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Math for Liberal Studies

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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

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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)

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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”

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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

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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

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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?

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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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

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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”)

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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

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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?

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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!

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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

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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…

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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)?

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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!

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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

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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…

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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?

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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!

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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

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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

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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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