Choice Under Uncertainty

1
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

2
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?

3
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.

4
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
½150½-1
½300½-150
½25000½-10000
150-7575
12500-5000 7500
75-0.574.50
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)
5
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 1/271/2(-7)
• 3.5-3.5 0

6
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

7
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(M0payoff of event)
• (probability of event 2) U(M0payoff of
  event 2)

M is income M0 is your initial income!
8
Risk Averse

• Defining Characteristic
• Prefers certain income over uncertain income

9
Risk Averse Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
v0
0
1-01
v1
1
1.41-10.41
v2
1.41

• Peter with UvM 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.

v9
3
v16
4
10
Risk Averse

• Defining Characteristic
• Prefers certain income over uncertain income

• Decreasing MU
• In other words, U increases at a decreasing rate

11
Risk Averse Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
v0
0
1-01
v1
1
1.41-10.41
v2
1.41
How would you describe Peters feelings about
winning vs. losing?
He hates losing more than he loves winning.
v9
3
What is Peters U at M9?
3
By how much does Peters utility increase if M
increases by 7?
4-31
By how much does Peters utility decrease if M
decreases by 7?
3-1.411.59
v16
4
12
Risk Seeker

• Defining Characteristic
• Prefers uncertain income over certain income

13
Risk Seeker Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
02
0
1-01
12
1
4-13
22
4

• Spidey with UM2 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.

92
81
162
256
14
Risk Seeker

• Defining Characteristic
• Prefers certain income over uncertain income

• Increasing MU
• In other words, U increases at an increasing rate

15
Risk Seeker Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
02
0
1-01
12
1
4-13
22
4
How would you describe Spideys feelings about
winning vs. losing?
He loves winning more than he hates losing.
92
81
What is Spideys U at M9?
81
256-81 175
By how much does Spideys utility increase if M
increases by 7?
81-477
By how much does Spideys utility decrease if M
decreases by 7?
162
256
16
Risk Neutral

• Defining Characteristic
• Indifferent between uncertain income and certain
  income

17
Risk Neutral Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
0
1-01
1
1
2-11
2
2

• Jane with UM 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
18
Risk Neutral

• Defining Characteristic
• Indifferent between uncertain income and certain
  income

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

19
Risk Neutral Example
M U MU
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0
0
1-01
1
1
2-11
2
2
How would you describe Janes feelings about
winning vs. losing?
She loves winning as much as she hates losing.
9
9
What is Janes U at M9?
9
16-9 7
By how much does Janes utility increase if M
increases by 7?
9-27
By how much does Janes utility decrease if M
decreases by 7?
16
16
20
Summary
Risk Averse Risk Seeker Risk Neutral
MU
Shape of U
Fair Gamble
increasing
constant
decreasing
21
Shape of U
Below concave
Above convex
On linear
Chord line connecting two points on U
22
Summary
Risk Averse Risk Seeker Risk Neutral
MU
Shape of U
Fair Gamble
increasing
constant
decreasing
concave
convex
linear
(.5)162 (.5)22 130
(.5)v16 (.5)v2 2.7
(.5)16 (.5)2 9
gt81, Yes
lt3, NO
9, indifferent
EUgamble
Uno gamble
M09 Coin toss to win or lose 7
23
Intuition check

• Why wont 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.

24
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
25
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)
• 52.25-1.75-4
• 1.5
• Fair?
• No, more than fair!

Yes!
Would a risk seeker take this gamble?
Yes!
Would a risk neutral take this gamble?
Would a risk averse take this gamble?
26
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 vM , will you take the
  gamble?

• What is your utility without the gamble?
• Uno gamble vM
• v16
• 4

27
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 vM , will you take the
  gamble?

• What is your EXPECTED utility with the gamble?

• EU ¼v(1620) ¼v(169) ¼v(16-7)¼v(16-16)
• EU ¼v(36) ¼v(25) ¼v(9)¼v(0)
• EU ¼6 ¼5 ¼3¼0
• EU 1.51.250.750
• EU 3.5

28
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 gamble4
• EUgamble 3.5
• What do you do?
• Uno gamblegtEUgamble
• Therefore, dont take the gamble!

29
What is insurance?

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

30

• Example Mia Dribble has a utility function of
  UvM. 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 womens
  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.

31
Mias utility

• If M0, U
• v00

• If M10000, U
• v10000100

• If M1000000, U
• v10000001000

10000
32
Mias utility

• If M250000, U
• v250000500

• If M640000, U
• v640000800

• If M810000, U
• v810000900

• If M1210000, U
• v12100001100

10000
33
Mias utility
UvM

• Utility if income is certain!

• Risk averse?
• Yes

34
Mias utility
UvM
Unot injured

• U if not injured?
• v10000001000
• Label her income and utility if she is not
  injured.

• Label her income and utility if she is injured.
• v10000100

Uinjured
10000
Minjured
M not injured
35
What is Mias expected Utility?

