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

MA 110: Finite Math Dr. Maria Byrne Instructional Laboratory 0345. Lecture 10/29/2008. "Democracy is the worst form of government” “Except for all those others that have been tried.“ ~ Winston Churchill. Democracy. “Of the people, by the people, and for the people.” – Abraham Lincoln

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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 10/29/2008

  2. "Democracy is the worst form of government” “Except for all those others that have been tried.“ ~ Winston Churchill

  3. Democracy • “Of the people, by the people, and for the people.” – Abraham Lincoln • Democracy is a form of government in which the supreme power is held completely by the people under a free electoral system. • Voting is the cornerstone of any electoral system.

  4. “Why are we learning about democracy in math class?” (or) “What’s mathematically the problem with democracy?”

  5. Arrow’s Impossibility Theorem It is mathematically impossible to create any system of voting involving 3 or more candidates that is “fair” in all possible scenarios.

  6. “Fair” means: If one candidate receives more than half the votes, that candidate should win. Majority Win Rule If a candidate would beat another candidate in a head-to-head match, it should not be possible for the first candidate to lose to the second candidate. Head-to-head Winning Rule If a candidate will win in one scenario with a certain distribution of votes, they should win in any other scenario in which they have the same votes and more. Keep Winning Rule If a losing candidate drops out of the race, it should not affect who the winner is. Irrelevant Loser Rule

  7. Majority verses Plurality We need to make a distinction between “majority” and “plurality”. Majority: A subset that has more than half of a group. (>50%) To avoid confusion, also known as an absolute majority. Plurality: A subset that has more than any other subset in a group. To create confusion, also known as a majority.

  8. 100 Doughnuts: • 40 doughnuts are glazed. • 30 doughnuts have sprinkles. • 30 doughnuts are old-fashioned. There is no majority. Glazed doughnuts represent the plurality.

  9. 100 Cats: • 58 cats are Siamese. • 12 cats are Persian. • 30 cats are Calico cats. Siamese cats are the plurality and the majority.

  10. A Typical Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars?

  11. A Typical Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars?

  12. A Typical Voting Scenario Where to go for dinner? McDonalds or Mellow Mushroom? By majority rule, they should go to Mellow Mushroom.

  13. Another Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars?

  14. Another Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars? The Pillars may win. However, less than half want to go there.

  15. Same Voting Scenario… More Info Where to go for dinner? McDonalds, MM or The Pillars?

  16. Same Voting Scenario… More Info Where to go for dinner? McDonalds, MM or The Pillars? Peter and Claire would like a head-to-head match between McDonalds and the Pillars.

  17. (out of many) We’ll Study:Two Voting Systems 1. Plurality Method 2. Pairwise Comparison Method

  18. Plurality and Pairwise Comparison Methods • Both satisfy the majority wins rule. • Both fail the irrelevant loser rule. • Only pairwise comparison satisfies the head-to-head winning rule.

  19. 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.

  20. 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.

  21. Countries using Plurality Voting United States! Canada India United Kingdom

  22. Pairwise Comparison MethodAlso known as condorcet methods. • Considers all ranked choices of each voter. • If one candidate beats every other candidate in a head-to-head natch, 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.

  23. Comparing Plurality and Pairwise Comparison

  24. Comparing Plurality and Pairwise Comparison

  25. Condorcet methods are not currently in use in government elections anywhere in the world.

  26. Condorcet methods are not currently in use in government elections anywhere in the world. Why not?

  27. Examples

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