MA 110: Finite Math Dr. Maria Byrne Instructional Laboratory 0345 - PowerPoint PPT Presentation

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MA 110: Finite Math Dr. Maria Byrne Instructional Laboratory 0345

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  1. MA 110: Finite MathDr. Maria ByrneInstructional Laboratory 0345 Lecture 10/31/2008

  2. We’ll Study:Two Voting Systems 1. Plurality Method 2. Pairwise Comparison Method

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. Comparing Plurality and Pairwise Comparison

  8. Comparing Plurality and Pairwise Comparison Methods

  9. 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.

  10. Plurality can fail the head-to-head criterion: Example 14 from book Page 429

  11. Comparing Plurality and Pairwise Comparison

  12. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? 9 votes:

  13. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? With cherry pie, Blueberry pie wins!

  14. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? With cherry pie, Blueberry pie wins! If there’s no cherry pie, 2 people revote:

  15. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? With cherry pie, Blueberry pie wins! If there’s no cherry pie, 2 people revote:

  16. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? With cherry pie, Blueberry pie wins! If there’s no cherry pie, 2 people revote: Without cherry pie, apple pie wins!

  17. 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."

  18. Failing Irrelevant Losers Rulewith Plurality Voting What kind of pie to have? With cherry pie, Blueberry pie wins! If there’s no cherry pie, 2 people revote: Without cherry pie, apple pie wins!

  19. Voting Systems And Us I. Who would have been president if other voting systems were used?

  20. 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

  21. What happened in 2000?

  22. 2000 Presidential Election George W. Bush Al Gore

  23. Did Nader spoil the election? Impossible to say but…

  24. 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.

  25. A Voting Game

  26. 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)

  27. 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.

  28. Plurality Voting • Vote #1 • Vote #2 • Vote #3

  29. Pairwise Comparison Voting • Vote #1