§ 1.6 Rankings

1. § 1.6 Rankings “Rome’s biggest contribution to American government was probably its legal system [. . . ] [which] would later form the basis of both the Bill of Rights and a mind-numbing quantity of Law and Order scripts.” - America (The Book)

2. Elections With Rankings • In Lawrence city commission elections, the candidate with the highest number of votes becomes mayor while other candidates are simply commissioners. • This is a simple example of an election where more than just the ‘winner’ is important--in these instances we must consider the ranking of each vote-getter.

3. Extended Ranking Methods • Each of the four counting methods described earlier this week has a natural extension.

4. Let’s look at the Muppet example again; this time supposing that they are voting for a President, Vice- President and Treasurer. Let us first use the Extended Plurality Method. (This method--along with the weighting of the Electoral College--was originally used in US Presidential Elections.) Example: Counting the first-place votes we get the following results:Office Place Candidate VotesPresident 1st Piggy 21Vice-Pres. 2nd Gonzo 15Treasurer 3rd Fozzie 12- None - 4th Kermit 7

5. Now let us see what happens with the Extended Borda Count Method. Example: Tallying the points we find:Office Place Candidate PointsPresident 1st Kermit 160Vice-Pres. 2nd Gonzo 152Treasurer 3rd Fozzie 120- None - 4th Piggy 118

6. Note: If a majority appears before all candidates have been ranked, we will simply continue the process of elimination until all candidates have been ranked. Now let us see what happens with the Extended Plurality-with-Elimination Method. Example: Extending Instant-Runoff Voting is a bit more subtle--we will rank candidates based on when they were eliminated. The first choice that is eliminated will be ranked last. Office Place Candidate Eliminated In President 1st Fozzie ---------------------------Vice-Pres. 2nd Piggy 3rd RoundTreasurer 3rd Gonzo 2nd Round- None - 4th Kermit 1st Round

7. Now showing: Extended Pairwise Comparison Method. Example: After examining all of the possible head-to-head pairings of candidates and awarding points we get:Office Place Candidate PointsPresident 1st Kermit 3Vice-Pres. 2nd Gonzo 2Treasurer 3rd Fozzie 1- None - 4th Piggy 0

8. Recursive Ranking Methods • The four methods we have discussed can also be used to rank candidates in a recursive manner. • The Idea: Suppose we use some voting method to find the winner of an election. We will then remove the winner from our preference schedule and find the winner of this ‘new’ election--this candidate will be ranked second. We repeat this process until all candidates have been ranked.

9. Recursive Plurality Method. Example: Step 1. (Choose 1st place.) We have already seen that Piggy wins in a plurality system with 21 votes.

10. Recursive Plurality Method. Example: Step 1. (Choose 1st place.) We have already seen that Piggy wins in a plurality system with 21 votes.Step 2. (Choose 2nd place.) First we remove Piggy from our preference schedule. In this new schedule the winner is Kermit with 28 votes.

11. Recursive Plurality Method. Example: Step 1. (Choose 1st place.) We have already seen that Piggy wins in a plurality system with 21 votes.Step 2. (Choose 2nd place.) First we remove Piggy from our preference schedule. In this new schedule the winner is Kermit with 28 votes.Step 3. (Choose 3rd place.) First remove Kermit from the preference schedule. In this new preference schedule Gonzo wins with 36 votes.

12. Recursive Plurality Method. Example: Under this recursive method we have:Office Place Candidate President 1st Piggy Vice-Pres. 2nd Kermit Treasurer 3rd Gonzo - None - 4th Fozzie

13. Recursive Plurality-with-Elimination Method. Example: Step 1. (Choose 1st place.) Using the Plurality-with-Elimination we have already seen that Fozzie would win.

14. Recursive Plurality-with-Elimination Method. Example: Step 1. (Choose 1st place.) Using the Plurality-with-Elimination we have already seen that Fozzie would win.Step 2. (Choose 2nd place.) First remove Fozzie from the preference schedule. Now we use the plurality-with-elimination method to find a winner.

15. Recursive Plurality-with-Elimination Method. Example: Step 1. (Choose 1st place.) Using the Plurality-with-Elimination we have already seen that Fozzie would win.Step 2. (Choose 2nd place.) First remove Fozzie from the preference schedule. Now we use the plurality-with-elimination method to find a winner. In this case, it is Gonzo.

16. Recursive Plurality-with-Elimination Method. Example: Step 1. (Choose 1st place.) Using the Plurality-with-Elimination we have already seen that Fozzie would win.Step 2. (Choose 2nd place.) First remove Fozzie from the preference schedule. Now we use the plurality-with-elimination method to find a winner. In this case, it is Gonzo.Step 3. (Choose 3rd place.) First remove Gonzo from the schedule.

17. Recursive Plurality-with-Elimination Method. Example: Step 1. (Choose 1st place.) Using the Plurality-with-Elimination we have already seen that Fozzie would win.Step 2. (Choose 2nd place.) First remove Fozzie from the preference schedule. Now we use the plurality-with-elimination method to find a winner. In this case, it is Gonzo.Step 3. (Choose 3rd place.) First remove Gonzo from the schedule. Now Kermit has a majority of the first-place votes in this schedule so he wins third place.

18. Recursive Plurality-with-Elimination Method. Example: Under this recursive method we find:Office Place Candidate President 1st Fozzie Vice-Pres. 2nd GonzoTreasurer 3rd Kermit - None - 4th Piggy

19. A Final Note: Arrow’s Impossibility Theorem • All of the voting methods we have seen so far have violated some form of fairness. • The natural question to ask is: “Is there a counting method that can be guaranteed to be both democratic and fair?” • Unfortunately, under rigorous definitions of “democratic and fair,” such social choices were shown by economist Kenneth Arrow to be impossible.