Choice Under Uncertainty

Choice Under Uncertainty

1. Introduction to uncertainty

1. Law of large Numbers
2. Expected Value
3. Fair Gamble

II. Von-Neumann Morgenstern Utility Expected Utility

1. Model
2. Risk Averse
3. Risk Lover
4. Risk Neutral
5. Applications
6. Gambles
7. Insurance – paying to avoid uncertainty
8. Adverse Selection

III. Full disclosure/Unraveling

I. Introduction to uncertainty

1. Law of large Numbers

What is the probability that if I toss a coin in the air that it will come up heads?

Does that mean that if I toss it up 2 times, one will be heads and one will be tails?

law of large numbers - a statistical law that says that if an event happens independently (one event is not related to the next) with probability p every time the event occurs, the proportion of cases in which the event occurs approaches p as the number of events increases

B. Expected Value

Which of the following gambles will you take?

Gamble 1: Heads you win $150, tails you lose $1

Gamble 2: Heads you win $300, tails you lose $150

Gamble 3: Heads you win $25,000, tails you lose $10,000

expected value of gamble=EV=

(probability of event 1)*(payoff of event 1)+(probability of event 2)*(payoff of event2)

C. Fair Gamble

• gamble whose expected value is 0
• expected income from gamble = expected income with out the gamble

Example:

II. Von-Neumann Morgenstern Utility Expected Utility

1. Model

1. Relates your income to your utility

Marginal Utility

2. Prediction - we will take a gamble only if…

Expected Utility:
B. Risk Averse

Defining characteristic

Example: Peter with could be many different formulas, this is one representation.

What is Peter's Utility at M = 9?

By how much does Peter's utility increase if M increases by 7?

By how much does Peter's utility decrease if M decreases by 7?

How would you describe Peter's feelings about winning vs. losing?

C. Risk Seeker

Defining characteristic

Example: Spidey with

What is Spidey's Utility at M = 9?

By how much does Spidey's utility increase if M increases by 7?

By how much does Spidey's utility decrease if M decreases by 7?

How would you describe Spidey's feelings about winning vs. losing?

D. Risk Neutral

Defining characteristic

Example: Jane with

What is Jane’s Utility at M = 9?

By how much does Jane’s utility increase if M increases by 7?

By how much does Jane’s utility decrease if M decreases by 7?

How would you describe Jane’s feelings about winning vs. losing?

Summary:

E. Applications

1. Gambles

Suppose a fair coin is flipped twice and the following payoffs are assigned to each of the 4 possible outcomes:

HH: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16

What is the expected value of the gamble?

If your initial income is $16 and your VNM utility function is , will you take the gamble?

2. Insurance:

Mia Dribble has a utility function of . In addition, Mia is a basketball star starting her senior year. If she makes it through her senior year without a serious injury, she will receive a $1,000,000 contract for playing in the new professional women’s basketball league (the $1,000,000 includes endorsements). If she injures herself, she will receive a $10,000 contract for selling concessions at the basketball arena. There is a 10 percent chance that Mia will injure herself badly enough to end her career.

What is Mia’s expected utility?

If Mia pays $p for an insurance policy that would give her $1,000,000 if she suffered a career-ending injury while in college, then she would be sure to have an income of $1,000,000-p, not matter what happened to her. What is the largest price Mia would pay for this insurance policy? (Hint: it is over $100,000).

Leah Shooter also has a utility function of . Lea is also starting college and she has the same options as Mia after college. However, Leah is notoriously clumsy and knows that there is a 50 percent chance that she will injure herself badly enough to end her career.

What is Leah’s expected utility?

What is the largest price Leah would pay for the above insurance policy?

Thea Thorough runs an insurance agency. Unfortunately, she is unable to distinguish between coordinated players and clumsy players, but she knows that half of all players are clumsy. If she insures both Lea and Mia, what is her expected value of claims/payouts (remember, she has to pay whenever either player gets injured)?

Suppose Thea is unable to distinguish among clutzy and non-clutzy basketball players and therefore has to change the same premium to everyone. If she sets her premium equal to the expected value of claims will both Lea and Mia buy insurance from Thea? Explain.

What do you expect to happen in this insurance market?

Adverse Selection

What is the source of the problem?

In reality, what do insurance companies do to avoid adverse selection?

Role for the government?

Other Examples of Adverse Selection

Used Cars

Why does your new car drop in value the minute you drive it off the lot?

First assume that there are two kinds of used cars - lemons and peaches. Lemons are worth $5,000 to consumers and peaches are worth $10,000. Assume also that demand is perfectly elastic and consumers are risk neutral. There is a demand for both kinds of cars and a supply of both kinds of cars.

Is the supply of lemons or peaches higher?

PeachesLemons

Assume there is perfect information

Buyers are willing to pay ______for a lemon and ______for a peach.

Now assume there is not perfect information. Sellers know the true condition of the car and buyers don’t.

Case 1: Assume that buyers think that there is a 50% chance that the car is a peach. What is their expected value of any car they see?

If they are risk neutral, how much are they willing to pay for the car?[1]

Case 2: Will the ratio of peaches to lemons stay at 50/50? If not, what will happen to the expected value?

Ultimately:

What could you do to signal to someone that your car is not a lemon?

Role for the Government?
III. Full disclosure/Unraveling

You’re on a job interview and the interviewer knows what the distribution of GPAs are for MSU graduates:

Expected/Average grade for everyone:

The job counselor at MSU advises anyone who had a B average to volunteer their GPA. Is this a stable outcome?

1

[1] If they are risk averse, are they willing to pay more or less than the risk neutral consumer? If they are risk seeking, are they willing to pay more or less than the risk neutral consumer?