2.2 Choice under Uncertainty:
Lotteries and Risk Aversion (2)
Outline
2.1 Basic concepts: Preferences, and utility
2.2 Choice under uncertainty: Lotteries and risk
aversion
2.3 Value of information: Decision trees and
backward induction
We saw that we should use a criteria different from “Mathematical
Expectation” to value lotteries
Because the Expectation might produce counterintuitive
results
Because it is a purely mathematical computation that does
not take into account the individual's preferences and
attitude towards risk
The expectation is just a convenient way of summarizing a
lottery
Expected Utility
In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote,
"the mathematicians estimate money in proportion to its quantity, and
men of good sense in proportion to the usage that they may make of it."
Instead of working with the amount of money (x) a
lottery pays, we will work with the utility (u(x)) this
money gives you.
This approach was formally proposed and justified by John vonNeumann
and Oskar Morgenstern in 1944, and has ben adopted as the main
tecnique to analyze decisions under risk ever since
Comparison
Expected Value
E(L) = p1 · x1 + p2 · x2 + p3 · x3 + · · ·
Expected Utility
E(u(L)) = p1 · u(x1) + p2 · u(x2) + p3 · u(x3) + · · ·
Notice !
Where u(x) represents the individual's preferences
over money as in:
u
utility
u(x)
x
money
Example
Expected value of the lottery L
1/3
1/3
1/3

2
1
1
1
E(L)  2  3 4  3
3
3
3
3
Now, if
4
u(x)  x2
1 2 1 2 1 2 29
E(u(L))  2  3  4 
3
3
3
3
Expected utility of the lottery L

The Expected Utility Theory
According to the Expected Utility Theory, individuals will
prefer that lottery that produces the highest “expected utility”
according to their own utility function over money u·
That is, given two lotteries L1 and L2,
L1 is preferred to L2
if E(u(L1 )) > E(u(L2 ))
L2 is preferred to L1
if E(u(L2 )) > E(u(L1 ))
L1 and L2 are indifferent if E(u(L1 )) = E(u(L2 ))
Notice !
The “expected utility” of a sure amount of money is just
the utility of such amount.
Indeed, a sure amount of $x is equivalent to a lottery (L)
that gives you $x with probability 1.
1
x
0
Then,
0
E(u(L)) = 1 · u(x) + 0 · u(0) = u(x)
Attitudes towards Risk
Remember the following (modified) example,
If we flip a coin 1000 times, and we get $2 with heads
and lose $1 with tails . . . How much will we earn at
the end ? (approximately)
Most likely, we well get around 500 heads and 500 tails.
That is, we will earn $1,000 and lose $500. At the end, we
expect to make around $500
What do you prefer, playing the gamble or getting $500 for sure ?
Some individuals would say:
“I now that most likely I would get $500 if I play the gamble.
Nevertheless, I'd better take the $500 because of the risk
of getting less than $500 if I play the gamble”
Others might say:
“I now that most likely I would get $500 if I play the gamble.
But I prefer to play it rather than getting $500 for sure
because of the possibility of earning more than $500”
Finally, some could say:
“I now that most likely I would get $500 if I play the gamble.
Hence, for me is the same to play the gamble than to get
$500 for sure.
The First kind of individuals are said to be risk avers. They
prefer to have the expected value of the gamble for sure
rather than playing the gamble itself. They do not like the risk
of getting less than the expected value
The Second kind of individuals are said to be risk lovers. They
like the chances of getting more than the expected value
of the gamble better than getting it for sure
The Third kind of individuals are said to be risk neutral. They
value the same both the expected value of the gamble for
sure and playing the gamble. They do not exhibit any special
attitude in front of a risky situation
In terms of expected utilities ...
Risk avers individuals are those such that:
u(E(L)) > E(u(L))
utility expected value lottery
expected utility lottery
Risk lovers individuals are those such that:
u(E(L)) < E(u(L))
utility expected value lottery
expected utility lottery
Risk neutral individuals are those such that:
u(E(L)) = E(u(L))
utility expected value lottery
expected utility lottery
Mathematical fact
Risk avers individuals have utility functions u that
are “concave”
Risk lovers individuals have utility functions u that
are “convex”
Risk neutral individuals have utility functions u that
are “linear”
Summary
Risk avers
Risk lover
Prefers the expected
value of the gamble
rather than the gamble
Prefers the gamble better
than the expected value of
the gamble
Indifferent
u(E(L)) > E(u(L))
u(E(L)) < E(u(L))
u(E(L)) = E(u(L))
u
Risk neutral
u
u
x
x
x
Example
Jack and Mary are thinking of playing the following
gamble. With probability ½ it gives your money back plus
a gain of $16, and with probability of ½ you lose
everything. The game costs $9, which is exactly the
amount of money each of them has.
Jack is risk neutral: u(x) = x, and Mary has the following
utility function: u(x)  x
Represent the lottery and try to guess each one`s choice.

