# Choice Under Uncertainty

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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?
Takers EV ½*150+½*-1 ½*300+½*-150 ½*25000+½*-10000 =150-75=\$75 = = \$7500 =75-0.5=\$74.50 What influences your decision to take the gamble? Expected value = EV =(probability of event 1)*(payoff of event 1)+ (probability of event 2)*(payoff of event2)

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 income M0 is your initial income!

Risk Averse Defining Characteristic
Prefers certain income over uncertain income

M U MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Averse Example: √0 =0 1-0=1 √1 =1 1.41-1=0.41 √2 =1.41 Peter with U=√M could be many different formulas, this is one representation What is happening to U? Increasing What is happening to MU? Decreasing Each dollar gives less satisfaction than the one before it. √9 =3 √16 =4

Risk Averse Defining Characteristic
Prefers certain income over uncertain income Decreasing MU In other words, U increases at a decreasing rate

Risk Averse Example: What is Peter’s U at M=9? 3
MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Averse Example: √0 =0 1-0=1 √1 =1 1.41-1=0.41 √2 =1.41 How would you describe Peter’s feelings about winning vs. losing? He hates losing more than he loves winning. √9 =3 What is Peter’s U at M=9? 3 By how much does Peter’s utility increase if M increases by 7? 4-3=1 By how much does Peter’s utility decrease if M decreases by 7? 3-1.41=1.59 √16 =4

Risk Seeker Defining Characteristic
Prefers uncertain income over certain income

M U MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Seeker Example: 02 =0 1-0=1 12 =1 4-1=3 22 =4 Spidey with U=M2 could be many different formulas, this is one representation What is happening to U? Increasing What is happening to MU? Each dollar gives more satisfaction than the one before it. 92 =81 162 =256

Risk Seeker Defining Characteristic
Prefers certain income over uncertain income Increasing MU In other words, U increases at an increasing rate

Risk Seeker Example: What is Spidey’s U at M=9? 81 256-81= 175
MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Seeker Example: 02 =0 1-0=1 12 =1 4-1=3 22 =4 How would you describe Spidey’s feelings about winning vs. losing? He loves winning more than he hates losing. 92 =81 What is Spidey’s U at M=9? 81 256-81= 175 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? 81-4=77 162 =256

Risk Neutral Defining Characteristic
Indifferent between uncertain income and certain income

M U MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Neutral Example: =0 1-0=1 1 =1 2-1=1 2 =2 Jane with U=M could be many different formulas, this is one representation What is happening to U? Increasing What is happening to MU? Constant Each dollar gives the same additional satisfaction as the one before it. 9 =9 16 =16

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: What is Jane’s U at M=9? 9
MU 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Risk Neutral Example: =0 1-0=1 1 =1 2-1=1 2 =2 How would you describe Jane’s feelings about winning vs. losing? She loves winning as much as she hates losing. 9 =9 What is Jane’s U at M=9? 9 By how much does Jane’s utility increase if M increases by 7? 16-9= 7 By how much does Jane’s utility decrease if M decreases by 7? 9-2=7 16 =16

Summary Risk Averse Risk Seeker Risk Neutral MU Shape of U Fair Gamble
decreasing increasing constant

Shape of U Below = concave Above = convex On = linear
Chord – line connecting two points on U

Summary Risk Averse Risk Seeker Risk Neutral MU Shape of U Fair Gamble
decreasing increasing constant concave convex linear (.5)162+ (.5)22 =130 (.5)√16+ (.5)√2 =2.7 (.5)16+ (.5)2 =9 <3, NO >81, Yes =9, indifferent EUgamble Uno gamble M0=\$9 Coin toss to win or lose \$7

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 1/4 ½* ½ = ¼=.25 1/4 1/4 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? The probability of 2 independent events is the product of the probabilities of each event. 1/2 T H 1/2 T H H 1/2 1/2 T 1/2 1/2

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)= = 1.5 Fair? No, more than fair! Would a risk seeker take this gamble? Yes! Yes! Would a risk neutral take this gamble? 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 = 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
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
√ =1000 10000

Mia’s utility If M=250000, U= √250000=500 If M=640000, U= √640000=800
√810000=900 If M= , U= √ =1100 10000

Mia’s utility U=√M Utility if income is certain! Risk averse? Yes

Mia’s utility U=√M U if not injured? √1000000=1000
Unot injured U if not injured? √ =1000 Label her income and utility if she is not injured. Label her income and utility if she is injured. √10000=100 Uinjured 10000 Minjured M not injured

