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

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

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**MA 110: Finite MathDr. Maria ByrneInstructional Laboratory**0345 Lecture 10/31/2008**We’ll Study:Two Voting Systems**1. Plurality Method 2. Pairwise Comparison Method**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.**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.**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.**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.**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.**Plurality can fail**the head-to-head criterion: Example 14 from book Page 429**Failing Irrelevant Losers Rulewith Plurality Voting**What kind of pie to have? 9 votes:**Failing Irrelevant Losers Rulewith Plurality Voting**What kind of pie to have? With cherry pie, Blueberry pie wins!**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:**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:**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!**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."**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!**Voting Systems And Us**I. Who would have been president if other voting systems were used?**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**2000 Presidential Election**George W. Bush Al Gore**Did Nader spoil the election?**Impossible to say but…**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.**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)**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.**Plurality Voting**• Vote #1 • Vote #2 • Vote #3**Pairwise Comparison Voting**• Vote #1