Section 9.1Great Expectations

Deciding How to Weigh the Unknown Future

Chance favors only the prepared mind.Louis Pasteur

Question of the Day

If your bicycle is worth $1000, does it make sense to buy theft insurance that costs $50 per year?

Expected Value

Expected value the average net gain or loss that we would expect per game if we played the game many times.

Expected Value

Computing Expected Value:

To compute the expected value, we multiply the value of each outcome with its probability of occurring and then add up all those products.

Expected Value

A game is called a fair game if the expected value equals zero.

Paradox

A paradox presents a situation that has two possible interpretations or resolutions.

Each view appears irrefutable, and yet the views are diametrically opposed to each other.

Newcomb’s Paradox

Section 9.2Risk

Deciding Personal and Public Policy

The moral:Beware of unintended consequences.

Question of the Day

An HIV test is 95% accurate for infected people. Suppose your roommate’s test result is positive. What are the chances your roommate has HIV?

Goal

When facing issues, we want to take steps to help us make informed decisions.

Risk

How do we measure risk?

Consider Unintended Consequences

Section 9.3Money Matters

Deciding Between Faring Well and Welfare

Lack of money is the root of all evil.Mark Twain

Question of the Day

Adam and Eve invest one penny in a bankaccount paying 3% compounded annually.How much money will the account hold after1000 years: $10,000? $100,000? $1 million?$1 billion?

A Compounding Pattern

Section 9.4Peril at the Polls

Deciding Who Actually Wins an Election

… Democracy is the worst form of government except all those others

that have been tried from time to time.Winston Churchill

Question of the Day

How do you pick the winner of a democraticelection?

An Election Conundrum

Simple Voting Methods

Plurality VotingEach voter votes for one person, and the

candidate with the most votes wins.

Simple Voting Methods

Vote-for-TwoEach voter must vote for two different

candidates and the candidate with the most votes wins.

Simple Voting Methods

Borda CountEach voter ranks all the candidates: 1, 2, 3,

and so on. The highest ranking is 1. The rankings are then tallied for each candidate, and the candidate with the lowest total wins.

Condorcet’s Paradox

The cumulative ranking of the group as a whole may not be transitive – that is, the ranking may have a circle of preferences.

Arrow’s Election Disaster Theorem

Section 9.5Cutting Cake for Greedy PeopleDeciding How to Slice Up Scarce Resources

Choose a convenient representation of an issue.

Question of the Day

Can you always cut a cake so that everyonegets his or her favorite piece?

Cake-Cutting Question

Given a cake and three people, is there a method of cutting the cake equitably?

Greedy Division Question

Given a cake and three people, is there a method for cutting cake into three pieces so that each person gets the piece that he or she believes has the greatest value?

In other words, can the cake be divided into three pieces so that, of the resulting slices, everyone gets their favorite piece?

Greedy Division Theorem

Suppose three preference diagrams are superimposed. Then there will be a point where the three people have indicated that they all prefer different pieces.

Four or More People

What happens if we want to divide a cake among four people?