Warm-Up
Rank the following soft drinks according to your
preference (1 being the soft drink you like
best and 4 being the one you like least)




Dr. Pepper
Pepsi
Mt. Dew
Sprite
Election Theory
Basics of Election Theory
How do we elect officials?


Sometimes it is necessary to rank candidates
instead of selecting a single candidate.
We can summarize votes into a preference
schedule.
Preference Ballot: a ballot in which voters are asked to
rank the candidates in order
There are 37 ballots, therefore 37 people voted
Preference Schedule: a table that organizes the ballots
Number of
voters
14
10
8
4
1
1st choice
A
C
D
B
C
2nd choice
B
B
C
D
D
3rd choice
C
D
B
C
B
4th choice
D
A
A
A
A
The Methods
1.
2.
3.
4.
5.
6.
Plurality
Borda Count
Pairwise Comparison
(Copeland)
Plurality with
Elimination (Hare)
Approval
Sequential Pairwise


You will work with a
group to prepare a
lesson on your method.
Must include:




Explanation
Example done for the
class
Example for the class to
do that you will go over.
What fairness criteria is
broken?
The Mathematics of Voting


Majority
The candidate with a more than half the votes
should be the winner.
Majority candidate
The candidate with the majority of 1st place
votes .
7
The Plurality Method: if X has the most first-place
votes, then X is the winner. X does not have to have a
majority of 1st place votes.
A is the winner
with 14 votes
R is the winner
with 49 votes
Example
10 6
5
4
2
How many candidates?
4
2) How many people voted?
27
3) Which candidate has the most firstplace votes? Is it a majority or
plurality?
B, Plurality
1)
1st A
B
B
C
D
2nd C
D
C
A
C
3rd B
C
A
D
B
4th D
A
D
B
A
The Mathematics of Voting
In the Borda Count Method each place on a
ballot is assigned points. In an election with
N candidates we give 1 point for last place, 2
points for second from last place, and so on.
10
The Mathematics of Voting
Borda Count Method
At the top of the ballot, a first-place vote is
worth N points. The points are tallied for
each candidate separately, and the candidate
with the highest total is the winner. We call
such a candidate the Borda winner.
11
The Mathematics of Voting
Borda Count Method
A gets 4(14)+1(10)+1(8)+1(4)+1(1)
56 + 10 + 8 + 4 + 1 = 81 points
B gets 3(14)+3(10)+2(8)+4(4)+2(1)
42 + 30 + 16 + 16 + 2 = 106 points
C gets 2(14)+4(10)+3(8)+2(4)+4(1)
28 + 40 + 24 + 8 + 4 = 104 points
D gets 1(14)+2(10)+4(8)+3(4)+3(1)
14 + 20 + 32 + 12 + 3 = 81 points
B is the
winner!!!
12
The Plurality-with Elimination Method (Hare)
 Steps:
1) Count the first place votes for each candidate. If a
candidate has a majority of the first-place votes, that
candidate is the winner.
2) If there isn’t a candidate that has the majority of votes
then, Cross out the candidate (or candidates if there is a
tie) with the fewest first-place votes
37 people voted so the majority
would need 19 votes
3) Move other candidates up and count the number of the
first-place votes again. If a candidate has a majority
votes, that candidate is the winner. Otherwise, continue
the process of crossing names and counting the firstplace votes.
The Plurality-with Elimination Method
Number of
voters
14
10 8
4
1
1st choice
A
C
D B
C
2nd choice
B
B
C D
D
3rd choice
C
D
B C
B
4th choice
D
A
A A
A
Number of
voters
14
10 8
1st choice
A
C
D D
2nd choice
C
D
3rd choice
D
A
4
Example 1: 37 VOTERS, need
19 votes for majority winner
Step 1: No one receives 19 votes, so
eliminate B and rewrite the table
1
14
10 8
C
Number of
voters
4
1
C C
D
1st choice
A
D
D D
D
A A
A
2nd choice
D
A
A A
A
4th choice
Step 2: No one with 19 votes
yet, so eliminate C and re-write
the table
Step 3: D has 23 votes so D is
the winner
The Mathematics of Voting
The Method of Pairwise Comparisons
(Copeland)
In a pairwise comparison between between X
and Y every vote is assigned to either X or Y,
the vote got in to whichever of the two
candidates is listed higher on the ballot. The
winner is the one with the most votes; if the
two candidates split the votes equally, it ends
in a tie.
15
The Mathematics of Voting
The Method of Pairwise Comparisons
The winner of the pairwise comparison gets 1
point and the loser gets none; in case of a tie
each candidate gets ½ point. The winner of
the election is the candidate with the most
points after all the pairwise comparisons are
tabulate.
16
The Mathematics of Voting
The Method of Pairwise Comparisons
There are 10 possible pairwise comparisons:
A vs. B, A vs. C, A vs. D, A vs. E, B vs. C,
B vs. D, B vs. E, C vs. D, C vs. E, D vs. E
17
The Mathematics of Voting
The Method of Pairwise Comparisons
A vs. B: B wins 15-7. B gets 1 point.
A vs. C: A wins 16-6. A gets 1 point. etc.
Final Tally: A-3, B-2.5, C-2, D-1.5, E-1. A
wins.
18

