Transcript Choice Under Uncertainty

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: $150
T: -$1
Gamble 2:
H: $300
T: -$150
Gamble 3:
H: $25,000
T: -$10,000
Takers
EV
½*150+½*-1
½*300+½*-150
=75-0.5=$74.50 =150-75=$75
½*25000+½*-10000
=12500-5000=
$7500
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

Risk Averse
Example:
M
U
0
1
√0 =0
√1 =1
2
√2 =1.41
3






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.
4
5
6
7
8
9
√9 =3
10
11
12
13
14
15
16
√16=4
MU
1-0=1
1.41-1=0.41
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
0
1
√0 =0
√1 =1
2
√2 =1.41
MU
1-0=1
1.41-1=0.41
3
U M
Peter
4 about winning vs. losing?
How would you describe Peter’s feelings
He4.0hates losing more than he loves
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0 1 2 3 4 5 6 7 8 9 10111213 141516
5
winning.
6
7
8
9
√9 =3
10
11
12
What is Peter’s U atMM=9? 3
13
By how much does Peter’s utility increase
4-3=1
14 if M increases by 7?
By how much does Peter’s utility decrease
15 if M decreases by 7? 3-1.41=1.59
16
√16=4
Risk Seeker
Defining Characteristic
 Prefers uncertain income over certain
income

Risk Seeker
Example:
M
U
0
1
02 =0
12 =1
2
22 =4
3






Spidey with U=M2 could be
many different formulas, this
is one representation
What is happening to U?
Increasing
What is happening to MU?
Increasing
Each dollar gives more
satisfaction than the one
before it.
4
5
6
7
8
9
92 =81
10
11
12
13
14
15
16
162 =256
MU
1-0=1
4-1=3
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
0
1
02 =0
12 =1
2
22 =4
MU
1-0=1
4-1=3
3
How would you describe Spidey’s feelings
about winning vs. losing?
2
UM
4
Spideymore than he hates
He loves winning
losing.
275
250
225
200
175
150
125
100
75
50
25
0
5
6
7
8
9
92 =81
10
11
0 1 2 3 4 5 6 7 8 9 10111213141516
12
M M=9? 81
What is Spidey’s U at
256-81=
13
By how much does Spidey’s utility increase
if M increases by 7? 175
14
By how much does Spidey’s utility decrease
if M decreases by 7? 81-4=77
15
16
162 =256
Risk Neutral
Defining Characteristic
 Indifferent between uncertain income and
certain income

Risk Neutral
Example:
M
U
MU
0
1
0
1
=0
=1
2
2
=2
3






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.
4
5
6
7
8
9
9
=9
10
11
12
13
14
15
16
16 =16
1-0=1
2-1=1
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
0
1
=0
=1
2
2
=2
1-0=1
2-1=1
3
How would you describe Jane’s feelings
about winning vs. losing?
4
U =She
M
Jane as much as she hates losing.
loves winning
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
5
6
7
8
9
9
=9
10
0 1 2 3 4 5 6 7 8 9 10111213141516
11
M
12
What is Jane’s U at M=9?
9
13
By how much does Jane’s utility increase
14 if M increases by 7? 16-9= 7
9-2=7
By how much does Jane’s utility decrease
15 if M decreases by 7?
16
16 =16
Summary
MU
Shape of U
Fair
Gamble
Risk Averse Risk
Seeker
Risk
Neutral
decreasing increasing
constant
Shape of U
Below = concave
U M
Peter
UM
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0 1 2 3 4 5 6 7 8 9 10111213141516
Above = convex
On = linear
2
Spidey
U=M
Jane
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
275
250
225
200
175
150
125
100
75
50
25
0
0 1 2 3 4 5 6 7 8 9 10111213141516
0 1 2 3 4 5 6 7 8 9 10111213141516
M
M
M
Chord – line connecting two points on U
Summary
MU
Risk Averse Risk
Seeker
Risk
Neutral
decreasing increasing
constant
Shape of U concave
Fair
Gamble
convex
linear
2
2
(.5)√16+ (.5)√2 (.5)16 + (.5)2 (.5)16+ (.5)2
=2.7 <3, NO =130 >81, Yes =9 =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
½* ½ = ¼=.25 1/4
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?


