MA 110: Finite Math Dr. Maria Byrne Instructional Laboratory 0345
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MA 110: Finite MathDr. Maria Byrne
Instructional Laboratory 0345
Lecture 10/31/2008
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1. Plurality Method
2. Pairwise Comparison Method
We’ll Study:Two Voting Systems
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Plurality Method
• Only considers voter’s top choice.
• The candidate with the most votes wins.
• A tie can be handled in any way the voting system agrees upon.
• If there is an absolute majority vote for one candidate, that candidate wins.
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Plurality Method
• Only considers voter’s top choice.
• The candidate with the most votes wins.
• A tie can be handled in any way the voting system agrees upon.
• If there is an absolute majority vote for one candidate, that candidate wins.
Simple and easy voting method.
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Pairwise Comparison MethodAlso known as condorcet method.
• Considers all ranked choices of each voter.
• If one candidate beats every other candidate in a head-to-head match, that candidate is the winner.
• If one candidate beats one candidate but is beaten by another candidate, a point-system is used to find the winner that will satisfy people the most.
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Plurality and Pairwise Comparison Methods
• Both satisfy the majority wins rule.
• Only pairwise comparison satisfies the head-to-head winning rule.
• Both fail the irrelevant loser rule.
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Comparing Plurality and Pairwise Comparison
Fairness Rules Plurality Pairwise Comparison
Majority Wins Rule X X
Head-to-head Winning Rule X
Irrelevant Loser Rule
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Comparing Plurality and Pairwise Comparison Methods
Fairness Rules Plurality Pairwise Comparison
Majority Wins Rule X X
Head-to-head Winning Rule X
Irrelevant Loser Rule
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The Head-to-Head Criterion
If a candidate wins when compared head to head (individually) with each
of the other candidates, then that candidate should be declared the
winner.
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Plurality can fail
the head-to-head criterion:
Example 14 from book
Page 429
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Comparing Plurality and Pairwise Comparison
Fairness Rules Plurality Pairwise Comparison
Majority Wins Rule X X
Head-to-head Winning Rule X
Irrelevant Loser Rule
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have? 9 votes:
Apple Blueberry Cherry
3 4 2
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have?
Apple Blueberry Cherry
3 4 2
With cherry pie, Blueberry pie wins!
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have?
Apple Blueberry Cherry
3 4 2
With cherry pie, Blueberry pie wins!
If there’s no cherry pie, 2 people revote:
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have?
Apple Blueberry Cherry
3 4 2
With cherry pie, Blueberry pie wins!
Apple Blueberry Cherry
3+2=5 4 0
If there’s no cherry pie, 2 people revote:
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have?
Apple Blueberry Cherry
3 4 2
With cherry pie, Blueberry pie wins!
Apple Blueberry Cherry
3+2=5 4 0
Without cherry pie, apple pie wins!
If there’s no cherry pie, 2 people revote:
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On the independence of irrelevant alternatives:
After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."
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Failing Irrelevant Losers Rulewith Plurality Voting
What kind of pie to have?
Apple Blueberry Cherry
3 4 2
With cherry pie, Blueberry pie wins!
Apple Blueberry Cherry
3+2=5 4 0
Without cherry pie, apple pie wins!
If there’s no cherry pie, 2 people revote:
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Voting Systems And Us
I. Who would have been president
if other voting systems were used?
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Often, details of the voting system don’t matter:
• 1984, 1988, 1992, 1996
• Different voting schemes (popular vote, the Electoral College, the pairwise comparison), all selected the same candidate.
Yoshie Furuhashi , 2004
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What happened in 2000?
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2000 Presidential Election
George W. Bush Al Gore
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Did Nader spoil the election?
Impossible to say but…
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Did Nader spoil the election?
Impossible to say but…
In the 2000 presidential election in Florida. Bush beat Gore by only 537 votes. Nader received 97,421 votes.
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A Voting Game
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A Voting Game
• Each student is an elf, a hobbit or a dwarf living in Middle Earth.
• Middle Earth is a democracy, but the inhabitants haven’t decided how to vote.
• Running for president:• Gandalf (elf)
• Gimli (dwarf)
• Bilbo Baggins (hobbit)
• Aragorn (man)
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Winning
• If Gandalf gets elected, elves and men win.
• If Gimli gets elected, dwarves and hobbits win.
• If Bilbo gets elected, elves and hobbits win.
• If Aragorn gets elected, man and dwarves win.
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Plurality Voting
• Vote #1
• Vote #2
• Vote #3
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Pairwise Comparison Voting
• Vote #1