Microeconomics I: Game Theory
Lecture 2:
Expected Utility Functions
(see Osborne, 2009, Sect 4.12.1-4.12.2)
Dr. Michael Trost
Department of Applied Microeconomics
November 1, 2013
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
1 / 38
The theory of rational choice
The theory of rational choice applies to any kind of decision
problem.
The set of imaginable actions could, for example, be the set
- of consumption bundles (discussed in consumer theory)
- of in- and output vectors (discussed in production theory)
- of quantities of information (discussed in search theory)
- of policies (discussed in social choice theory)
- ......
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
2 / 38
Decision problems under risk
In this lecture, we focus on decision problems under risk.
Decision problems under risk are situations in which the
outcomes of some of the decision maker’s actions are
non-deterministic in the sense that they are associated with
different mutually exclusive outcomes which occur according to
some objectively known probability measure.
Decision problems under risk are sometimes referred to as risky
situations.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
3 / 38
Examples of decision problems under risk
Decision problems under risk are, for example,
- choosing between monetary gambles on the spin of an
unbiased roulette wheel.
- deciding how many lottery tickets to buy.
- deciding which random device to take in order to allocate a
limited quantity of resources to individuals of a population
- ......
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
4 / 38
Lottery
Let X be a set of outcomes, then a lottery on X means nothing
but a probability distribution on X . The set of all lotteries on X
is usually denoted by ∆(X ).
Suppose X := {x1 , . . . , xK } consist of K distinguishable
outcomes. Then a lottery on X is representable by a vector
(p1 , . . . , pK ),
where pk is the probability that outcome xk will occur.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
5 / 38
Axioms of probability
Suppose X := {x1 , . . . , xK } consist of K distinguishable
outcomes. Since lottery (p1 , . . . , pK ) represents a probability
distribution on X , it has the following two properties:
1
2
pk ≥ 0 for every k ∈ {1, . . . , K },
PK
k=1 pk = 1.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
6 / 38
Decision problem under risk
Suppose X be a set of outcomes, A := ∆(X ) be the set of all
lotteries on X and C ⊆ A be a subset of A consisting of all
lotteries being available for the decision maker.
A decision problem under risk is a situation in which the
decision maker is required to select one lottery from the set C of
available lotteries.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
7 / 38
Rational choice under risk
Suppose a decision maker facing a decision problem under risk
rank the possible lotteries according to preference relation %.
Then her choice is called a rational choice under risk whenever
her chosen lottery belongs to set
{p ∈ C : p % p 0 for every lottery p 0 ∈ C } .
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
8 / 38
A simple decision problem under risk
Let X := {e 0, e 1, e 4} be a set of monetary prizes.
A decision maker is offered two lotteries, lottery p := ( 12 , 0, 12 )
and lottery p 0 := (0, 34 , 41 ) whose ith number indicates the
probability of the ith lowest monetary prize.
Her decision problem is to choose one out of the two lotteries.
Q UESTION : Describe formally both the set A of possible lotteries
on X and the decision maker’s restriction C !
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
9 / 38
A simple decision problem under risk
Suppose the decision maker facing previous decision problem
be an expected payoff maximizer (i.e., if a lottery has a higher or
equally high expected payoff than some other lottery, then she
weakly prefers the former one to the latter one).
Q UESTION : Which of the two available lotteries is chosen by
such decision maker?
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
10 / 38
St. Petersburg Lottery
Suppose you be offered the following lottery which has been
introduced by Daniel Bernoulli (1700-1782) and is known
nowadays as the St. Petersburg Lottery.
A fair coin is repeatedly tossed until tail appears. The pot starts at e 2
and is doubled every time a head appears. The first time a tail appears,
the game ends and you win whatever is in the pot. Thus you win e 2
if a tail appears on the first toss, e 4 if a head appears on the first toss
and tail on the second, e 8 if a head appears on the first two tosses and
a tail on the third, and so on.
Q UESTION : How much would you, at most, pay for taking part
in this lottery?
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
11 / 38
Are you an expected payoff maximizer?
The utmost amount of money you are willing to pay for taking
part in the St. Petersburg Lottery is
.........
The expected payoff of the St. Petersburg Lottery is
...............................................................
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
12 / 38
St. Petersburg Paradox
The observation that decision makers are willing to pay for the
St. Petersburg Lottery less than its expected value is known as
the St. Petersburg Paradox.
This paradox can be resolved if the assumption of expected
payoff maximization is replaced with the weaker assumption of
expected utility maximization, as Daniel Bernoulli did to
explain this paradox .
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
13 / 38
Utility function on lotteries
Let X := {x1 , . . . , xK } be a set of K distinguishable outcomes.
A preference relation % on ∆(X ) is said to be representable by a
utility function U whenever, for every lotteries p := (p1 , . . . , pk )
and p 0 := (p10 , . . . , pk0 ),
p % p 0 holds if and only if U(p) ≥ U(p 0 ) holds .
If decisions under risk are explained or predicted, theory often
assumes that the utility function of the decision maker takes the
form of an expected utility function.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
14 / 38
Expected utility function
Let X := {x1 , . . . , xK } be a set of K distinguishable outcomes.
