§ 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)

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.

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

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy 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.) OfficePlaceCandidateVotes Counting the first-place votes we get the following results: OfficePlaceCandidateVotes President1stPiggy21 Vice-Pres.2ndGonzo15 Treasurer3rdFozzie12 - None -4thKermit7

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy Now let us see what happens with the Extended Borda Count Method. OfficePlaceCandidatePoints Tallying the points we find: OfficePlaceCandidatePoints President1stKermit160 Vice-Pres.2ndGonzo152 Treasurer3rdFozzie120 - None -4thPiggy118

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy Now let us see what happens with the Extended Plurality-with-Elimination Method. OfficePlace CandidateEliminated In 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. OfficePlace CandidateEliminated In President1st Fozzie Vice-Pres.2nd Piggy3rd Round Treasurer3rd Gonzo2nd Round - None -4th Kermit1st Round 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.

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy Now showing: Extended Pairwise Comparison Method. OfficePlaceCandidatePoints After examining all of the possible head-to-head pairings of candidates and awarding points we get: OfficePlaceCandidatePoints President1stKermit3 Vice-Pres.2ndGonzo2 Treasurer3rdFozzie1 - None -4thPiggy0

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.

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy Recursive Plurality Method. Step 1. (Choose 1st place.) We have already seen that Piggy wins in a plurality system with 21 votes.

Example: Recursive Plurality Method. Number of voters st Choice KermitGonzoFozzieKermit 2nd Choice GonzoKermitGonzoFozzie 3rd Choice Fozzie KermitGonzo 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.

Example: Recursive Plurality Method. Number of voters st Choice GonzoFozzie 2nd Choice FozzieGonzo 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.

Example: Recursive Plurality Method. Number of voters st Choice GonzoFozzie 2nd Choice FozzieGonzo OfficePlaceCandidate Under this recursive method we have: OfficePlaceCandidate President1stPiggy Vice-Pres.2ndKermit Treasurer3rdGonzo - None -4thFozzie

Example: Number of voters st Choice PiggyGonzoFozzieKermit 2nd Choice Kermit GonzoFozzie 3rd Choice GonzoFozzieKermitGonzo 4th Choice FozziePiggy Recursive Plurality-with-Elimination Method. Step 1. (Choose 1st place.) Using the Plurality- with-Elimination we have already seen that Fozzie would win.

Example: Number of voters st Choice PiggyGonzoKermit 2nd Choice Kermit Gonzo 3rd Choice GonzoPiggy Recursive Plurality-with-Elimination Method. 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.

Example: Number of voters st Choice PiggyGonzoKermit 2nd Choice Kermit Gonzo 3rd Choice GonzoPiggy Recursive Plurality-with-Elimination Method. 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.

Example: Number of voters st Choice PiggyKermit 2nd Choice KermitPiggy Recursive Plurality-with-Elimination Method. 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.

Example: Number of voters st Choice PiggyKermit 2nd Choice KermitPiggy Recursive Plurality-with-Elimination Method. 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.

Example: Number of voters st Choice PiggyKermit 2nd Choice KermitPiggy Recursive Plurality-with-Elimination Method. OfficePlaceCandidate Under this recursive method we find: OfficePlaceCandidate President1stFozzie Vice-Pres.2ndGonzo Treasurer3rdKermit - None -4thPiggy

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.