Math for Liberal Studies

We have studied the plurality and Condorcet
methods so far

In this method, once again voters will be
allowed to express their complete preference
order

Unlike the Condorcet method, we will assign
points to the candidates based on each ballot

We assign points to the candidates based on
where they are ranked on each ballot

The points we assign should be the same for
all of the ballots in a given election, but can
vary from one election to another

The points must be assigned nonincreasingly:
the points cannot go up as we go down the
ballot

Suppose we assign points like this:
 5 points for 1st place
 3 points for 2nd place
 1 point for 3rd place
Number of
Voters
Preference Order
6
Milk > Soda > Juice
5
Soda > Juice > Milk
4
Juice > Soda > Milk

Determine the winner by multiplying the
number of ballots of each type by the number
of points each candidate receives
Number of
Voters
Preference Order
6
Milk > Soda > Juice
5
Soda > Juice > Milk
4
Juice > Soda > Milk



5 points for 1st place
3 points for 2nd place
1 point for 3rd place
Number of
Voters
Preference Order
6
Milk > Soda > Juice
5
Soda > Juice > Milk
4
Juice > Soda > Milk
Milk
Soda
Juice



5 points for 1st place
3 points for 2nd place
1 point for 3rd place
Number of
Voters
Preference Order
Milk
6
Milk > Soda > Juice
30
5
Soda > Juice > Milk
5
4
Juice > Soda > Milk
4
Soda
Juice



5 points for 1st place
3 points for 2nd place
1 point for 3rd place
Number of
Voters
Preference Order
Milk
Soda
6
Milk > Soda > Juice
30
18
5
Soda > Juice > Milk
5
25
4
Juice > Soda > Milk
4
12
Juice



5 points for 1st place
3 points for 2nd place
1 point for 3rd place
Number of
Voters
Preference Order
Milk
Soda
Juice
6
Milk > Soda > Juice
30
18
6
5
Soda > Juice > Milk
5
25
15
4
Juice > Soda > Milk
4
12
20



Milk gets 39 points
Soda gets 55 points
Juice gets 41 points
 Soda wins!
Number of
Voters
Preference Order
Milk
Soda
Juice
6
Milk > Soda > Juice
30
18
6
5
Soda > Juice > Milk
5
25
15
4
Juice > Soda > Milk
4
12
20

Sports
 Major League Baseball MVP
 NCAA rankings
 Heisman Trophy

Education
 Used by many universities (including Michigan and
UCLA) to elect student representatives

Others
 A form of rank voting was used by the Roman Senate
beginning around the year 105

The Borda Count is a special kind of rank
method


With 3 candidates, the scoring is 2, 1, 0
With 4 candidates, the scoring is 3, 2, 1, 0
With 5 candidates, the scoring is 4, 3, 2, 1, 0
etc.

Last place is always worth 0





Rank methods do not satisfy the Condorcet
winner criterion
In this profile, the
Condorcet winner is A
Voters
Preference Order
4
A>B>C
3
B>C>A
However, the Borda count winner is B

Notice that C is a loser either way

If we get rid of C, notice
what happens…
Voters
Preference Order
4
A>B>C
3
B>C>A

Notice that C is a loser either way

If we get rid of C, notice
what happens…

…now the Borda count
winner is A
Voters
Preference Order
4
A>B
3
B>A



If we start with this profile, A is the clear
winner
But adding C into the mix
causes A to lose using the
Borda count
In this way, C is a “spoiler”
Voters
Preference Order
4
A>B
3
B>A

Voters prefer A over B

A third candidate C shows up

Now voters prefer B over A





After finishing dinner, you and your friends
decide to order dessert.
The waiter tells you he has two choices: apple pie
and blueberry pie.
You order the apple pie.
After a few minutes the waiter returns and says
that he forgot to tell you that they also have
cherry pie.
You and your friends talk it over and decide to
have blueberry pie.

In the 2000 Presidential election, if the
election had been between only Al Gore and
George W. Bush, the winner would have been
Al Gore

However, when we add Ralph Nader into the
election, the winner switches to George W.
Bush

The spoiler effect is sometimes called the
independence of irrelevant of alternatives
condition, or IIA for short

In a sense, the third candidate (the “spoiler”)
is irrelevant in the sense that he or she cannot
win the election

Look at a particular profile and try to identify
a candidate you think might be a spoiler

Determine the winner of the election with the
spoiler, and also determine the winner if the
spoiler is removed

If the winner switches between two nonspoiler candidates, then the method you are
using suffers from the spoiler effect

A beats B, but when C shows up, B wins
C is a spoiler!

A beats B, but when C shows up, A still wins
No spoiler!

A beats B, but when C shows up, C wins
No spoiler!

We now have two criteria for judging the
fairness of an election method
 Condorcet winner criterion (CWC)
 Independence of irrelevant alternatives (IIA)

We still haven’t found an election method
that satisfies both of these conditions

Well, actually, the Condorcet method satisfies
both conditions

But as we have seen, Condorcet’s method will
often fail to decide a winner, so it’s not really
usable

Ideally, we want an election method that
always gives a winner, and satisfies our
fairness conditions

In the next section we will consider several
alternative voting methods, and test them
using these and other conditions