CRITERIA FOR A FAIR ELECTION
1st criteria for a fair election:
The Majority Criterion (page 6) - If a choice receives a majority
of the first place votes in an election, then that choice should be
the winner of the election.
2nd criteria for a fair election
The Condorcet Criterion (page 8) - If there is a choice that
in a head-to-head comparison is preferred by the voters
over every other choice, then that choice should be the
winner of the election.
3rd criteria for a fair election:
The Monotonicity Criterion (page 15). If choice X is a winner of
an election and, in a reelection, the only changes in the ballots
are changes that only favor X, then X should remain a winner
of the election.
4th criteria for a fair election:
The Independence-of-Irrelevant-Alternatives Criterion
(page 18).
If choice X is a winner of an election and one
(or more) of the other choices is removed and the ballots
recounted, then X should still be a winner of the election.
Arrow’s Impossibility Theorem (page 3) - A method for
determining election results that is democratic and always
fair is a mathematical impossibility.
Methods used to find the winner of an election:
1. Plurality Method
2. Borda Count Method
3. Plurality-with-Elimination Method
4. Method of Pairwise Comparison
Example 1.2: The Math Club Election (Page 6)
I.
THE PLURALITY METHOD
plurality method (page 6) - the candidate (or candidates) with the
most first place votes wins.
A plurality does not imply a majority but a majority does
imply a plurality.
What’s wrong with the plurality method?
Example 1.3. The Band Election (page 7)
If we compare the Hula Bowl to any other bowl on a head-tohead basis, the Hula Bowl is always the preferred choice.
Y
Plurality
Borda Count
Plurality w elimination
Pairwise comparison
N
Indep of Irrel Alter
Monoticity
Condorcet
Majority
Which methods
satisfy which
criterion?
II.
THE BORDA COUNT METHOD
The Borda count method (page 10) - Each place on a ballot is
assigned points. In an election with N candidates we give 1
point for last place, 2 points for next to last place, …, and N
points for first place. The points are tallied for each candidate
separately and the candidate with the most points is the
winner.
4 points
56
40
32
16
4
3 points
42
30
24
12
3
2 points
28
20
16
8
2
1 point
14
10
8
4
1
Math Appreciation Society
A gets 56+10+8+4+1=79 points
B gets 42+30+16+16+2=106 points
C gets 28+40+24+8+4=104 points
D gets 14+20+32+12+3=81 points
What’s wrong with the Borda count method?
Example 1.6: The School Board Principal Election (page 11)
TABLE 1-5
Example 1.6. The School Board Principal Election
TABLE 1-5
The School Board Principal Election
A gets 24+2+3=29 points
B gets 18+8+6=32 points
C gets 12+6+12=30 points
D gets 6+4+9=19 points
The Borda count method violates the majority criterion.
Y N
Plurality
N N
Borda Count
Plurality w elimination
Pairwise comparison
Indep of Irrel Alter
Monoticity
Condorcet
Majority
Which methods
satisfy which
criterion?
N
III.
THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. (page 12) Count the first-place votes for each
candidate. If a candidate has a majority of first-place
votes, that candidate is declared the winner. Otherwise
eliminate the candidate (or candidates if there is a tie) with
the fewest first place votes.
TABLE 1-6
37 voters so a majority requires at least 19 votes
III.
THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. (page 12) Count the first-place votes for each
candidate. If a candidate has a majority of first-place
votes, that candidate is declared the winner. Otherwise
eliminate the candidate (or candidates if there is a tie) with
the fewest first place votes.
Round 2. Cross out the name(s) of the candidate
eliminated from the preference schedule and recount the
first-place votes. If a candidate has a majority of firstplace votes, declare that candidate the winner.
Otherwise, eliminate the candidate with the fewest first
place votes.
TABLE 1-6
37 voters so a majority requires at least 19 votes
III.
THE PLURALITY-WITH-ELIMINATION METHOD
Round 1. Count the first-place votes for each candidate.
If a candidate has a majority of first-place votes, that
candidate is declared the winner. Otherwise eliminate the
candidate (or candidates if there is a tie) with the fewest
first place votes.
Round 2. Cross out the name(s) of the candidate
eliminated from the preference schedule and recount the
first-place votes. If a candidate has a majority of firstplace votes, declare that candidate the winner.
Otherwise, eliminate the candidate with the fewest first
place votes.
Round 3,4, etc. Repeat the process, each time
eliminating one or more candidates, until there finally is a
candidate with a majority of first-place votes, which is then
declared a winner.
TABLE 1-6
37 voters so a majority requires at least 19 votes
Example 1.9: (page 15)
TABLE 1-9
11 voters so a majority requires at least 6 votes
Winner MAS
Alisha
Boris
Dave
Example:
Voting Method
Plurality
Borda Count
Plurality-with-elimination
26 voters so a majority requires at least 14 votes
Example 1.10. The summer Olympic games (page 15)
TABLE 1-9
 Round 1
Candidate
A
Number of first place votes 11
 Round 2
Candidate
A
Number of first place votes 11
B C
8 10
B
C
18
29 voters so a
majority requires
at least 15 votes
TABLE 1-10
29 voters so a
majority requires
at least 15 votes
 Round 1
Candidate
A
Number of first place votes 7
B C
8 14
 Round 2
Candidate
A B C
Number of first place votes
15 14
What’s wrong with the plurality-with-elimination method?
3th criteria for a fair election:
The Monotonicity Criterion. If choice X is a winner of an
election and, in a reelection, the only changes in the ballots
are changes that only favor X, then X should remain a winner
of the election.
The plurality-with-elimination method violates the
monotonicity criterion.
Indep of Irrel Alter
Monoticity
Condorcet
Majority
Which methods
satisfy which
criterion?
Y N
Plurality
N N
N
Borda Count
Plurality w elimination Y N N
Pairwise comparison
Homework
• Read Pages 6 – 23
• Page 30: 18, 19, 24, 28, 29, 30, 33, 59,
60, 71