A preference relation % on ∆(X ) is said to be representable by
an expected utility function whenever there is a utility function
on X (referred to as the Bernoulli utility function) so that, for
every lotteries p := (p1 , . . . , pk ) and p 0 := (p10 , . . . , pk0 ),
0
p % p holds if and only if
K
X
pk u(xk ) ≥
|k=1 {z
}
Expected utility
of lottery p
Dr. Michael Trost
Microeconomics I: Game Theory
K
X
pk0 u(xk ) holds .
|k=1 {z
}
Expected utility
of lottery p 0
Lecture 2
15 / 38
Bernoulli vs. Expected utility function
. The Bernoulli utility function u is a utility function on the
set X of outcomes.
. The expected utility function EU is a utility function on the
set ∆(X ) of lotteries.
The expected utility value EU(p) of lottery p is the with
probabilities p weighted average of the Bernoulli utility values u
of the possible outcomes.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
16 / 38
A simple decision problem under risk
Consider the previous decision problem under risk where
- X := {e 0, e 1, e 4} has been the set of outcomes,
- C := {p, p 0 } has been the constraint consisting of lotteries
p := ( 21 , 0, 21 ) and p 0 := (0, 34 , 14 ).
Suppose the decision maker’s preferences be representable by
an expected utility function with Bernoulli utility function u
u(0) = 0,
u(1) = 1,
u(4) = 4.
Q UESTION : Which of the two lotteries p and p 0 does such
decision maker choose?
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
17 / 38
A simple decision problem under risk
Now, suppose the decision maker facing this decision problem
have preferences being representable by an expected utility
function with Bernoulli utility function v
v (0) = −100,
v (1) = 0,
v (4) = 300.
Q UESTION : Which of the two lotteries p and p 0 does such
decision maker choose?
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
18 / 38
A simple decision problem under risk
Now, suppose the decision maker facing this decision problem
have preferences being representable by an expected utility
function with Bernoulli utility function w
w (0) = 0,
w (1) = 1,
w (4) = 2.
Q UESTION : Which of the two lotteries p and p 0 does such
decision maker choose?
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
19 / 38
Resolving the St. Petersburg Paradox
Suppose we observe that a decision maker is willing to pay at
most e 4 for playing the St. Petersburg Lottery. Such maximal
willingness to pay can be explained as if she maximizes
expected utility with Bernoulli utility function u(x) := ln(x).
E XPLANATION : (Hint:
Dr. Michael Trost
P∞
k
k=1 2k
= 2 and ln(x k ) = k ln(x) hold.)
Microeconomics I: Game Theory
Lecture 2
20 / 38
Allais Experiment
The following choice experiment has been designed by Maurice
Allais and its known as the Allais Experiment.
Maurice Félix Charles Allais (May 31, 1911 October 9, 2010) was a French economist and was
the 1988 winner of the Alfred Nobel Memorial
Prize in Economic Sciences “for his pioneering
contributions to the theory of markets and
efficient utilization of resources.”
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
21 / 38
Allais Experiment - Decision problem I
Suppose there be three possible monetary prizes:
first prize
e 5 000 000
second prize
e 1 000 000
third prize
e 0
In decision problem I, you are required to choose between the
lotteries
A:
B:
receive the second prize with certainty


 the first prize with probability of 0.1
receive
the second prize with probability of 0.89


the third prize with probability of 0.01
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
22 / 38
Allais Experiment - Decision problem II
Suppose there be three possible monetary prizes:
first prize
e 5 000 000
second prize
e 1 000 000
third prize
e 0
In decision problem II, you are required to choose between the
lotteries
(
the second prize with probability of 0.11
C: receive
the third prize with probability of 0.89
(
the first prize with probability of 0.1
D : receive
the third prize with probability of 0.9
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
23 / 38
Allais Experiment
Q UESTION : Which of the following combinations of lotteries
have you chosen? Mark your choices with a cross.
Decision problem I
Decision problem II
Your choice
Dr. Michael Trost
A
C
Microeconomics I: Game Theory
A
D
B
C
B
D
Lecture 2
24 / 38
Allais Paradox
The choice combinations . . . . . . . . . and . . . . . . . . . are not
consistent with expected utility.
A RGUMENTATION :
Experiments show that many participants choose combination
. . . . . . . . . . This contradiction with the expected utility
hypothesis is known as the Allais Paradox.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
25 / 38
A simple decision problem under risk
Review the behavior in the decision problem under risk where
- X := {e 0, e 1, e 4} has been the set of outcomes,
- C := {p, p 0 } has been the constraint consisting of lotteries
p := ( 21 , 0, 21 ) and p 0 := (0, 34 , 14 ).
We have observed the following choices.
Bernoulli utility function
Decision maker’s choice
Dr. Michael Trost
u
p
Microeconomics I: Game Theory
v
p
w
p0
Lecture 2
26 / 38
A simple decision problem under risk
Note that
= 100u(x) − 100
p
w (x) =
u(x)
v (x)
are satisfied for every monetary prize x ∈ {e 0, e 1, e 4}.
