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

MA 110: Finite MathDr. 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 ScenarioWhere to go for dinner? McDonalds, Mellow

Mushroom or The Pillars?

A Typical Voting ScenarioWhere to go for dinner? McDonalds, Mellow

Mushroom or The Pillars?McDonalds Mellow Mushroom The Pillars

Peter X

Claire X

Hiro X

Suresh X

Silar X

Niki X

Jessica X

A Typical Voting ScenarioWhere to go for dinner?

McDonalds or Mellow Mushroom?

McDonalds Mellow Mushroom

Peter X

Claire X

Hiro X

Suresh X

Silar X

By majority rule, they should go to Mellow Mushroom.

Another Voting ScenarioWhere to go for dinner?

McDonalds, Mellow Mushroom or The Pillars?McDonalds Mellow Mushroom The Pillars

Peter X

Claire X

Hiro X

Suresh X

Silar X

Niki X

Jessica X

Another Voting ScenarioWhere to go for dinner?

McDonalds, Mellow Mushroom or The Pillars?McDonalds Mellow Mushroom The Pillars

Peter X

Claire X

Hiro X

Suresh X

Silar X

Niki X

Jessica X

The Pillars may win. However, less than half want to go there.

Same Voting Scenario… More InfoWhere to go for dinner? McDonalds, MM or The Pillars?

McDonalds MM The Pillars Comments

Peter X I can’t afford Pillars!

I would rather go to McDonalds!

Claire X I can’t afford Pillars!

I would rather go to McDonalds!

Hiro X

Suresh X

Silar X

Niki X I have lots of $$ today!

Jessica X I have lots of $$ today!

Same Voting Scenario… More InfoWhere to go for dinner? McDonalds, MM or The Pillars?

McDonalds MM The Pillars Comments

Peter X I can’t afford Pillars!

I would rather go to McDonalds!

Claire X I can’t afford Pillars!

I would rather go to McDonalds!

Hiro X

Suresh X

Silar X

Niki X I have lots of $$ today!

Jessica X I have lots of $$ today!

Peter and Claire would like a head-to-head match between McDonalds and the Pillars.

(out of many)

1. Plurality Method2. 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 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.

Comparing Plurality and Pairwise Comparison

Fairness Rules Plurality Pairwise Comparison

Majority Wins Rule X X

Head-to-head Winning Rule X

Keep Winning Rule X X

Irrelevant Loser Rule

Comparing Plurality and Pairwise Comparison

Fairness Rules Plurality Pairwise Comparison

Majority Wins Rule X X

Head-to-head Winning Rule X

Keep Winning Rule X X

Irrelevant Loser Rule

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

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

Why not?

Examples