MA 110: Finite Math Dr. Maria Byrne Instructional Laboratory 0345 Lecture 10/31/2008

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

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 Method Also 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.

Comparing Plurality and Pairwise Comparison Fairness RulesPluralityPairwise Comparison Majority Wins RuleXX Head-to-head Winning RuleX Irrelevant Loser Rule

Comparing Plurality and Pairwise Comparison Methods Fairness RulesPluralityPairwise Comparison Majority Wins RuleXX Head-to-head Winning RuleX 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

Comparing Plurality and Pairwise Comparison Fairness RulesPluralityPairwise Comparison Majority Wins RuleXX Head-to-head Winning RuleX Irrelevant Loser Rule

Failing Irrelevant Losers Rule with Plurality Voting What kind of pie to have? 9 votes: AppleBlueberryCherry 342

Failing Irrelevant Losers Rule with Plurality Voting What kind of pie to have? AppleBlueberryCherry 342 With cherry pie, Blueberry pie wins!

Failing Irrelevant Losers Rule with Plurality Voting What kind of pie to have? AppleBlueberryCherry 342 With cherry pie, Blueberry pie wins! If there’s no cherry pie, 2 people revote:

Failing Irrelevant Losers Rule with Plurality Voting What kind of pie to have? AppleBlueberryCherry 342 With cherry pie, Blueberry pie wins! AppleBlueberryCherry 3+2=540 If there’s no cherry pie, 2 people revote:

Failing Irrelevant Losers Rule with Plurality Voting What kind of pie to have? AppleBlueberryCherry 342 With cherry pie, Blueberry pie wins! AppleBlueberryCherry 3+2=540 Without cherry pie, apple pie wins! If there’s no cherry pie, 2 people revote:

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 Rule with Plurality Voting What kind of pie to have? AppleBlueberryCherry 342 With cherry pie, Blueberry pie wins! AppleBlueberryCherry 3+2=540 Without cherry pie, apple pie wins! If there’s no cherry pie, 2 people revote:

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

What happened in 2000?

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