Section 2.2: The Number of Candidates Matters

Math for Liberal Studies

Preference Lists

In most US elections, voters can only cast a single ballot for the candidate he or she likes the best

However, most voters will have “preference lists”: a ranking of the candidates in order of most preferred to least preferred

Preference Lists

For example, there are three candidates for Congress from the 19th district of Pennsylvania (which includes Carlisle, York, and Shippensburg): Todd Platts (R) Ryan Sanders (D) Joshua Monighan (I)

Preference Lists

When you go into the voting booth, you can only choose to vote for one candidate

However, even if you vote for, say, Platts, you might still prefer Monighan over Sanders

Platts is your “top choice,” but in this example, Monighan would be your “second choice”

Preference Lists

As another example, many (but not all) of the people who voted for Ralph Nader in 2000 would have had Al Gore as their second choice

If those “second place” votes had somehow counted for something, Al Gore might have been able to win the election

A Simple Example

Suppose a class of children is trying to decide what drink to have with their lunch: milk, soda, or juice

Each child votes for their top choice, and the results are: Milk 6 Soda 5 Juice 4

Milk wins a plurality of the votes, but not a majority

Considering Preferences

Now suppose we ask the children to rank the drinks in order of preference

We know 6 students had milk as their top choice because milk got 6 votes

But what were those students’ second or third choices?

Considering Preferences

Here are the preference results: 6 have the preference Milk > Soda > Juice 5 have the preference Soda > Juice > Milk 4 have the preference Juice > Soda > Milk

Is the outcome fair? If we choose Milk as the winner of this election, 9 of the 15 students are “stuck” with their last choice

Rules for Preference Lists

We will not allow ties on individual preference lists, though some methods will result in an overall tie

All candidates must be listed in a specific order

Two Candidates

When there are only two candidates, things are simple

There are only two preferences: A > B and B > A

Voters with preference A > B vote for A

Voters with preference B > A vote for B

The candidate with the most votes wins

This method is called majority rule

Majority Rule

Notice that one of the two candidates will definitely get a majority (they can’t both get less than half of the votes)

Majority rule has three desirable properties anonymous neutral monotone

Anonymous

If any two voters exchange (filled out) ballots before submitting them, the outcome of the election does not change

In this way, who is casting the vote doesn’t impact the result of the vote; all the voters are treated equally

Neutral

If a new election were held and every voter reversed their vote (people who voted for A now vote for B, and vice versa), then the outcome of the election is also reversed

In this way, one candidate isn’t being given preference over another; the candidates are treated equally

Monotone

If a new election is held and the only thing that changes is that one or more voters change their votes from a vote for the original loser to a vote for the original winner, then the new election should have the same outcome as the first election

Changing your vote from the loser to the winner shouldn’t help the loser

Other Methods

Majority rule satisfies all three of these conditions

But majority rule is not the only way to determine the winner of an election with two candidates

Let’s consider some other systems

Some Examples of Other Methods Patriarchy: only the votes of men count Dictatorship: there is a certain voter called

the dictator, and only the dictator’s vote counts (all other ballots are ignored)

Oligarchy: there is a small council of voters, and only their votes count

Imposed rule: a certain candidate wins no matter what the votes are

Other Methods

Those other methods may not seem “fair”

But “fairness” is subjective

The three conditions (anonymous, neutral, and monotone) give us a way to objectively measure fairness

An Example

Suppose we have an election between two candidates, A and B

Let’s say there are 5 voters: Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

In a majority rule election, B wins, 3 to 2

Changing the Rules

Let’s see how things change if we don’t use majority rule, but instead use a different system

Changing the Rules

Suppose we are using the matriarchy system: only the votes of women count

Now our votes look like this Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

Wally and Xander still vote, but their votes don’t count; B still wins, 2 to 1

Why is this unfair?

Aside from the obvious unfairness in the matriarchy system, our “official” measure of fairness is the three conditions we discussed earlier

anonymous (“the voters are treated equally”) neutral (“the candidates are treated equally”) monotone (“changing your vote from the

loser to the winner shouldn’t help the loser”)

Is Matriarchy Anonymous?

To test if a system is anonymous, we need to consider what might happen if two of the voters switch ballots before submitting them

In an anonymous system, this should not change the results

However, we can change the results in our example

Is Matriarchy Anonymous?

Here’s our original election: Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

Do you see a way that two voters could swap ballots that could change the election result?

Matriarchy Is Not Anonymous

Here’s our original election: Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

If Wally and Zelda switch ballots… Ursula and Zelda prefer A Xander, Yolanda, and Wally prefer B

Now A wins, 2 to 1!

Changing the Rules Again

Let’s consider another “unfair” system

Suppose in this new system, votes for A are worth 2 points, and votes for B are only worth 1 point Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

A wins, 4 to 3

Is This System Neutral?

Again, we can see that the system is obviously “unfair,” but we want to see that using those three conditions

This time we want to know if our system is neutral

If we make every voter reverse his or her ballot, the winner of the election should also switch…

Is This System Neutral?

Here’s the original election: Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

What will happen if everyone reverses their ballot (i.e., everyone votes for the candidate they didn’t vote for the first time around)?

This System Is Not Neutral

Here’s the original election: Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

And now we reverse everyone’s ballot: Ursula and Wally prefer B Xander, Yolanda, and Zelda prefer A

But A still wins, 6 to 2!

Changing the Rules Again (Again)

This time we’ll use minority rule: the candidate with the fewest votes wins

You could imagine this system being used on a game show, where the person with the fewest votes doesn’t get “voted off the island”

Here’s the original election Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

A wins, 2 to 3

Is Minority Rule Monotone?

Again, for a normal election, it seems patently unfair to have the person with the fewest votes win, but let’s consider the three conditions

To test monotone, we need to see if it is possible to have voters change their votes from the loser to the winner and change the outcome

In a “fair” election, this should not change the outcome…

Is Minority Rule Monotone?

Here’s the original election Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

Do you see a way that one or more voters who voted for the loser could change their votes so that the loser now wins?

Minority Rule Is Not Monotone

Here’s the original election Ursula and Wally prefer A Xander, Yolanda, and Zelda prefer B

Now we’ll have Zelda change her vote from B (the original loser) to A (the original winner): Ursula, Wally, and Zelda prefer A Xander and Yolanda prefer B

But now B wins 2 to 3!

“Unfair”

What seems unfair to one person might seem fair to another

An election method for two candidates that satisfies all three conditions is “fair,” and a method that does not isn’t

May’s Theorem

In 1952, Kenneth May proved that majority rule is the only “fair” system with two candidates

This fact is known as May’s Theorem

No matter what system for two candidates we come up with (other than majority rule), it will fail at least one of the three conditions

Looking Ahead

May’s Theorem gives us a way to consider the fairness of a system objectively

The situation gets significantly more complex with more than two candidates

However, we will still use these kinds of conditions to consider the issue of “fairness”