Choice Under Uncertainty Introduction to uncertainty

Law of large Numbers Expected Value Fair Gamble

Von-Neumann Morgenstern Utility Expected Utility Model Risk Averse Risk Lover Risk Neutral Applications

Gambles Insurance – paying to avoid uncertainty Adverse Selection

Full disclosure/Unraveling

Introduction to uncertainty

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

50% Does that mean that if I toss it up 2 times,

one will be heads and one will be tails?

Introduction to uncertainty

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.

Which of the following gambles will you take?

Gamble 1: H: $150T: -$1

Gamble 2: H: $300 T: -$150

Gamble 3: H: $25,000 T: -$10,000

Takers

EV

Expected value = EV =(probability of event 1)*(payoff of event 1)+ (probability of event 2)*(payoff of event2)

What influences your decision to take the gamble?

½*150+½*-1=75-0.5=$74.50

½*300+½*-150=150-75=$75

½*25000+½*-10000=12500-5000=$7500

Fair Gamble

a gamble whose expected value is 0 or, a gamble where the expected income from

gamble = expected income without the gamble

Ex: Heads you win $7, tails you lose $7 EV = 1/2*$7+1/2*(-$7) = $3.5+-$3.5 = $0

Von-Neumann Morgenstern Utility Expected Utility Model Utility and Marginal Utility Relates your income to your utility/satisfaction Utility – cardinal or numerical representation of

the amount of satisfaction - each indifference curve represented a different level of utility or satisfaction

Marginal Utility - additional satisfaction from one more unit of income

Von-Neumann Morgenstern Utility Expected Utility Model: Prediction we will take a gamble only if the expected utility

of the gamble exceeds the expected utility without the gamble.

• EU = Expected Utility = • (probability of event 1)*U(M0+payoff of event)

• +(probability of event 2)* U(M0+payoff of event 2)M is incomeM0 is your initial income!

Risk Averse

Defining Characteristic Prefers certain income over uncertain

income

Risk Averse Example: Peter with U=√M could be

many different formulas, this is one representation

M U MU

0

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9

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14

15

16

√0 =0√1 =1

1-0=1

√2 =1.411.41-1=0.41

√9 =3

√16=4

What is happening to U? Increasing What is happening to MU? Decreasing Each dollar gives less

satisfaction than the one before it.

Risk Averse

Defining Characteristic Prefers certain income over uncertain

income Decreasing MU• In other words, U increases at a

decreasing rate

Risk Averse Example:

M U MU

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

√0 =0√1 =1

1-0=1

√2 =1.411.41-1=0.41

√9 =3

√16=4

Peter

0.00.51.01.52.02.53.03.54.0

0 1 2 3 4 5 6 7 8 9 10111213141516

M

U M

What is Peter’s U at M=9? 3By how much does Peter’s utility increase if M increases by 7? 4-3=1By how much does Peter’s utility decrease if M decreases by 7? 3-1.41=1.59

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

He hates losing more than he loves winning.

Risk Seeker

Defining Characteristic Prefers uncertain income over certain

income

Risk Seeker Example: Spidey with U=M2 could be

many different formulas, this is one representation

M U MU

0

1

2

3

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5

6

7

8

9

10

11

12

13

14

15

16

02 =012 =1

1-0=1

22 =44-1=3

92 =81

162 =256

What is happening to U? Increasing What is happening to MU? Increasing Each dollar gives more

satisfaction than the one before it.

Risk Seeker

Defining Characteristic Prefers certain income over uncertain

income Increasing MU• In other words, U increases at an

increasing rate

Risk Seeker Example:

M U MU

0

1

2

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5

6

7

8

9

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12

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14

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16

02 =012 =1

1-0=1

22 =44-1=3

92 =81

162 =256

Spidey

0255075

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0 1 2 3 4 5 6 7 8 9 10111213141516

M

U M 2

What is Spidey’s U at M=9? 81By how much does Spidey’s utility increase if M increases by 7?

256-81= 175

By how much does Spidey’s utility decrease if M decreases by 7? 81-4=77

How would you describe Spidey’s feelings about winning vs. losing?He loves winning more than he hates losing.

Risk Neutral

Defining Characteristic Indifferent between uncertain income and

certain income

Risk Neutral Example: Jane with U=M could be

many different formulas, this is one representation

M U MU

0

1

2

3

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5

6

7

8

9

10

11

12

13

14

15

16

0 =01 =1

1-0=1

2 =22-1=1

9 =9

16 =16

What is happening to U? Increasing What is happening to MU? Constant Each dollar gives the same

additional satisfaction as the one before it.