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 Probability of NO injury = 90 percent = 9/10=0.9 E(U) = 9/10*√( )+1/10* √(10000)= 9/10*1000+1/10*100= = 910

What is Mia’s expected Income?
No injury: M = \$1,000,000 Injury: M = \$10,000 Probability of injury = 10% = 1/10=0.1 Probability of NO injury = 90% = 9/10=0.9 E(M) = 9/10*( )+1/10* (10000)= = 901,000

Mia’s utility U=√M Label her E(M) and E(U). E(U) Is her E(U) certain?
Unot injured Label her E(M) and E(U). Is her E(U) certain? No, therefore, not on U=√M line E(U)=910 E(U) Uinjured 10000 E(M)=901000 Minjured Mnot injured

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 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 Square both sides Solve for p Interpret: If the premium is less than \$171,000, Mia will purchase insurance

Mia’s utility U=√M U = 910 Unot injured What certain income gives her the same U as the risky income? 1,000, ,900 \$828,100 E(U)=910 E(U) Uinjured 10000 E(M)=901000 828,100 Minjured Mnot injured

Leah Shooter also has a utility function of U=√M
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
√ =1000 10000

Leah’s utility If M=250000, U= √250000=500 If M=640000, U= √640000=800
√810000=900 If M= , U= √ =1100 10000

Leah’s utility U=√M U if not injured? √1000000=1000
Unot injured U if not injured? √ =1000 Label her income and utility if she is not injured. Label her income and utility if she is injured. √10000=100 Uinjured 10000 Minjured M not injured

What is Leah’s expected Utility?
No injury: M = \$1,000,000 Injury: M = \$10,000 Probability of injury = 50 % =0.5 Probability of NO injury = 0.5 E(U) = 1/2*√( )+1/2*√(10000)= 550

What is Leah’s expected income?
No injury: M = \$1,000,000 Injury: M = \$10,000 Probability of injury = 50% = 0.5 Probability of NO injury = 0.5 E(M) = 1/2*( )+1/2* (10000)= = 55,000

Leah’s utility U=√M Label her E(M) and E(U). E(U) E(U)=550
Unot injured Label her E(M) and E(U). E(U) E(U)=550 Uinjured 10000 E(M)=550,000 Minjured Mnot injured

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 p < 697,500

Leah’s utility U=√M E(U) U = 550
Unot injured What certain income gives her the same U as the risky income? 1,000, ,500= \$302,500 E(U) U = 550 E(U)=550 Uinjured 10000 E(M)=550,000 302,500 Minjured Mnot injured

Thea Thorough runs an insurance agency
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
Probability of non-risky player Expected claim from Mia = \$99,000 Expected claim from Leah= \$495,000 Thea’s expected value of claims = 0.5*99, *495,000 =\$297,000 Probability of risky player

Premium=\$297,000 Willingness 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

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 S P P S 10,000 D D 5,000 Q of Lemons Q of Peaches 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. 5,000 10,000 Peaches Lemons P S P S 10,000 D D 5,000 Q of Lemons Q of Peaches 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*\$ *\$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 Ratio shifts to fewer peaches and more lemons Expected value falls as beliefs about # of lemons increases More peaches drop out. Peaches Lemons S P P S 10,000 D D(50/50) D(50/50) 7,500 7,500 D 5,000 Q of Lemons Q of Peaches Q* (new) Q* (p.i.) Q* (p.i.) Q* (new)

Another Adverse Selection Example – Used Cars
Ultimately In the extreme case, no peaches, all lemons Peaches Lemons S P P S 10,000 D D(50/50) D(50/50) 7,500 7,500 D 5,000 Q of Lemons Q of Peaches Q* (new) Q* (p.i.) Q* (p.i.) 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? (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 GPA 1.0 2.0 3.0 4.0 What does the potential employer believe about the people who stay quiet? 3.0 They know their GPA is below a 3.0, but how far below? or better

III. Full disclosure/Unraveling
Those who don’t reveal: Original percent divided by what share of students remain 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 0.20/.50 0.30/.50 =0.40 =0.60 Intuitively, those who are above the expected average don’t want employers to think they are average, so they disclose! 0.4*1+0.6*2 =1.6 Therefore, those w/ a 2.0 should reveal…unravels so that there is full disclosure.

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 college 1800 students
Mean SAT score > 1300 (out of 1600) 1020 is the mean SAT score of those who take it