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 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.

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

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.

“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

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.

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

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

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

A Typical Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars? McDonaldsMellow MushroomThe Pillars PeterX ClaireX HiroX SureshX SilarX NikiX JessicaX

A Typical Voting Scenario Where to go for dinner? McDonalds or Mellow Mushroom? McDonaldsMellow Mushroom PeterX ClaireX HiroX SureshX SilarX By majority rule, they should go to Mellow Mushroom.

Another Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars? McDonaldsMellow MushroomThe Pillars PeterX ClaireX HiroX SureshX SilarX NikiX JessicaX

Another Voting Scenario Where to go for dinner? McDonalds, Mellow Mushroom or The Pillars? McDonaldsMellow MushroomThe Pillars PeterX ClaireX HiroX SureshX SilarX NikiX JessicaX The Pillars may win. However, less than half want to go there.

Same Voting Scenario… More Info Where to go for dinner? McDonalds, MM or The Pillars? McDonaldsMMThe PillarsComments PeterXI can’t afford Pillars! I would rather go to McDonalds! ClaireXI can’t afford Pillars! I would rather go to McDonalds! HiroX SureshX SilarX NikiXI have lots of $$ today! JessicaXI have lots of $$ today!

Same Voting Scenario… More Info Where to go for dinner? McDonalds, MM or The Pillars? McDonaldsMMThe PillarsComments PeterXI can’t afford Pillars! I would rather go to McDonalds! ClaireXI can’t afford Pillars! I would rather go to McDonalds! HiroX SureshX SilarX NikiXI have lots of $$ today! JessicaXI have lots of $$ today! Peter and Claire would like a head-to-head match between McDonalds and the Pillars.

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

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.

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.

Countries using Plurality Voting United States! Canada India United Kingdom

Pairwise Comparison Method Also 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.

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

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

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

Why not?

Examples