1.6 Rankings

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1.6 Rankings

• Romes 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)

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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.

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Extended Ranking Methods

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

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Lets 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 resultsOffice Place Candidate Votes
President 1st Piggy 21Vice-Pres. 2nd Gonzo 15
Treasurer 3rd Fozzie 12- None - 4th Kermit 7
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Now let us see what happens with the Extended
Borda Count Method.
Example
Tallying the points we findOffice Place Candi
date PointsPresident 1st Kermit 160Vice-Pres.
2nd Gonzo 152Treasurer 3rd Fozzie 120- None
- 4th Piggy 118
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Now let us see what happens with the Extended
Plurality-with-Elimination Method.
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
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
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Now showing Extended Pairwise Comparison Method.
Example
After examining all of the possible head-to-head
pairings of candidates and awarding points we
getOffice Place Candidate PointsPresident 1s
t Kermit 3Vice-Pres. 2nd Gonzo 2Treasurer 3rd
Fozzie 1- None - 4th Piggy 0
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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.

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Recursive Plurality Method.
Example
Step 1. (Choose 1st place.) We have already seen
that Piggy wins in a plurality system with 21
votes.
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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.
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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.
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Recursive Plurality Method.
Example
Under this recursive method we haveOffice Plac
e Candidate President 1st Piggy Vice-Pres. 2n
d Kermit Treasurer 3rd Gonzo - None
- 4th Fozzie
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Recursive Plurality-with-Elimination Method.
Example
Step 1. (Choose 1st place.) Using the
Plurality-with-Elimination we have already seen
that Fozzie would win.
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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-eliminati
on method to find a winner.
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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-eliminati
on method to find a winner. In this case, it is
Gonzo.
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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-eliminati
on method to find a winner. In this case, it is
Gonzo.Step 3. (Choose 3rd place.) First remove
Gonzo from the schedule.
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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-eliminati
on 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.
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Recursive Plurality-with-Elimination Method.
Example
Under this recursive method we findOffice Plac
e Candidate President 1st Fozzie Vice-Pres. 2
nd GonzoTreasurer 3rd Kermit - None
- 4th Piggy
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A Final Note Arrows 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.