Risk Neutral

Defining Characteristic Indifferent between uncertain income and

certain income Constant MU• In other words, U increases at a constant

rate

Risk Neutral Example:

M U MU

0

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6

7

8

9

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12

13

14

15

16

0 =01 =1

1-0=1

2 =22-1=1

9 =9

16 =16

Jane

0123456789

10111213141516

0 1 2 3 4 5 6 7 8 9 10111213141516M

U = M

What is Jane’s U at M=9? 9By how much does Jane’s utility increase if M increases by 7? 16-9= 7By how much does Jane’s utility decrease if M decreases by 7? 9-2=7

How would you describe Jane’s feelings about winning vs. losing?She loves winning as much as she hates losing.

Summary

Risk Averse

Risk Seeker

Risk Neutral

MU

Shape of U

Fair Gamble

decreasing increasing constant

Shape of U

Spidey

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M

U M 2

Jane

0123456789

10111213141516

0 1 2 3 4 5 6 7 8 9 10111213141516M

U = M

Chord – line connecting two points on U

Peter

0.00.51.01.52.02.53.03.54.0

0 1 2 3 4 5 6 7 8 9 10111213141516

M

U M

Below = concave Above = convex On = linear

Summary

Risk Averse

Risk Seeker

Risk Neutral

MU

Shape of U

Fair Gamble

decreasing increasing constant

concave convex linear

M0=$9Coin toss to win or lose $7

(.5)√16+ (.5)√2=2.7 <3, NO

(.5)162+ (.5)22

=130>81, Yes(.5)16+ (.5)2=9 =9, indifferent

EUgamble Uno gamble

Intuition check…

Why won’t Peter take a gamble that, on average, his income is no different than without the gamble?

Dislikes losing more than likes winning. The loss in utility from the possibility of losing is greater than the increase in utility from the possibility of winning.

Gambles

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

H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 What is the expected value of the gamble? First, what is the probability of each event?

H 1/2 T1/2

H 1/2 T1/2 H1/2

T1/2

The probability of 2 independent events is the product of the probabilities of each event.

½* ½ = ¼=.25 1/4 1/4 1/4

Problem 1:

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

H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 What is the expected value of the gamble? ¼ *(20)+ ¼ *(9) + ¼ *(-7)+ ¼*(-16)= 5+2.25-1.75-4= 1.5 Fair? No, more than fair!

Would a risk seeker take this gamble? Yes!Would a risk neutral take this gamble? Yes!Would a risk averse take this gamble?

Gambles Suppose a fair coin is flipped twice and the following payoffs

are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 If your initial income is $16 and your VNM utility

function is U= √M , will you take the gamble? What is your utility without the gamble? Uno gamble = √M• = √16• = 4

Gambles Suppose a fair coin is flipped twice and the following payoffs

are assigned to each of the 4 possible outcomes: H-H: win 20; H-T: win 9; T-H: lose 7; T-T: lose 16 If your initial income is $16 and your VNM utility

function is U= √M , will you take the gamble? What is your EXPECTED utility with the gamble?

• EU = ¼*√(16+20)+ ¼*√(16+9)+ ¼*√(16-7)+¼*√(16-16)• EU = ¼*√(36)+ ¼*√(25)+ ¼*√(9)+¼*√(0)• EU = ¼*6+ ¼*5+ ¼*3+¼*0• EU = 1.5+1.25+0.75+0• EU = 3.5

Von-Neumann Morgenstern Utility Expected Utility Prediction - we will take a gamble only if the

expected utility of the gamble exceeds the expected utility without the gamble.

Uno gamble=4 EUgamble = 3.5 What do you do? Uno gamble>EUgamble Therefore, don’t take the gamble!

What is insurance?

Pay a premium in order to avoid risk and Smooth consumption over all possible

outcomes Magahee

Example: Mia Dribble has a utility function of U=√M. 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.

Mia’s utility

If M=0, U= √0=0 If M=10000, U= √10000=100 If M=1000000, U= √1000000=1000

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Mia’s utility

If M=250000, U= √250000=500 If M=640000, U= √640000=800 If M=810000, U= √810000=900 If M=1210000, U= √1210000=1100

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Mia’s utility

Utility if income is certain!

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U=√M

Risk averse? Yes

Mia’s utility U if not injured? √1000000=1000 Label her income

and utility if she is not injured.

