Mathematics for Liberal Arts Chapter 1 The Mathematics of Voting

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Mathematics for Liberal Arts

Mathematics for Liberal ArtsChapter 1 The Mathematics of VotingLecture 1Preference Ballots and Preference Schedules

The Plurality Method

The Borda Count Method

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


Everyone will see the same movie.Rank your top four choices.Rank the top four choices for the whole class. No ties allowed!Everybody votes.Explain how you decided.

Do you feel that the decision was fair?

Movies at AMC Showplace Inver Grove 16:

Gangster SquadZero Dark ThirtyA Haunted HouseParental GuidanceDjango UnchainedLes MisrablesTexas Chainsaw 3DJack ReacherThis is 40LincolnThe HobbitPromised LandThe ImpossibleSilver Linings PlaybookMonsters, Inc. 3DArgoWhat is voting theory?Every person has their own preferences, interests, and values.How can we make decisions that take everyones 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 BallotsA 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.

LincolnDjango UnchainedThe HobbitZero Dark ThirtyPreference SchedulesA preference schedule lists the preferences for all voters.Example:

Lets make a preference schedule for the class movie preferences!What is the maximum number of columns?53321st choiceACDC2nd choiceCAAB3rd choiceBBBD4th choiceDDCAAssumptions about PreferencesEvery 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. Cyclic PreferencesEquivalently, there are no cyclic preferences. You cant like A better than B, B better than C, and C better than A.

This is equivalent to the assumption that preferences are transitive.

Eliminating optionsIf 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

LincolnDjango UnchainedThe HobbitZero Dark ThirtyNew Ballot

LincolnThe HobbitZero Dark ThirtyIn 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 criterionIt 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?14751stABC2ndBCB3rdCAD4thDDA6541stABC2ndBDB3rdCAA4thDCDA 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 doesnt tell us who should win.

The Plurality MethodThe 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 paradoxSuppose 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?6541stABC2ndBCA3rdCABThis is called Condorcets paradox (after Marquis de Condorcet, 1743-1794).The Condorcet CriterionA 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.

86551stABBC2ndDCCA3rdCADD4thBDABCondorcet Criterion ExampleIs there a Condorcet candidate?4321st ABC2ndBCB3rdCAAA 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 MethodThe 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?

867731stABBCB2ndCCAAA3rdBACBCDoes the Borda Count satisfy the Condorcet Criterion?

55431stBACB2ndABAC3rdCCBABorda Count and MathIf 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 votingInsincere 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 EliminationIf 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.ExamplesWho is the winner using Plurality with Elimination?

867731stABBCB2ndCCAAA3rdBACBC55431stBDCB2ndAAAC3rdCBBD4th DCDAHow fair is IRV?IRV satisfies the majority criterion. (Why?)

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

2431stABC2ndBAA3rdCCBIRV is unfair in another way: It is not monotone.

The Monotonicity CriterionThe 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 shouldnt make you LOSE. Thats 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?87331stABCC2ndCCAB3rdBABA852331stABACC2ndCCBAB3rdBACBAThis 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.Its 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.SummaryMajorityCondorcetMonotonePluralityYesNoYesBorda CountNoNoYesInstant RunoffYesNoNoBe sure you understand the following concepts.Preference ballots and preference schedulesTransitive preferencesVoting methods: Plurality, Borda Count, Instant RunoffFairness criteria: Majority, Condorcet, Monotonicity

Homework: Page 30: #4, 8, 12, 16, 20, 24, 28