Preference
Schedule:
Number
of voters |
3 |
2 |
points |
1st
choice |
A |
B |
2 |
2nd
choice |
B |
C |
1 |
3rd
choice |
C |
A |
0 |
A
receives 3 * 2 + 2 * 0 = 6 points.
B
receives 3 * 1 + 2 * 2 = 7 points.
C
receives 3 * 0 + 2 * 1 = 2 points
So B
wins by the Borda count method, although A receives a majority of the
first place votes.
Thus,
the Borda count method doesn’t satisfy the majority criterion.
(There
are many preference schedules for which a candidate receiving a majority also
wins by the Borda count method, however there is at least one example where it
fails).
The plurality, Borda count,
and pairwise comparisons methods all satisfy this criterion (think about why).
However,
look at the following example with 3 candidates and 5 voters:
Preference
Schedule:
Number
of voters |
6 |
5 |
3 |
7 |
1st
choice |
A |
B |
B |
C |
2nd
choice |
C |
A |
C |
B |
3rd
choice |
B |
C |
A |
A |
A
has the fewest first place votes, so is eliminated. The preference schedule for the remaining candidates is:
Number
of voters |
8 |
13 |
1st
choice |
B |
C |
2nd
choice |
C |
B |
So C wins by the
plurality-with-elimination (easily, in fact).
Now we’ll help C
out by changing some of the ballots in C’s favor. Switch C and B in the third column:
Number
of voters |
6 |
5 |
3 |
7 |
1st
choice |
A |
B |
C |
C |
2nd
choice |
C |
A |
B |
B |
3rd
choice |
B |
C |
A |
A |
This
time B has the fewest first place votes, so is eliminated. The preference schedule for the remaining
candidates is:
Number
of voters |
11 |
10 |
1st
choice |
A |
C |
2nd
choice |
C |
A |
So A wins by the
plurality-with-elimination.
Thus plurality-with-elimination fails to satisfy the monotonicity criterion. Another example is given in the text on page 13.
So far we’ve seen drawbacks to the Borda count and plurality-with-elimination methods. Do the plurality and pairwise comparison methods have any drawbacks?
Condorcet
Criterion: If a candidate wins all pairwise
comparisons, it should win the election.
(The “cet” in Condorcet is pronounced “say”).
The pairwise comparisons
method satisfies this criterion (think about why).
However,
look at the following example with 3 candidates and 7 voters:
Preference
Schedule:
Number
of voters |
3 |
2 |
4 |
1st
choice |
A |
B |
C |
2nd
choice |
B |
A |
A |
3rd
choice |
C |
C |
B |
C wins by the plurality method. However, A beats both B and C in pairwise comparisons. Thus, the plurality method does not satisfy the Condorcet criterion. Actually the Borda count and plurality-with-elimination methods fail to satisfy this criterion also (the book gives examples).
Only the pairwise
comparisons method satisfies all three of the fairness criteria we’ve looked
at; it is looking good right now!
This criterion can be violated by all of the voting methods discussed in the text. An example for pairwise comparisons is given in the text (it’s kind of complicated, so I’ll let you read it at your leisure). Here is a simpler example showing that the Borda method fails to satisfy this criterion:
Preference
Schedule:
Number
of voters |
2 |
2 |
3 |
points |
1st
choice |
A |
B |
C |
3 |
2nd
choice |
D |
A |
B |
2 |
3rd
choice |
C |
D |
A |
1 |
4th
choice |
B |
C |
D |
0 |
Check
that A receives 13 points, B 12, C 11 and D 6.
So A is the Borda method winner.
D is an “irrelevant alternative”; suppose we eliminate D:
Number
of voters |
2 |
2 |
3 |
points |
1st
choice |
A |
B |
C |
2 |
2nd
choice |
C |
A |
B |
1 |
3rd
choice |
B |
C |
A |
0 |
Now
A receives 6 points, B 7 and C 8. So C now wins.
We
see that none of the election methods we have studied satisfies all four
fairness criteria. Is there some more
ingenious election method, which does?
Arrow’s
Impossibility Theorem: It is impossible to devise an
election method, which satisfies all four fairness criteria.
Kenneth
Arrow was an economist with a background in mathematics. He proved this theorem
in 1952, as part of his Ph.D. Thesis.
For this and related work, he received the Nobel Prize in Economics in
1972.