Transcript 1.1 Preference Ballots and Preference Schedules

§ 1.1 Preference Ballots and
Preference Schedules
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
Example:
• Suppose the ballots broke down as follows:
We could also represent this information with a table:
Number of
Ballot
voters
21
Ballot 15
1st Piggy
1st Gonzo
1st Choice
Piggy 2nd Kermit
Gonzo
2nd Kermit
3rd Gonzo
3rd Fozzie
4th Fozzie
2nd Choice
Kermit 4th Piggy
Kermit
3rd Choice
Gonzo
4th Choice
Fozzie
21
Fozzie
15
Piggy
Ballot12
1st Fozzie
Fozzie
2nd Gonzo
3rd Kermit
4th Piggy
Gonzo
7
Ballot
1st Kermit
2ndKermit
Fozzie
3rd Gonzo
4thFozzie
Piggy
Kermit
Gonzo
Piggy
Piggy
12
7
Example: (cont’d)
• 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.
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
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.
§ 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
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.
The Majority Criterion
 If a choice receives a majority of firstplace votes in an election, then that
choice should be the winner of the
election.
The Majority Criterion
 If a choice receives a majority of firstplace votes, but does not win then we
have a violation of the majority criterion.
 Does the plurality method satisfy the
Majority Criterion?
The Majority Criterion
 If a choice receives a majority of firstplace 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.)
The Plurality Method
 What’s wrong with the plurality
method?
Example:
Let’s look at the Muppet example again.
Number of
voters
21
15
12
7
1st Choice
Piggy
Gonzo
Fozzie
Kermit
2nd Choice
Kermit
Kermit
Gonzo
Fozzie
3rd Choice
Gonzo
Fozzie
Kermit
Gonzo
4th Choice
Fozzie
Piggy
Piggy
Piggy
The Condorcet Criterion
 If there is a choice that in a head-tohead 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-tohead comparison against each of the
other candidates is called the Condorcet
The Plurality Method
 What’s wrong with the plurality
method?
 It violates the Condorcet Criterion.
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
``Don`t blame me - I voted forcertain
Kodos.`` candidate.
- Homer Simpson, Treehouse of Horror VII
§ 1.3 The Borda Count
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
Example:
Let’s look at the Muppet example again.
Number of
voters
21
15
12
7
If we tally up the points we find:
1st Choice
Piggy
Gonzo
Fozzie
Kermit
Piggy
gets
21(4)
+
15(1)
+
12(1)
+7(1)
=
118
4 points
Kermit gets 21(3) + 15(3) + 12(2) + 7(4) = 160
2nd Choice
Kermit 21(2) + Kermit
Gonzo gets
15(4) + 12(3)Gonzo
+ 7(2) = 152 Fozzie
3 points
Fozzie gets 21(1) + 15(2) + 12(4) + 7(3) = 120
3rd Choice
2 points
Gonzo
Fozzie
Kermit
Gonzo
4th Choice
1 point
Fozzie
Piggy
Piggy
Piggy
The Borda Count Method
 What’s wrong with this method?
Example: The Springfield
Republican primary
Krusty the Clown, Sideshow Bob, Dracula and Mr. Burns
are running in the
primary to be Springfield’s congressional
Number
of voters 18 Suppose9 the vote breaks
6
representative.
down like
this:
1st Choice
4 points
Krusty
Bob
Dracula
2nd Choice
3 points
Bob
Dracula
Burns
3rd Choice
2 points
Dracula
Burns
Bob
4th Choice
1 point
Burns
Krusty
Krusty
The Borda Count Method
 What’s wrong with this method?
 It violates the Majority Criterion. . .
The Borda Count Method
 What’s wrong with this method?
 It violates the Majority Criterion. . .
 . . .and the Condorcet Criterion.