Rank Voting

1. MAT 105 Spring 2008 Rank Voting

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 Soda wins with 55 points!

5. 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 • Used by some academic departments to elect members to committees • Others • A form of rank voting was used by the Roman Senate beginning around the year 105

6. A Special Kind of Rank Method • The Borda Count is a special kind of rank method • Each candidate is given a number of points equal to the number of candidates ranked below them • So with 3 candidates, in the Borda count 1st place is worth 2 points, 2nd place is worth 1 point, and 3rd place is worth 0 points • With 4 candidates, the scoring is 3, 2, 1, 0

7. Problems with Rank Methods • Suppose we have an election where A is the winner, B is not, and there are possibly other candidates • Suppose now that we have a new election, and some of the voters change their ballots • However, no one who had A ranked above B changed their ballot to have B above A • What should the outcome of the new election be?

8. Irrelevant Alternatives • Let’s look at an example • We’ll use the Borda countto find the winner of thiselection • A gets 11 points • B gets 6 points • C gets 4 points • A is the winner, and B is not • We will have a new election, and no one who had A above B will change to have B above A

9. Irrelevant Alternatives • Notice that every voter changed his ballot • However, no one changed the order that they had A and B ranked, they only moved C • B wins the new election! • We say that C was “irrelevant” to the question of A versus B, but moving C around affected the outcome

10. Independence of Irrelevant Alternatives (IIA) • After finishing dinner, Sidney decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Sidney says "In that case I'll have the blueberry pie.“ • In our example, A is apple pie, B is blueberry pie, and C is cherry pie

11. Independence of Irrelevant Alternatives (IIA) • This gives us a way to tell if a voting system is fair • Here’s the process: • We have an original election, where A wins and B does not • We hold a new election, and while the voters can change their ballots, no one changes from having A above B to having B above A • The outcome of the election should not change

12. Independence of Irrelevant Alternatives (IIA) • If it is not possible to change the outcome of the election by this process, we say the voting method satisfies IIA • If it is possible to change the outcome of the election by this process, we say the voting method does not satisfy IIA • Borda count does not satisfy IIA because of the example we had (so Borda count is “unfair” in this way)

13. 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 • The voters did not change their preference regarding Bush vs. Gore, but the winner changed • Plurality also does not satisfy IIA