H
H
1/2
1/2
1/2 T
1/2
H
1/2
TThe probability of 2 independent
events is the product of the
probabilities of each event.
1/2 T
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=
Would a risk seeker take this gamble? Yes!
1.5
Would a risk neutral take this gamble? Yes!
Would a risk averse take this gamble?
Fair?
No, more than fair!
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
1300
1200
1100
500
400
300
200
100
M, E(M)
1300000
1200000
1100000
1000000
900000
800000
700000
600000
500000
400000
0
300000

600
200000

700
100000

800
0

900
10000

If M=0, U=
√0=0
If M=10000, U=
√10000=100
If M=1000000, U=
√1000000=1000
U, E(U)

1000
Mia’s utility
1300
1200
1100
300
200
100
M, E(M)
1300000
1200000
1100000
1000000
900000
800000
700000
600000
0
500000

If M=1210000, U=
√1210000=1100
400
400000

500
300000

600
200000

700
100000

800
0

900
10000

If M=250000, U=
√250000=500
If M=640000, U=
√640000=800
If M=810000, U=
√810000=900
U, E(U)

1000
Mia’s utility
300
200
M, E(M)
1300000
1200000
1100000
1000000
900000
800000
700000
600000
500000
400000
100
0
300000

600
500
400
200000
Risk averse?
Yes
100000

900
800
700
0
Utility if income is
certain!
U=√M
1100
1000
U, E(U)

1300
1200
Mia’s utility
300
200
Uinjured 100
Minjured
M, E(M)
M not injured
1300000
1200000
1100000
1000000
900000
800000
700000
600000
500000
400000
300000
0
200000

Label her income
and utility if she is
injured.
√10000=100
100000

600
500
400
0

10000

U if not injured?
√1000000=1000
Label her income
and utility if she is
not injured.
U=√M
1100
1000
Unot injured
900
800
700
U, E(U)

1300
1200
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*√(1000000)+1/10* √(10000)=
9/10*1000+1/10*100=
900+10 = 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*(1000000)+1/10* (10000)=
900000+1000 = 901,000
Mia’s utility
U=√M
1100
1000
Unot injured
E(U)=910
900
800
700
U, E(U)
E(U)
600
500
400
Minjured
M, E(M)
Mnot injured
1300000
1200000
1100000
800000
700000
600000
500000
400000
300000
200000
100000
0
0
900000
Uinjured 100
E(M)=901000
1000000
300
200
10000
Label her
E(M) and
E(U).
 Is her E(U)
certain?
 No,
therefore,
not on
U=√M line

1300
1200
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 careerending 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 careerending 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
1,000,000  p  910
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
 1,000,000  p   910
2
2
Square both sides
1,000,000  p  828,100 Solve for p
1,000,000  828,100  p
Interpret: If the premium is
$171,900  p less than $171,000, Mia will
purchase insurance
Mia’s utility
U=√M
1100
1000
Unot injured
E(U)=910
900
800
700
U, E(U)
E(U)
600
500
400
Minjured
M, E(M)
Mnot injured
1300000
1200000
1100000
700000
600000
500000
400000
300000
200000
100000
0
0
800000
828,100
900000
Uinjured 100
E(M)=901000
1000000
300
200
10000
WhatUcertain
= 910
income
gives her
the same U
as the risky
income?
 1,000,000171,900
 $828,100

1300
1200

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
1300
1200
1100
500
400
300
200
100
M, E(M)
1300000
1200000
1100000
1000000
900000
800000
700000
600000
500000
400000
0
300000

600
200000

700
100000

800
0

900
10000

If M=0, U=
√0=0
If M=10000, U=
√10000=100
If M=1000000, U=
√1000000=1000
U, E(U)