Obviously, the two Bernoulli utility functions v and w prove to
be positive monotone transformations of Bernoulli utility
function u.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
27 / 38
A simple decision problem under risk
However, as the previous table demonstrates, choices are
different under Bernoulli utility function u and w .
C ONCLUSION : Expected utility representation is not unique up
to any monotone transformation of its Bernoulli utility function.
However, as the following theorem reveals, expected utility is
unique up to any positive linear transformation of its Bernoulli
utility function.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
28 / 38
Positive linear transformation
A transformation of a real-valued function u on X to a
real-valued function v on X is called positive linear whenever
there exist α, β ∈ with α > 0 so that
R
v (x) = αu(x) + β
hold for every x ∈ X .
Positive linear transformation are also known as affine
transformations.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
29 / 38
Irrelevance of positive linear transformations
Theorem 2.1
Suppose the set X of outcomes have at least three elements and
preference relation % on ∆(X ) be representable by an expected utility
function with Bernoulli utility function u.
Preference relation % is also representable by expected utility function
with Bernoulli utility function v if and only if there exist α, β ∈
with α > 0 so that
v (x) = αu(x) + β
R
is satisfied for every outcome x ∈ X .
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
30 / 38
Cardinal concepts of measurement
A cardinal concept of measurement (also known as the interval
concept of measurement) is a concept of measurement that is
unique up to positive linear transformations.
Such concepts have the property that ratios of differences are
interpretable.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
31 / 38
Unique ratio of differences
Let X be a set of outcomes of some attribute which is cardinally
measurable.
Then, for every measures u and v on X ,
u(x 0 ) − u(x)
u(y 0 ) − u(y )
|
{z
}
ratio of differences
in measure u
=
v (x 0 ) − v (x)
v (y 0 ) − v (y )
|
{z
}
ratio of differences
in measure v
holds for every outcomes x, x 0 , y , y 0 ∈ X .
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
32 / 38
Example: Temperature
Temperature is a cardinal concept of measurement.
It can be measured, for example, by Celsius scale (°C ) or by
Fahrenheit scale (°F ). Any positive linear transformation of the
Celsius scale like
°F = 1.8 · °C + 32.
is also a valid measure of temperature.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
33 / 38
Example: Temperature
Suppose in the morning on the day before yesterday the temperature were
5 °C (or 41 °F ) and in the afternoon on the day before yesterday it were 10
°C (or 50 °F ). Yesterday morning a temperature of 10 °C (or 50 °F ) has been
measured and yesterday afternoon a temperature of 20 °C (or 68 °F ).
I NCORRECT: To say that yesterday afternoon it was twice as hot as at
the afternoon on the day before yesterday.
C ORRECT: To say that the increase in temperature from yesterday
morning to yesterday afternoon was twice as high as the increase in
temperature from morning to afternoon on the day before yesterday.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
34 / 38
Example: Bernoulli utility
As the previous theorem reveals, the Bernoulli utility is a
cardinal concept of measurement of preferences.
Suppose the decision maker preferences on the set ∆(X ) of all
lotteries on X be representable by an expected utility function.
Then the ratio of differences in Bernoulli utility values is unique and
meaningful.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
35 / 38
Example: Bernoulli utility
Let X be the set of possible monthly incomes of some individual
where x, x 0 , y , y 0 ∈ X are given by
x := e 1000, x 0 := e 2000, y := e 1000000, y 0 := e 1001000.
Suppose her preferences on the set ∆(X ) of all lotteries on X be
representable by an expected utility function.
Then statements like “the increase in her well-being induced by a
switch of monthly income of e 1000 to monthly income e 2000 would
be twelfth times higher than the increase in her well-being induced by
a switch of monthly income e 1000000 to monthly income
e 1001000” are admissible.
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
36 / 38
Normalization of Bernoulli utility
Corollary 2.2
Consider a set X of outcomes and a preference relation % on ∆(X )
which is representable by an expected utility function. Suppose x∗ ∈ X
be one of the least preferred outcomes, x ∗ ∈ X be one the most preferred
outcomes and x ∗ x∗ hold.
Then preference relation % is also representable by an expected utility
function whose Bernoulli utility function u has the properties
u(x∗ ) = 0
and
u(x ∗ ) = 1 .
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
37 / 38
Exercise: Normalization of Bernoulli utility
Suppose there be three possible outcome x1 , x2 and x3 and a
decision maker preferring x3 to x2 and x2 to x1 . Moreover, she
prefers lottery (0.5, 0, 0.5) to lottery (0.1, 0.5, 0.4) and lottery
(0.4, 0.2, 0.4) to lottery (0.55, 0, 0.45). Note, as usual, the ith
number of the lottery indicates the probability that outcome xi
will occur.
E XERCISE : Are the preferences of this decision maker
representable by an expected utility function? Explain your
answer! (Hint: Make use of the normalization theorem just
discussed.)
Dr. Michael Trost
Microeconomics I: Game Theory
Lecture 2
38 / 38