Jack and Mary
Mary: u(x)  x
Jack: u(x)=x
u(9)  9  3
u(9)=9
1/2
1/2

9 + 16 = 25


9–9=0
1/2
9 + 16 = 25
1/2
9–9=0
1 1
E
(
L
)
25
0

12
.
5
2 2
1 1
E
(
L
)
25
0

12
.
5
2 2
1
1
E(u(L))  25 0 12.5
2
2
E(u(L)) 
Jack will play as
E(u(L)) 12.5 9  u(9)

1
1
25  0  2.5
2
2
Mary won’t play as
E(u(L))  2.5  3  u(9)
How to find out the utility function?
We will try to draw our utility functions. For
being able to do that we need to answer some
questions.
In each of the following cases, what is the
amount of money that makes you indifferent
between playing the proposed lottery and
getting that amount for sure?
1/2
100
1)
1
A
Indifferent to
1/2
1/2
0
A
1
B
Indifferent to
2)
1/2
1/2
0
100
3)
1
Indifferent to
1/2
A
C
Notice . . .
assuming u(0) = 0 and u(100) = 1
(Just a normalization of units of measurament)
1
1
1 1
u(A)  u(100)  u(0)  1 0  0.5
2
2
2 2
1
1
1
1
u(B)  u(A)  u(0)  0.5  0  0.25
2
2
2
2
1
1
1 1
u(C)  u(100)  u(A)  1 0.5  0.75
2
2
2 2

Hence, C is preferred to A which is preferred to B
If A=70, B=40, and C=95
U(C)
U(A)
U(B)
B
A
C
If A=50, B=25, and C=75
U(C)
U(A)
U(B)
B
A
C
If A=30, B=5, and C=60
U(C)
U(A)
U(B)
B
A
C
Thus, depending on the answers to the questions
we might be risk lover, risk neutral, or risk avers
In many applications, it is assumed that people is risk
lover when there are possibilities of earning money
and risk avers when they might lose money
utility
wealth
“Status Quo”
money
Example
Mary just paid $20,000 for a new car and now is thinking
about the best insurance for it. Mary has a probability of 0.01
of having an accident that would result in the total loss of the
car (and hence the need to buy a new one for $20,000).
The insurance company offers Mary two different policies:
FC (Full Coverage) In the event of an accident, Mary will be
reimbursed $20,000. It costs $200
PC (Partial Coverage) In the event of an accident, Mary will be
reimbursed $10,000. It costs $100
Mary also has the possibility of not buying any insurance (NC).
Her wealth after buying the car is $50,000 and her utility function
over money is u(x)  x What will she choose ?

Mary has three lotteries to choose from
0.01
50,000-20,000 = 30,000
0.99
50,000
0.01
50,000-20,000+20,000-200 = 49,800
0.99
50,000-200 = 49,800
0.01
50,000-20,000+10,000-100 = 39,900
0.99
50,000-100 = 49,900
NC
FC
PC
The expected utility in each case is:
remember,
u(x)  x
E(u(NC))=0.01 u(30,000) + 0.99 u(50,000) = 223.10

E(u(FC))=0.01 u(49,800) + 0.99 u(49,800) = 223.16
E(u(PC))=0.01 u(39,900) + 0.99 u(49,900) = 223.14
Hence, FC is preferred to PC which is preferred to NC
This is a particular example. In general, it can be shown that
Risk Avers individuals will always prefer Full Coverage
Summary
• We have seen that we must use a criteria different from
“Mathematical Expectation” to value lotteries
• Instead of working with the amount of money (x) a lottery
pays, we will work with the utility (u(x)) this money gives you.
E(u(L)) = p1 · u(x1) + p2 · u(x2) + p3 · u(x3) + · · ·
• According to the Expected Utility Theory, individuals will
prefer that lottery that produces the highest “expected utility”
according to their own utility function over money u
Summary
• Different individuals, with different preferences (that is, different utility
functions) will exhibit different attitudes in front of risk
Risk avers individuals prefer to have the expected value of the gamble for sure
rather than playing the gamble itself. They do not like the risk of getting less than
the expected value. They have concave utility functions
•
Risk loving individuals prefer the chances of getting more than the expected
value of the gamble better than getting it for sure. They have convex utility
functions
•
Risk neutral individuals value the same both the expected value of the gamble
for sure and playing the gamble. They do not exhibit any special attitude in front
of a risky situation. They have linear utility functions
•