• No injury M 1,000,000
• Injury M 10,000
• Probability of injury 10 percent 1/100.1

• Probability of NO injury
• 90 percent 9/100.9

• E(U)
• 9/10v(1000000)1/10 v(10000)
• 9/1010001/10100
• 90010 910

36
What is Mias expected Income?

• No injury M 1,000,000
• Injury M 10,000
• Probability of injury 10 1/100.1

• Probability of NO injury
• 90 9/100.9

• E(M)
• 9/10(1000000)1/10 (10000)
• 9000001000 901,000

37
Mias utility
UvM
Unot injured

• Label her E(M) and E(U).
• Is her E(U) certain?
• No, therefore, not on UvM line

E(U)910
E(U)
Uinjured
10000
E(M)901000
Minjured
Mnot injured
38
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

39
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 v(1,000,000-p)

40
Remember prediction will take a gamble only if
the expected utility of the gamble exceeds the
utility without the gamble.

• Buy insurance if
• Uv(1,000,000-p) gt 910 E(U)
• Solve

Square both sides
41
Remember prediction will take a gamble only if
the expected utility of the gamble exceeds the
utility without the gamble.

• Buy insurance if
• Uv(1,000,000-p) gt 910 E(U)
• Solve

Square both sides
Solve for p
Interpret If the premium is less than
171,000, Mia will purchase insurance
42
Mias utility
UvM
Unot injured
U 910

• What certain income gives her the same U as the
  risky income?
• 1,000,000-171,900
• 828,100

E(U)910
E(U)
Uinjured
10000
E(M)901000
828,100
Minjured
Mnot injured
43

• Leah Shooter also has a utility function of UvM
  . 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.

44
Leahs utility

• If M0, U
• v00

• If M10000, U
• v10000100

• If M1000000, U
• v10000001000

10000
45
Leahs utility

• If M250000, U
• v250000500

• If M640000, U
• v640000800

• If M810000, U
• v810000900

• If M1210000, U
• v12100001100

10000
46
Leahs utility
UvM
Unot injured

• U if not injured?
• v10000001000
• Label her income and utility if she is not
  injured.

• Label her income and utility if she is injured.
• v10000100

Uinjured
10000
Minjured
M not injured
47
What is Leahs 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)
• 1/2v(1000000)1/2v(10000)
• 550

48
What is Leahs 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)
• 5000005000 55,000

49
Leahs utility
UvM
Unot injured

• Label her E(M) and E(U).

E(U)
E(U)550
Uinjured
10000
E(M)550,000
Minjured
Mnot injured
50
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?

51
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 v(1,000,000-p)
• Buy insurance if
• Uv(1,000,000-p) gt 550 E(U)

52
Remember prediction will take a gamble only if
the expected utility of the gamble exceeds the
utility without the gamble.

• Buy insurance if
• Uv(1,000,000-p) gt 550 E(U)
• Solve

p lt 697,500
53
Leahs utility
UvM
Unot injured

• What certain income gives her the same U as the
  risky income?
• 1,000,000-697,500
• 302,500

E(U)
E(U)550
U 550
Uinjured
10000
E(M)550,000
302,500
Minjured
Mnot injured
54

• 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)?

55
Theas 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
• 0.1990000
• 99,000
• Expected claim from Leah
• 0.5990000
• 495,000

56
Theas expected value of claims/payouts
Probability of non-risky player

• Expected claim from Mia 99,000
• Expected claim from Leah 495,000

• Theas expected value of claims
• 0.599,000 0.5495,000

• 297,000

Probability of risky player
57
Premium297,000Willingness 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

58
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.

59
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

60
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

61
Other examples of adverse selection
62
Another Adverse Selection Example

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

63
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
D
10,000
D
5,000
Q of Lemons
Q of Peaches
Q (perfect info)
Q (perfect info)
64
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
P
S
S
D
10,000
D
5,000
Q of Lemons
Q of Peaches
Q (perfect info)
Q (perfect info)
65
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?
• 0.50100000.505000
• 7500
• If they are risk neutral, how much are they
  willing to pay for the car?
• 7500, indifferent between certain and uncertain
  income

66
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
D
10,000
D(50/50)
7,500
D(50/50)
7,500
D
5,000
Q of Lemons
Q of Peaches
Q (p.i.)
Q (p.i.)
Q (new)
Q (new)
67
Another Adverse Selection Example Used Cars

• Ultimately
• In the extreme case, no peaches, all lemons

Peaches
Lemons
S
P
P
S
D
10,000
D(50/50)
7,500
D(50/50)
7,500
D
5,000
Q of Lemons
Q of Peaches
Q (p.i.)
Q (p.i.)
Q (new)
Q (new)
68
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.

69
Role for the Government?

• Does the asymmetric info mean the govt
  can/should be involved?
• http//www.oag.state.ny.us/consumer/cars/qa.html
• (look up the Lemon Law for MI)

70
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?

71
III. Full disclosure/Unraveling

• Youre on a job interview and the interviewer
  knows what the distribution of GPAs are for MSU
  graduates
• Expected/Average grade for everyone
• 0.210.320.330.24
• 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
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
72
III. Full disclosure/Unraveling
Those who dont 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 dont
  reveal

Percent
GPA 0.1 0.2
0.30/.50
0.20/.50
0.60
0.40
Intuitively, those who are above the expected
average dont want employers to think they are
average, so they disclose!

• 0.410.62
• 1.6
• Therefore, those w/ a 2.0 should revealunravels
  so that there is full disclosure.

73
Intuition check

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

74
Voluntary disclosure and SAT scores

• Institutional Details
• Voluntary disclosure question
• Data
• Results

75
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

76
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 dont reveal.
• Is it only the students with very low SAT scores
  that dont reveal?

77
Data

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