My guy lost? What’s up with that…
In the 1950’s, Kenneth Arrow, a
mathematical economist, proved that
a method for determining election
results that is democratic and always
fair is mathematically impossible.
 Basically, any system that we could
ever create will have inherent flaws.

What does “democratic” and “fair”
mean?
 There are four criteria that
mathematicians and political scientists
have agreed a fair voting system
should meet.


A majority of first place votes (over
50%) is different from a plurality,
which is just the largest quantity of
first place votes.
Number of Votes
6
3
2
First Choice
E
G
F
Second Choice
F
H
G
Third Choice
G
F
H
Fourth Choice
H
E
E
Clearly, Candidate E should win since they have more
than half of the 1st place votes. But under the Borda
count, the results are much different.
E = 29 pts, F = 32 pts, G = 30 pts, and H = 19 pts
This puts candidate E in third place, with candidate F
winning the election – A clear violation of the Majority
Criterion.
“Daffy the duck and Jock the West Highland Terrier.” Virginmedia.com


If candidate is favored when compared
separately – that is, head-to-head – with
every other candidate, then that candidate
should win the election.
In a pairwise comparison, it is possible to
have a three-way tie, where A beats B, B beats
C, and C beats A. This was shown by the
Marquis de Condorcet whose name is
sometimes associated with the head-to-head
criterion (the Condorcet Criterion).
Number of Votes
8
6
4
4
First Choice
A
C
C
B
Second Choice
B
B
A
A
Third Choice
C
A
B
C
In a Pairwise Comparison, A beats B (12 to
10), and A beats C (12 to 10). This makes A
the winner in this method.
But if the Plurality method were used, C
would win the election with 10 votes, more
than A’s 8 or B’s 4.

Who knew math could look this impressive…
Ross, Chip, Prof. “Julia Set.” Abacus.bates.edu,

If a candidate wins an election and, in a
reelection, the only changes are changes that
favor the candidate, then that candidate
should win the reelection.


An initial poll is taken to see where people
stand (before the actual election):
Number of Votes
20
16
14
8
1st place
W
V
G
G
2nd place
G
W
V
W
3rd place
V
G
W
V
If this initial poll was run under plurality with
elimination, then V would be eliminated and
W would beat G with 36 votes (to 22).

Because of the results of the initial “election”,
the 8 people in the last column change their
votes to match those in the 1st column:
Number of Votes
20
16
14
8
1st place
W
V
G
G
2nd place
G
W
V
W
3rd place
V
G
W
V
Number of Votes
28
16
14
1st place
W
V
G
2nd place
G
W
V
3rd place
V
G
W


When this new configuration is run (for the
actual election), the results are now different
Number of Votes
28
16
14
1st place
W
V
G
2nd place
G
W
V
3rd place
V
G
W
Now, candidate G will be eliminated (instead
of V), and V will win the election with 30
votes (to W’s 28).

If a candidate wins an election and, in a
recount, the only changes are that one or
more of the other candidates are removed
from the ballot, then that candidate should
still win the election.

Here is an election run with pairwise
comparison
Number of Votes

160 100
80
20
1st place
E
G
H
H
2nd place
F
F
E
E
3rd place
G
H
G
F
4th place
H
E
F
G
After running the comparisons, E has 2 pts, F
and G tie with 1.5 each, and H has 1 pt.


Because they lost the initial voting,
candidates F and G pull out of the election.
This is what remains of the preference table…
Number of votes
160
100
80
20
1st place
E
H
H
H
2nd place
H
E
E
E
Now, because of the removed candidates, H
now defeats E in the reelection.
Fairness
Criteria
Plurality
Method
Borda Count
Plurality with
Elimination
Pairwise
Comparison
Majority
Criterion
Always
satisfies
May not
satisfy
Always
satisfies
Always
satisfies
Head-toHead
May not
satisfy
May not
satisfy
May not
satisfy
Always
satisfies
Monotonicity
Always
satisfies
Always
satisfies
May not
satisfy
Always
satisfies
Irrelevant
Alternatives
May not
satisfy
May not
satisfy
May not
satisfy
May not
satisfy

P. 747-748; #1, 5, 7, 9