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U=√M

Label her income and utility if she is injured.

√10000=100

M not injured

Unot injured

1000

0

Minjured

Uinjured

What is Mia’s expected Utility? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 10 percent = 1/10=0.1

E(U) = 9/10*√(1000000)+1/10* √(10000)= 9/10*1000+1/10*100= 900+10 = 910

Probability of NO injury = 90 percent = 9/10=0.9

What is Mia’s expected Income? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 10% = 1/10=0.1

E(M) = 9/10*(1000000)+1/10* (10000)= 900000+1000 = 901,000

Probability of NO injury = 90% = 9/10=0.9

Mia’s utility Label her E(M) and

E(U). Is her E(U) certain? No, therefore, not

on U=√M line

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M, E(M)

U, E

(U)

U=√M

Mnot injured

Unot injured

1000

0

Minjured

Uinjured

E(U)

E(M

)=90

1000

E(U)=910

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. 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?

What is the E(U) without insurance? 910

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. 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?

What is the U with insurance? U = √(1,000,000-p)

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. Buy insurance if… U=√(1,000,000-p) > 910 = E(U) Solve

910000,000,1 p Square both sides

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. Buy insurance if… U=√(1,000,000-p) > 910 = E(U) Solve

22910000,000,1 p Square both sides

100,828000,000,1 p Solve for p

p 100,828000,000,1p900,171$ Interpret: If the premium is

less than $171,000, Mia will purchase insurance

Mia’s utility What certain income

gives her the same U as the risky income?

1,000,000-171,900 $828,100

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U, E

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U=√M

Mnot injured

Unot injured

1000

0

Minjured

Uinjured

E(U)

E(M

)=90

1000

E(U)=910

828,

100

U = 910

Leah Shooter also has a utility function of U=√M . 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.

Leah’s utility

If M=0, U= √0=0 If M=10000, U= √10000=100 If M=1000000, U= √1000000=1000

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Leah’s utility

If M=250000, U= √250000=500 If M=640000, U= √640000=800 If M=810000, U= √810000=900 If M=1210000, U= √1210000=1100

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Leah’s utility U if not injured? √1000000=1000 Label her income

and utility if she is not injured.

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U=√M

Label her income and utility if she is injured.

√10000=100

M not injured

Unot injured

1000

0

Minjured

Uinjured

What is Leah’s expected Utility? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 50 % =0.5

E(U) = 1/2*√(1000000)+1/2*√(10000)= 550

Probability of NO injury = 0.5

What is Leah’s expected income? No injury: M = $1,000,000 Injury: M = $10,000 Probability of injury = 50% = 0.5

E(M) = 1/2*(1000000)+1/2* (10000)= 500000+5000 = 55,000

Probability of NO injury = 0.5

Leah’s utility Label her

E(M) and E(U).

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U=√M

Mnot injured

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1000

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Uinjured

E(U)

E(M

)=55

0,00

0

E(U)=550

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. What is the largest price Leah would

pay for the above insurance policy? Intuition check: Will Leah be willing to pay

more or less?

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. What is the largest price Leah would pay

for the above insurance policy? What is the E(U) without insurance? 550 What is the U with insurance? U = √(1,000,000-p) Buy insurance if… U=√(1,000,000-p) > 550 = E(U)

Remember prediction: will take a gamble only if the expected utility of the gamble exceeds the

utility without the gamble. Buy insurance if… U=√(1,000,000-p) > 550 = E(U) Solve

550000,000,1 p

p < 697,500

Leah’s utility

What certain income gives her the same U as the risky income?

1,000,000-697,500=

$302,5000

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302,

500

U = 550

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)?

Thea’s expected value of claims/payouts What does Thea have to pay if the basketball player

gets injured? Difference in incomes w/ and w/o injury 1,000,000-10,000 = 990,000 Expected claim from Mia = 0.1*990000= $99,000 Expected claim from Leah= 0.5*990000= $495,000

Thea’s expected value of claims/payouts Expected claim from Mia = $99,000 Expected claim from Leah= $495,000 Thea’s expected value of claims = 0.5*99,000 + 0.5*495,000 =$297,000

Probability of risky player

Probability of non-risky player

Premium=$297,000Willingness to pay:

Mia: $171,900, Leah: $697,500 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?

Only Leah will buy insurance. Mia will not because she is only willing to pay $171,900

Adverse Selection - undesirable members of a group are more likely to participate in a voluntary exchange

What do you expect to happen in this market? Only the risky players will buy insurance. Premiums will increase The low-risk players will not be able to buy

insurance.

