Mathematics for Liberal ArtsChapter 1 The Mathematics of Voting

Lecture 1• Preference Ballots and Preference Schedules

• The Plurality Method

• The Borda Count Method

• Instant Runoff Voting

OpenerImagine that you all decided to watch a movie together. Which movies do you want to see?

Rules:

1. Everyone will see the same movie.2. Rank your top four choices.3. Rank the top four choices for the

whole class. No ties allowed!4. Everybody votes.5. Explain how you decided.

Do you feel that the decision was fair?

Movies at AMC Showplace Inver Grove 16:

a) Gangster Squadb) Zero Dark Thirtyc) A Haunted Housed) Parental Guidancee) Django Unchainedf) Les Misérablesg) Texas Chainsaw 3Dh) Jack Reacheri) This is 40j) Lincolnk) The Hobbitl) Promised Landm) The Impossiblen) Silver Linings Playbooko) Monsters, Inc. 3Dp) Argo

What is voting theory?• Every person has their own preferences, interests, and values.• How can we make decisions that take everyone’s interests into

account?• In a democracy, we make decisions by voting.• Direct democracy: Vote for specific policies (e.g. referenda)• Representative democracy: Elect representatives who will make

policy decisions.• How can we know if an election is fair?• What is fairness, anyway? Can we define it mathematically?

Preference Ballots• A preference ballot is a ranking of the available choices. • A ranking without ties is called a linear ballot.• The most desirable choice is ranked first, and the least

desirable choice is ranked last.• Please list your preference ballot, choosing from the top four

movies for the whole class.• Here is my preference ballot.

1. Lincoln2. Django Unchained3. The Hobbit4. Zero Dark Thirty

Preference Schedules• A preference schedule lists the preferences for all voters.• Example:

• Let’s make a preference schedule for the class movie preferences!

• What is the maximum number of columns?

5 3 3 2

1st choice A C D C

2nd choice C A A B

3rd choice B B B D

4th choice D D C A

(n factorial)

Assumptions about Preferences

• Every voter has the same options.• Given a choice between two options, a voter will always prefer

one or the other. (Maybe not realistic?)• Preferences are transitive. If you prefer chocolate to vanilla,

and you prefer vanilla to strawberry, then you also prefer chocolate to strawberry.

http://webclipart.about.com/od/seasonsclipart/ss/Summer-Images-2_13.htm

No Cyclic Preferences

• Equivalently, there are no cyclic preferences. You can’t like A better than B, B better than C, and C better than A.

• This is equivalent to the assumption that preferences are transitive.

A

BC

Eliminating options• If an option is eliminated, then the ranking of the other

choices does not change.• For example, here is how my preference ballot would be

altered if there were no more tickets for Django Unchained.

Old Ballot

1. Lincoln2. Django Unchained3. The Hobbit4. Zero Dark Thirty

New Ballot

1. Lincoln2. The Hobbit3. Zero Dark Thirty

In other words, individual preferences are assumed to be rational.

Are these assumptions reasonable?

• Yes (probably). Most people act rationally, most of the time.

• Rational behavior is much easier to model.

• Social scientists are starting to take non-rationality seriously.

• Dan Ariely, Predictably Irrational was a New York Times bestseller in 2008.

• We will assume that voters are rational.

What is fairness?• It is not possible to give an exact mathematical definition of

fairness.

• A fair voting system should treat everybody equally, and take all interests into account.

• We attempt to make fairness into a precise concept by defining fairness criteria.

• A fairness criterion is any property that we believe that a voting system should have.

• Discuss: What makes a voting system fair?

The majority criterion• It is important to understand the difference between a

majority and a plurality.• A majority candidate is a candidate who receives more than

half of the first-place votes.• A plurality candidate is a candidate who receives more first-

place votes than any other candidate.• Is it possible to have a plurality but not a majority?• Is it possible to have a majority but not a plurality?

• The majority criterion says:

If there is a majority candidate, then she should win the election.

Who should win the election?14 7 5

1st A B C

2nd B C B

3rd C A D

4th D D A

6 5 4

1st A B C

2nd B D B

3rd C A A

4th D C D

A is a majority candidate.

The majority condition says that A should win the election.

A has a plurality but not a majority.

The majority condition doesn’t tell us who should win.

The Plurality Method• The plurality method is the simplest voting method.

• We simply declare that the plurality candidate is the winner.

• Also called “first-past-the-post”.

• Only first choices are counted.

• Does the plurality method satisfy the majority criterion? Why or why not?

• Is the plurality method fair?

A voting paradox• Suppose that an election results in the following preference

schedule.

• The majority of voters prefer A to B.• The majority of voters prefer B to C.• The majority of voters prefer C to A.• Does this mean that voters are irrational?• Who should win this election?

6 5 4

1st A B C

2nd B C A

3rd C A B

This is called Condorcet’s paradox (after Marquis de Condorcet, 1743-1794).

