Math for Liberal Studies

 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

 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)

 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”

 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

 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

 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?

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

 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

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

 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

 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”)

 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

 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?

 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!

 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

 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…

 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)?

 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!

 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

 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…

 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?

 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!

 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

 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

 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”