What is the source of the problem?

Asymmetric information – cannot tell how risky Is all information asymmetric? No, sex, age, health all observable (and cannot

fake) Therefore, insurance companies can charge

higher risk people higher rates Illegal to use certain characteristics, like race

and religion

How do insurance companies mitigate this problem? Offer different packages: 1. Deductibles – the amount of medical

expenditures the person has to pay before the plan starts paying benefit

risky people reveal themselves by choosing low deductibles

2. Do not cover preexisting condition

Other examples of adverse selection

Another Adverse Selection Example Used Cars Why does your new car drop in value the

minute you drive it off the lot?

Another Adverse Selection Example – used Cars 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?Peaches Lemons

P P

Q of Peaches Q of Lemons

D10,000

5,000 D

S

S

Q* (perfect info) Q*

(perfect info)

Another Adverse Selection Example – Used Cars Assume there is perfect information Buyers are willing to pay ___________ for

a lemon and ___________ for a peach.

Peaches LemonsP P

Q of Peaches Q of Lemons

D10,000

5,000 D

SS

5,000

10,000

Q* (perfect info) Q*

(perfect info)

Another Adverse Selection Example – Used Cars 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?

0.50*$10000+0.50*$5000 =$7500 If they are risk neutral, how much are they willing

to pay for the car? $7500, indifferent between certain and uncertain

income

Another Adverse Selection Example – Used Cars Case 2: Will the ratio of peaches to lemons stay at 50/50? If

not, what will happen to the expected value? Demand for peaches falls, demand for lemons rises

Peaches LemonsP P

Q of Peaches Q of Lemons

D10,000

5,000 D

SS

7,500 D(50/50) 7,500 D(50/50)

Q* (p.i.) Q*

(p.i.)

Ratio shifts to fewer peaches and more lemons Expected value falls as beliefs about # of lemons increases More peaches drop out.

Q* (new) Q*

(new)

Another Adverse Selection Example – Used Cars Ultimately In the extreme case, no peaches, all lemons

Peaches LemonsP P

Q of Peaches Q of Lemons

D10,000

5,000 D

SS

7,500 D(50/50) 7,500 D(50/50)

Q* (p.i.) Q*

(p.i.)Q* (new) Q*

(new)

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

Pay for a mechanic to inspect it. Offer a warranty on the car. Generally, offer something that is costly to

fake.

Role for the Government?

Does the asymmetric info mean the gov’t can/should be involved?

http://www.oag.state.ny.us/consumer/cars/qa.html

(look up the Lemon Law for MI)

Other examples of signaling

Brand names company advertising Dividends versus Capital gains Football players How can you signal how good of an

employee you will be?

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:

0.2*1+0.3*2+0.3*3+0.2*4 =2.5 The job counselor at MSU

advises anyone who had a B average to volunteer their GPA. Is this a stable outcome?

Per-cent

0.2 0.3 0.3 0.2

GPA 1.0 2.0 3.0 4.0

3.0

or better

What does the potential employer believe about the people who stay quiet? They know their GPA is below a 3.0, but how far below?

III. Full disclosure/Unraveling

Employers know their GPA is below a 3.0, but how far below?

Expected/Average grade for those who don’t reveal:

Percent

GPA 0.1 0.2

0.4*1+0.6*2 =1.6 Therefore, those w/ a 2.0

should reveal…unravels so that there is full disclosure.

Those who don’t reveal:Original percent divided by what share of students remain

0.20/.50=0.40

0.30/.50=0.60

Intuitively, those who are above the expected average don’t want employers to think they are average, so they disclose!

Intuition check

What does this full disclosure principle say about whether only peaches will provide a signal of their value?

Voluntary disclosure and SAT scores Institutional Details Voluntary disclosure question Data Results

Institutional Details

Increasing # of schools are adopting policies where submitting your SAT scores are optional I.e., students can submit high school G.P.A.,

extracurricular activities etc, and exclude standardized test score on their application

School will judge based on submitted material

Voluntary disclosure question

If it is fairly costless to reveal your scores, all by the students with the lowest scores should reveal to avoid being considered the “average” of those who don’t reveal.

Is it only the students with very low SAT scores that don’t reveal?

Data

Liberal arts college1800 studentsMean SAT score > 1300 (out of 1600) 1020 is the mean SAT score of those who

take it