Sequential Pairwise Voting



Sequential pairwise voting starts with an agenda and pits the first
candidate against the second in a one-on-one contest.
The loser is deleted and the winner then moves on to confront
the third candidate in the list, one on one.
This process continues throughout the entire agenda, and the
one remaining at the end wins.
Example: Who would be the winner using the agenda A, B, C, D for
the following preference list ballots of three voters?
Rank
Number of Voters (3)
First
A
C
B
Second
B
A
D
Third
D
B
C
Fourth
C
D
A
Using the agenda A, B, C, D, start with A
vs. B and record (with tally marks) who
is preferred for each ballot list (column).
A vs. B
A vs. C
II
I
I
A wins; B is
deleted.
II
C wins; A is
deleted.
C vs. D
Candidate D
I
II
wins for this
D wins; C is agenda.
deleted.

Approval Voting



Under approval voting, each voter is allowed to give one
vote to as many of the candidates as he or she finds
acceptable.
No limit is set on the number of candidates for whom an
individual can vote; however, preferences cannot be
expressed.
Voters show disapproval of other candidates simply by not
voting for them.

Approval Voting (cont)

The winner under approval voting is the candidate who
receives the largest number of approval votes.

This approach is also appropriate in situations where more
than one candidate can win,


EX: in electing new members to an exclusive society such as the National
Academy of Sciences or the Baseball Hall of Fame.

Approval voting is also used to elect the secretary general of the United
Nations.
Approval voting was proposed independently by several
analysts in 1970s.
First basic fairness criterion


The Majority Criterion: if X has the majority
of the first-place votes (more than half), then
X is the winner.
The plurality method satisfies the majority
criterion.
Second basic fairness criterion



The Condorcet Criterion was introduced in
1785 by the French mathematician Le
Marquis de Condorcet
If candidate X is preferred over other
candidates in a head-to-head comparison,
then X is the winner
If X is the winner under the Majority Criterion,
then X is also the Condorcet winner.
Third basic fairness criterion
If a candidate is winning & votes are changed in
FAVOR of the winner
3.

Monotonicity Criterion: If votes are changed in favor of
the winning candidate, the winner should not change.
Monotonicity
(The Hare system fails monotonicity.)
 Monotonicity says that if a candidate is a winner and a new
election is held in which the only ballot change made is for some
voter to move the former winning candidate higher on his or her
ballot, then the original winner should remain a winner.
 In a new election, if a voter moves a winner higher up on his
preference list, the outcome should still have the same winner.
Number of Voters (13)
Rank
5
4
3
1
First
A
C
B
A
Second
B
B
C
B
Third
C
A
A
C
Number of Voters (13)
Rank
5
4
3
1
First
A
C
C
A
Second
C
A
A
C
In this example, A won because A has the
most 1st place votes.
Round 1: B is deleted with Hare method
because B has the fewest 1st place
votes.
Round 2: C moves up to replace B on the
third column.
However, C wins because now has the
most 1st place votes—this is a glaring
defect!
Fourth basic fairness criterion
What if a non-winning candidate drops out?

Independence-of-Irrelevant Alternatives Criterion: If a
non-winning candidate drops out, or is disqualified, the
winner should not change.
Original Borda Score: A=11, B=10, C=9
Rank
Number of Voters (5)
First (3pts)
A
A
A
C
C
Second (2 pts)
B
B
B
B
B
Rank
Number of Voters (5)
Third (1 pt)
C
C
C
A
A
First (3 pts)
A
A
A
B
B
Second (2pts)
B
B
B
C
C
Third (1 pt)
C
C
C
A
A
Suppose the last two voters change
their ballots (reverse the order of just
the losers). This should not change the
winner.
New Borda Score: A= 11, B=12, C=8
B went from loser to winner
and did not switch with A!
Summary
Majority
Condercet
Monotonicity Independence
Plurality
Yes
No
Yes
No
Borda Count
No
No
Yes
No
Plurality
Elimination
Yes
No
No
No
Pairwise
Comparison
Yes
Yes
Yes
No