Section 2.4: Rank MethodsMath for Liberal Studies

Another Voting Method

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

Rank Method

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

An Example

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

An Example

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

An Example

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

5 Soda > Juice > Milk

4 Juice > Soda > Milk

An Example

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

5 Soda > Juice > Milk 5

4 Juice > Soda > Milk 4

An Example

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

5 Soda > Juice > Milk 5 25

4 Juice > Soda > Milk 4 12

An Example

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

An Example

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

Rank Methods are Common

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

A Special Kind of Rank Method

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

Are Rank Methods “Fair”?

Rank methods do not satisfy the Condorcet winner criterion

In this profile, the Condorcet winner is A

However, the Borda count winner is B

Voters Preference Order

4 A > B > C

3 B > C > A

Are Rank Methods “Fair”?

Notice that C is a loser either way

If we get rid of C, noticewhat happens… Voters Preference Order

4 A > B > C

3 B > C > A

Are Rank Methods “Fair”?

Notice that C is a loser either way

If we get rid of C, noticewhat happens…

…now the Borda countwinner is A

Voters Preference Order

4 A > B

3 B > A

Are Rank Methods “Fair”?

If we start with this profile, A is the clear winner

But adding C into the mixcauses A to lose using theBorda count

In this way, C is a “spoiler”

Voters Preference Order

4 A > B

3 B > A

The Spoiler Effect

Voters prefer A over B

A third candidate C shows up

Now voters prefer B over A

The Spoiler Effect With Pies

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.

Another Example

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

Independence of Irrelevant Alternatives Condition (IIA)

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

How do we tell if a method satisfies the IIA condition?

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 non-spoiler candidates, then the method you are using suffers from the spoiler effect

How do we tell if a method satisfies the IIA condition?

A beats B, but when C shows up, B winsC is a spoiler!

A beats B, but when C shows up, A still winsNo spoiler!

A beats B, but when C shows up, C winsNo spoiler!

Still Searching

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

Still Searching… No, Really!

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

Still Searching… No, Really!

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