Math for Liberal Studies Section 2 2 The
Math for Liberal Studies Section 2. 2: The Number of Candidates Matters
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 19 th 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”
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