1000
Leah’s utility
1300
1200
1100
300
200
100
M, E(M)
1300000
1200000
1100000
1000000
900000
800000
700000
600000
0
500000

If M=1210000, U=
√1210000=1100
400
400000

500
300000

600
200000

700
100000

800
0

900
10000

If M=250000, U=
√250000=500
If M=640000, U=
√640000=800
If M=810000, U=
√810000=900
U, E(U)

1000
Leah’s utility
300
200
Uinjured 100
Minjured
M, E(M)
M not injured
1300000
1200000
1100000
1000000
900000
800000
700000
600000
500000
400000
300000
0
200000

Label her income
and utility if she is
injured.
√10000=100
100000

600
500
400
0

10000

U if not injured?
√1000000=1000
Label her income
and utility if she is
not injured.
U=√M
1100
1000
Unot injured
900
800
700
U, E(U)

1300
1200
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*√(1000000)+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*(1000000)+1/2* (10000)=
500000+5000 = 55,000


Leah’s utility
E(U)
600
E(U)=550
500
400
Minjured
M, E(M)
Mnot injured
1300000
1200000
1100000
1000000
900000
800000
700000
400000
300000
200000
100000
0
0
500000
Uinjured 100
E(M)=550,000
600000
300
200
10000
Label her
E(M) and
E(U).
U=√M
1100
1000
Unot injured
900
800
700
U, E(U)

1300
1200
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
1,000,000  p  550
p < 697,500
U=√M
1100
1000
Unot injured
900
800
700
M, E(M)
Mnot injured
1300000
1200000
800000
700000
E(M)=550,000
600000
500000
300000
302,500
400000
Minjured
200000
100000
0

10000

1100000
E(U)
1000000
What certain
income gives
her the same
600
E(U)=550
U =risky
550
500
U as the
400
income?
300
200
1,000,000Uinjured 100
697,500=
0
$302,500
U, E(U)

1300
1200
900000
Leah’s utility

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*99,000 + 0.5*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

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
P
Lemons
S
P
S
10,000
D
D
5,000
Q* (perfect info)
Q of Peaches
Q* (perfect info)
Q of
Lemons
Another Adverse Selection
Example – Used Cars
Assume there is perfect information
5,000
 Buyers are willing to pay ___________
for
10,000
a lemon and ___________
for a peach.

Peaches
P
Lemons
S
10,000
P
S
D
D
5,000
Q* (perfect info)
Q of Peaches
Q* (perfect info)
Q of
Lemons
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
Ratio shifts to fewer peaches and more lemons
Expected value falls as beliefs about # of lemons increases
More peaches
drop out.
Peaches
S
P
10,000
Lemons
P
D
D(50/50) 7,500
7,500
5,000
Q* (new) Q* (p.i.)
Q of Peaches
S
D(50/50)
D
Q of
Q* (p.i.) Q* (new) Lemons
Another Adverse Selection
Example – Used Cars


Ultimately
In the extreme case, no peaches, all lemons
Peaches
P
10,000
S
Lemons
P
D
D(50/50) 7,500
7,500
5,000
Q* (new) Q* (p.i.)
Q of Peaches
S
D(50/50)
D
Q of
Q* (p.i.) Q* (new) Lemons
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 3.0
average to volunteer their GPA.
Is this a stable outcome?
or better
Percent
0.2 0.3 0.3 0.2
GPA
1.0 2.0 3.0 4.0
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?
Those who don’t reveal:
III. Full disclosure/Unraveling
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:
0.4*1+0.6*2
=1.6
Therefore, those w/ a 2.0
should reveal…unravels so
that there is full disclosure.
Percent
0.20/.50 0.30/.50
=0.60
=0.40
GPA
0.1
0.2
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 college
 1800
students
 Mean SAT score > 1300 (out of 1600)
 1020 is the mean SAT score of those who
take it