MAT 105 Spring 2008

There are many more methods for
determining the winner of an election with
more than two candidates

We will only discuss a few more:
 sequential pairwise voting
 Hare system
 plurality runoff

Idea: We like pairwise voting (where we can
use majority rule), but if we look at all
pairwise elections, we sometimes don’t get a
winner

In sequential pairwise voting, we put the
candidates in order on a list, called the
agenda

We pit the first two candidates on the agenda
against each other. The winner moves on to face the
next candidate on the list, and so on. The candidate
remaining at the end is the winner.

This process resembles a tournament bracket, and
has the advantage that, unlike Condorcet’s method,
we always get a winner

Let’s use sequential
pairwise voting with
this profile and the
agenda A, B, C, D
A
B
Voters
Preference Order
4
A>B>D>C
3
C>A>B>D
3
B>D>C>A
A beats B, 7-3
A
C beats A, 6-4
C
C
D beats C, 7-3
D
D

If we look closely at this
agenda, we notice that
every single voter prefers
B over D, and yet D was
our winner!
Voters
Preference Order
4
A>B>D>C
3
C>A>B>D
3
B>D>C>A

In fact, by cleverly choosing the right agenda, we
could make any of the four candidates win this
election

Sequential pairwise voting does not satisfy the
Pareto condition

If every voter prefers one candidate over
another, then the latter candidate should not
be among the winners of the election

Named for Vilfredo Pareto (1848-1923),
Italian economist

Does plurality satisfy the Pareto condition?

Also known as Instant Runoff Voting, this
system is used for various elections in the US,
Canada, the UK, Ireland, and Australia

Repeatedly delete candidates that are “least
preferred” in the sense of being at the top of
the fewest ballots. If there is a tie, eliminate
all of the tied candidates, until there is no one
left to eliminate



In this example, A has 5
first-place votes, B has 5
first-place votes, and C has
4 first-place votes, so C is
eliminated
Now A has 5 first-place
votes, and B has 9, so A is
eliminated
B is the only candidate left,
so B is the winner
Voters
Preference Order
5
A>B>C
4
C>B>A
3
B>C>A
2
B>A>C
Voters
Preference Order
5
A>B
4
B>A
3
B>A
2
B>A


This time, A has 5 first-place
votes, and B and C are tied
with 4, so B and C are both
eliminated at the same time
Voters
Preference Order
5
A>B>C
4
C>B>A
3
B>C>A
1
B>A>C
This leaves only A to win the election

Now let’s modify the profile from the previous
example, so that the 1 voter with preference
B > A > C now has preference A > B > C

Notice that this change moves the winner
higher on that voter’s ballot
Voters
Preference
Voters
Preference
6
A>B>C
6
A>C
4
C>B>A
4
C>A
3
B>C>A
3
C>A
C wins!

A was the winner of the original election, and
one of the voters changed his ballot to move
A higher, causing A to lose

This shows that the Hare system is not
monotone

A voting system is monotone if whenever a
candidate is a winner, and a new election is
held where the only change is for some voter
to move that winner higher on his/her ballot,
then the original winner should remain the
winner

The Hare system is not monotone, but despite
this drawback it is one of the more common
alternative voting systems in use today

Hold a plurality election, but if no candidate
receives a majority, we hold a runoff election

The runoff election is between the two
candidates who received the most first-place
votes in the original election

In case of ties, there might be more than two
candidates with the most first-place votes, so we
use plurality to decide a winner between those
candidates only

In this profile, A gets 5
first-place votes, B gets
5 first-place votes, and C
only gets 4
Voters
Preference Order
5
A>B>C
4
C>B>A
3
B>C>A
2
B>A>C

The runoff is between A and B

B wins the runoff 9 votes to 5

In this profile, A has 4
first-place votes, B has 3,
C has 3, and D has 2
Voters
Preference Order
4
A>B>C>D
3
C>D>B>A
3
B>C>D>A
2
D>B>A>C

The runoff is between A, B,
and C

We use plurality to decide the winner; keep in
mind that the 2 voters who like D best get to vote
in the runoff!

B wins the runoff with 5 votes