Exploring Rank Methods in Voting Systems

1. Math for Liberal Studies Section 2.4: Rank Methods

2. Another Voting Method • We have studied the plurality and Condorcet methods so far • In this method, once again voters will be allowed to express their complete preference order • Unlike the Condorcet method, we will assign points to the candidates based on each ballot

3. Rank Method • We assign points to the candidates based on where they are ranked on each ballot • The points we assign should be the same for all of the ballots in a given election, but can vary from one election to another • The points must be assigned nonincreasingly: the points cannot go up as we go down the ballot

4. An Example • Suppose we assign points like this: • 5 points for 1st place • 3 points for 2nd place • 1 point for 3rd place

5. An Example • Determine the winner by multiplying the number of ballots of each type by the number of points each candidate receives

6. An Example • 5 points for 1st place • 3 points for 2nd place • 1 point for 3rd place

7. An Example • 5 points for 1st place • 3 points for 2nd place • 1 point for 3rd place

8. An Example • 5 points for 1st place • 3 points for 2nd place • 1 point for 3rd place

9. An Example • 5 points for 1st place • 3 points for 2nd place • 1 point for 3rd place

10. An Example • Milk gets 39 points • Soda gets 55 points • Juice gets 41 points • Soda wins!

11. Rank Methods are Common • Sports • Major League Baseball MVP • NCAA rankings • Heisman Trophy • Education • Used by many universities (including Michigan and UCLA) to elect student representatives • Others • A form of rank voting was used by the Roman Senate beginning around the year 105

12. A Special Kind of Rank Method • The Borda Count is a special kind of rank method • With 3 candidates, the scoring is 2, 1, 0 • With 4 candidates, the scoring is 3, 2, 1, 0 • With 5 candidates, the scoring is 4, 3, 2, 1, 0 • etc. • Last place is always worth 0

13. Are Rank Methods “Fair”? • Rank methods do not satisfy the Condorcet winner criterion • In this profile, the Condorcet winner is A • However, the Borda count winner is B

14. Are Rank Methods “Fair”? • Notice that C is a loser either way • If we get rid of C, noticewhat happens…

15. Are Rank Methods “Fair”? • Notice that C is a loser either way • If we get rid of C, noticewhat happens… • …now the Borda countwinner is A

16. Are Rank Methods “Fair”? • If we start with this profile, A is the clear winner • But adding C into the mixcauses A to lose using theBorda count • In this way, C is a “spoiler”

17. The Spoiler Effect • Voters prefer A over B • A third candidate C shows up • Now voters prefer B over A

18. The Spoiler Effect With Pies • After finishing dinner, you and your friends decide to order dessert. • The waiter tells you he has two choices: apple pie and blueberry pie. • You order the apple pie. • After a few minutes the waiter returns and says that he forgot to tell you that they also have cherry pie. • You and your friends talk it over and decide to have blueberry pie.

19. Another Example • In the 2000 Presidential election, if the election had been between only Al Gore and George W. Bush, the winner would have been Al Gore • However, when we add Ralph Nader into the election, the winner switches to George W. Bush

20. Independence of Irrelevant Alternatives Condition (IIA) • The spoiler effect is sometimes called the independence of irrelevant of alternatives condition, or IIA for short • In a sense, the third candidate (the “spoiler”) is irrelevant in the sense that he or she cannot win the election

21. How do we tell if a method satisfies the IIA condition? • Look at a particular profile and try to identify a candidate you think might be a spoiler • Determine the winner of the election with the spoiler, and also determine the winner if the spoiler is removed • If the winner switches between two non-spoiler candidates, then the method you are using suffers from the spoiler effect

22. How do we tell if a method satisfies the IIA condition? • A beats B, but when C shows up, B winsC is a spoiler! • A beats B, but when C shows up, A still winsNo spoiler! • A beats B, but when C shows up, C winsNo spoiler!

23. Still Searching • We now have two criteria for judging the fairness of an election method • Condorcet winner criterion (CWC) • Independence of irrelevant alternatives (IIA) • We still haven’t found an election method that satisfies both of these conditions

24. Still Searching… No, Really! • Well, actually, the Condorcet method satisfies both conditions • But as we have seen, Condorcet’s method will often fail to decide a winner, so it’s not really usable

25. Still Searching… No, Really! • Ideally, we want an election method that always gives a winner, and satisfies our fairness conditions • In the next section we will consider several alternative voting methods, and test them using these and other conditions