1.1 Preference Ballots and Preference Schedules

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1.1 Preference Ballots and Preference Schedules
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Example

• Starting in the early 90s, the Henson
  production company started to pay the Muppets
  with stock options rather than a straight salary.
  Quietly, the Muppets, as a group, gained a
  controlling interest in Henson productions. In a
  move that shocked the world, the Muppets decided
  to elect one of their own as the CEO of the
  company.

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Example

• Suppose the ballots broke down as follows

We could also represent this information with a
table
Ballot1st Piggy2nd Kermit3rd Gonzo4th Fozzie
Ballot1st Gonzo2nd Kermit3rd Fozzie4th Piggy
Ballot1st Fozzie2nd Gonzo3rd Kermit4th Piggy
Ballot1st Kermit2nd Fozzie3rd Gonzo4th Piggy
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15
12
7
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Example (contd)

• This kind of ballot, in which the voters rank
  candidates in order of preference is called a
  preference ballot.
• If ties are disallowed then we have a linear
  ballot.
• The table we used is an example of a preference
  schedule for the election.

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Transitivity and Candidate Elimination

• Voter preferences are transitive--that is if a
  voter prefers candidate A to candidate B and
  prefers B over candidate C then the voter prefers
  A to C.
• This means that if we want to see which
  candidate someone would vote for in a two person
  election all we need to check is which candidate
  is placed higher on the ballot.

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Transitivity and Candidate Elimination

• Now suppose a candidate drops out of the race.
  In such a case, the relative preferences of a
  voter are not affected.

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1.2 The Plurality Method

• Kent Senator Dole, why should people vote for
  you instead of President Clinton?Kang/Dole It
  makes no difference which one of us you vote for.
  Either way, your planet is doomed. DOOMED!Kent
  Well, a refreshingly frank response there from
  senator Bob Dole.
• - The Simpsons, Treehouse of Horror VII

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The Plurality Method

• The plurality method says that the candidate
  (or candidates) with the most first-place votes
  wins a given election.
• This method is an extension of the concept of
  majority rule, which states that in an election
  between two candidates one with the majority of
  votes wins.

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The Majority Criterion

• If a choice receives a majority of first-place
  votes in an election, then that choice should be
  the winner of the election.

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The Majority Criterion

• If a choice receives a majority of first-place
  votes, but does not win then we have a violation
  of the majority criterion.
• Does the plurality method satisfy the Majority
  Criterion?

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The Majority Criterion

• If a choice receives a majority of first-place
  votes, but does not win then we have a violation
  of the majority criterion.
• Does the plurality method satisfy the Majority
  Criterion?
• Yes! (Since a candidate with a majority of
  first-place votes would automatically have a
  plurality as well.)

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The Plurality Method

• Whats wrong with the plurality method?

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Example
Lets look at the Muppet example again.
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The Condorcet Criterion

• If there is a choice that in a head-to-head
  comparison is preferred by the voters over each
  of the other choices, then that choice should be
  the winner of the election.
• A candidate that wins in every head-to-head
  comparison against each of the other candidates
  is called the Condorcet candidate.

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The Plurality Method

• Whats wrong with the plurality method?
• It violates the Condorcet Criterion.

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Insincere Voting

• Insincere voting occurs when a voter changes
  his or her true preferences on the ballot in an
  effort to influence the election against a
  certain candidate.

Dont blame me - I voted for Kodos. - Homer
Simpson, Treehouse of Horror VII
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1.3 The Borda Count
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The Borda Count Method

• The Idea Assign points to each ranking on the
  ballot--the candidate with the highest total
  wins. This method produces the best compromise
  candidate.
• If we have an election with N candidates we will
  give 1 point for last place, 2 points for second
  to last, . . . , and N points for first place.
  The candidate with the highest total number of
  points is the winner.

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Example
Lets look at the Muppet example again.
If we tally up the points we find Piggy gets
21(4) 15(1) 12(1) 7(1) 118Kermit
gets 21(3) 15(3) 12(2) 7(4) 160Gonzo
gets 21(2) 15(4) 12(3) 7(2) 152Fozzie
gets 21(1) 15(2) 12(4) 7(3) 120
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The Borda Count Method

• Whats wrong with this method?

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Example The Springfield Republican primary
Krusty the Clown, Sideshow Bob, Dracula and Mr.
Burns are running in theprimary to be
Springfields congressional representative.
Suppose the vote breaks down like this
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The Borda Count Method

• Whats wrong with this method?
• It violates the Majority Criterion. . .

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The Borda Count Method

• Whats wrong with this method?
• It violates the Majority Criterion. . .
• . . .and the Condorcet Criterion.