The Condorcet Criterion• A Condorcet candidate is a candidate who is preferred over

every other candidate in a head-to-head comparison.• The Condorcet criterion says that if there is a Condorcet

candidate, then he should win the election.• Does the plurality method satisfy the Condorcet criterion?

Explain why this criterion is satisfied, or give an example where it fails.

8 6 5 5

1st A B B C

2nd D C C A

3rd C A D D

4th B D A B

Condorcet Criterion ExampleIs there a Condorcet candidate?

4 3 2

1st A B C

2nd B C B

3rd C A A

A majority (5 to 4) prefer B to A.A majority (7 to 2) prefer B to C.Therefore B is a Condorcet candidate

Question: Is it possible to have TWO Condorcet candidates?

Borda Count Method• The Borda Count is a voting method named after Jean-Charles

de Borda (1733-1799).

• It is a point-based system. A candidate is awarded 1 point for being ranked in last place, 2 points for second to last, and so on.

• The candidate with the most Borda points wins.

• Which movie would have been selected if we used the Borda count?

Where is the Borda Count used?• Choosing the MVP in Major League Baseball

• NCAA sports rankings (AP and UPI)

• Heisman Trophy

• Eurovision song contest

Is the Borda Count Fair?

Does the Borda Count satisfy the majority criterion?

8 6 7 7 3

1st A B B C B

2nd C C A A A

3rd B A C B C

Does the Borda Count satisfy the Condorcet Criterion?

5 5 4 3

1st B A C B

2nd A B A C

3rd C C B A

Borda Count and Math• If there are N candidates, how many points are awarded by

each ballot?

• If there are N candidates and V voters, what is the minimum number of points that a winner could have?

• What is the minimum number of points that ensures a win?

• Is the Borda Count “less fair” than the plurality criterion?

Insincere voting• Insincere voting is when a voter does not express his true

preferences on a ballot. Also known as strategic voting or tactical voting.

• There are two forms of insincere voting:• Compromising – Ranking a candidate higher in hopes of getting

her elected.• Burying – Ranking a candidate lower in hopes of defeating her.

• This is very common in elections that use the plurality method. People will rarely vote for a third-party candidate even if they prefer that candidate.

• Gibbard-Satterthwaite theorem: All voting methods are susceptible to insincere voting, except for random methods and dictatorships.

Plurality with Elimination• If a candidate has a majority of first-place votes, then he is

elected.• Otherwise, the candidate with the fewest first place votes is

eliminated and the election is run again.• More commonly known as Instant Runoff Voting (IRV).• Equivalent to Single Transferable Vote when there is only one

winner.• IRV is becoming more popular, and it is often used for state

and local elections.• St. Paul MN has recently begun using IRV to elect the mayor

and city council.• Unlike the plurality method, IRV accommodates multiple

candidates and avoids the spoiler effect.

Examples

Who is the winner using Plurality with Elimination?

8 6 7 7 3

1st A B B C B

2nd C C A A A

3rd B A C B C

5 5 4 3

1st B D C B

2nd A A A C

3rd C B B D

4th D C D A

How fair is IRV?• IRV satisfies the majority criterion. (Why?)

• IRV does not satisfy the Condorcet criterion. (Why not?)

4 3 2

1st A B C

2nd B A A

3rd C C B

• IRV is unfair in another way: It is not monotone.

The Monotonicity Criterion• The monotonicity criterion says that if A wins an election, and

some ballots are changed to rank A higher (without altering the ranking of the other candidates), then A should still win the election.

• Becoming MORE popular shouldn’t make you LOSE. That’s weird (a paradox).

• The plurality method satisfies the monotonicity criterion.• Borda count also satisfies the monotonicity criterion.• IRV does not satisfy the monotonicity criterion. Can you think

of an example?

IRV is not monotone!Consider the following preference schedule.

Verify that A wins the election. But what if two voters changed their ballots from B > C > A to A > B > C?

8 7 3 31st A B C C

2nd C C A B

3rd B A B A

8 5 2 3 3

1st A B A C C

2nd C C B A B

3rd B A C B A

This time, C wins!

How did this happen?• A won the election, but most voters preferred C to A.• C was eliminated in the first round.• Most voters preferred B to C.• If A takes away votes from B, then B might not be able to

eliminate C.• It’s like Survivor. The enemy of your enemy is your friend –

keep him alive. (For now…)• This cannot happen when there is a Condorcet candidate.

Condorcet methods are always monotone.

A

BC

SummaryMajority Condorcet Monotone

Plurality Yes No Yes

Borda Count No No Yes

Instant Runoff Yes No No

Be sure you understand the following concepts.• Preference ballots and preference schedules• Transitive preferences• Voting methods: Plurality, Borda Count, Instant Runoff• Fairness criteria: Majority, Condorcet, Monotonicity• Homework: Page 30: #4, 8, 12, 16, 20, 24, 28, 32, 62, 64