Section 2.2: The Number of Candidates Matters

1. Math for Liberal Studies Section 2.2: The Number of Candidates Matters

2. 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

3. Preference Lists • For example, there are three candidates for Congress from the 19th district of Pennsylvania (which includes Carlisle, York, and Shippensburg): • Todd Platts (R) • Ryan Sanders (D) • Joshua Monighan (I)

4. 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”

5. 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

6. 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

7. 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?

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Changing the Rules • Let’s see how things change if we don’t use majority rule, but instead use a different system

20. 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

21. 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”)

22. 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

23. 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?

24. 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 Zeldaprefer A • Xander, Yolanda, and Wally prefer B • Now A wins, 2 to 1!

25. 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

26. 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…

27. 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)?

28. 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!

29. 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

30. 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…

31. 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?

32. 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!

33. “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

34. 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

35. 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”