choice under uncertainty
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
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
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
4
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
3
4
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
Spidey
0255075
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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
4
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
1
2
3
4
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
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
0255075
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0 1 2 3 4 5 6 7 8 9 10111213141516
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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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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M, E(M)
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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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M, E(M)
U, E
(U)
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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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
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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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, E
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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)
U=√M
Mnot injured
Unot injured
1000
0
Minjured
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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Unot injured
1000
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Uinjured
E(U)
E(M
)=55
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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