MAT 105 Spring 2008

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

Milk Soda Juice

30 18 6

5 25 15

4 12 20

39 55 41

Soda wins with 55 points!

Sports Major League Baseball MVP NCAA rankings Heisman Trophy

Education Used by many universities (including Michigan and UCLA) to

elect student representatives Used by some academic departments to elect members to

committees 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

Each candidate is given a number of points equal to the number of candidates ranked below them

So with 3 candidates, in the Borda count 1st place is worth 2 points, 2nd place is worth 1 point, and 3rd place is worth 0 points

With 4 candidates, the scoring is 3, 2, 1, 0

Suppose we have an election where A is the winner, B is not, and there are possibly other candidates

Suppose now that we have a new election, and some of the voters change their ballots

However, no one who had A ranked above B changed their ballot to have B above A

What should the outcome of the new election be?

Let’s look at an example We’ll use the Borda count

to find the winner of thiselection A gets 11 points B gets 6 points C gets 4 points

A is the winner, and B is not We will have a new election, and no one who

had A above B will change to have B above A

Voters Preference Order

4 A > C > B

3 B > A > C

Notice that every voter changed his ballot However, no one changed the order that they

had A and B ranked, they only moved C B wins the new election! We say that C was “irrelevant” to the question of

A versus B, but moving C around affected the outcome

Voters Preference Order

4 A > C > B

3 B > A > C

Voters Preference Order

4 A > B > C

3 B > C > A

After finishing dinner, Sidney decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Sidney says "In that case I'll have the blueberry pie.“

In our example, A is apple pie, B is blueberry pie, and C is cherry pie

This gives us a way to tell if a voting system is fair

Here’s the process: We have an original election, where A wins and B

does not We hold a new election, and while the voters can

change their ballots, no one changes from having A above B to having B above A

The outcome of the election should not change

If it is not possible to change the outcome of the election by this process, we say the voting method satisfies IIA

If it is possible to change the outcome of the election by this process, we say the voting method does not satisfy IIA

Borda count does not satisfy IIA because of the example we had (so Borda count is “unfair” in this way)

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 voters did not change their preference regarding Bush vs. Gore, but the winner changed

Plurality also does not satisfy IIA