### Section 2.4: Rank Methods

```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
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



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

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
```