gravelle 17 choice under uncertainty

Choice under uncertainty
A. Introduction
The analysis in the preceding chapters has assumed that all decisions are taken in
conditions of certainty. Any decision would result in one and only one outcome.
When a firm chooses a set of input quantities, there is only one level of output
which will result, and it knows the profit which it will receive from the sale of each
output, no matter how far in the future production and sale will take place. Likewise,
in planning their purchases of goods and services, and borrowing or lending decisions,
households are assumed to know with certainty the expenditure and utility
associated with each consumption vector.
But uncertainty is pervasive. There is technological uncertainty, when the firm is not
able to predict for sure the output level which would result from a given set of input
quantities. Machines may break down; crops may be affected by the weather. There
is market uncertainty when a single household or firm is not able to predict for sure
the prices at which it will buy or sell. Market uncertainty is associated with disequilibrium
and change: if an economy were permanently in long-run static equilibrium,
then firms and households would expect to trade at equilibrium prices, which,
by experience, become known. If, however, changes are taking place through time
which change equilibrium positions, the individual agents in the markets cannot
know the new equilibria in advance, and can only form expectations of prices which
they know may be wrong.
Extension of the theory to take account of uncertainty has two main aims. It
should first tell us something about the usefulness and validity of the concepts
and propositions already derived. What becomes of the conclusions about the
working of a decentralized price mechanism, for example? Can we still establish
existence and optimality of competitive equilibrium? Are the predictions about
households’ and firms’ responses to changes in parameters affected qualitatively?
The answers are important positively and normatively. Second, many important
aspects of economic activity cannot be adequately analysed without explicit recognition
of uncertainty. For example, the joint stock limited liability company, the basic
institutional form of the firm in capitalist economies, has no real rationale in a
world of certainty, and neither has the stock market. Insurance, futures markets and
speculation cannot be understood expect in the context of uncertainty. Relaxation
of the certainty assumption gives new insights into many other areas, for example
investment decisions.
As with models of an economy with certainty, we begin with the optimization
problem of a single decision-taker. The optimization problem under uncertainty has
the same basic structure as under certainty: objects of choice; objective function;
and constraints defining a feasible set of choice objects. The main interest centres on
the first two of these, and, in particular, the construction of a set of axioms which
B. A FORMALIZATION OF ‘UNCERTAINTY’ 447
allows us to define a preference ordering, representable by a utility function, over
the objects of choice.
B. A formalization of ‘uncertainty’
Uncertainty arises because the consequence of a decision is not a single sure outcome
but rather a number of possible outcomes. Our first task in developing a
theory of choice under uncertainty is to set out a precise formalization of the
decision-taking situation. We can begin by distinguishing three kinds of variables
which play a part in an economic system. These are:
(a) The choice variables of the decision-taker which are directly under his control.
Such variables are not only endogenous to the model of the economic system,
but are also endogenous to the model of the individual economic agent. Examples
in earlier chapters include firms’ output levels and consumers’ purchases.
(b) Variables whose values are determined by the operation of the economic system,
i.e. by the interaction of the choices of individual economic agents, and which
are regarded as parameters by them. Prices are an example in a competitive economy.
Such determined variables are endogenous to the model of the economic
system, but exogenous to the model of the individual economic agent.
(c) Environmental variables, whose values are determined by some mechanism
outside the economic system and which can be regarded as parameters of the
economic system. They influence its outcome, but are not in turn affected by
it. The weather is an example, at least for some problems, though, in the light
of such events as global warming, even this could be seen as endogenous in
some models.
Suppose that the economy operates over only two periods, period 1 (the present)
and period 2 (the future). In period 1, the environmental variables take on specific
values which are known to all economic agents. We assume that the economy
produces a resource allocation and a set of relative prices. If there were complete
independence between the decisions made in period 1 and those to be made in
period 2, then the state of knowledge at period 1 about the environmental variables
at period 2 is irrelevant. In this case, decisions for period 2 can be left until period
2, and do not affect decision-taking at period 1. We assume that this kind of temporal
separability of decision-taking does not exist. At period 1, economic agents will
have to choose values of variables such as investment (purchase of durable goods)
and financial assets (bonds and shares), which affect what they will be able to do in
period 2. Agents’ plans for the values of variables they will choose at period 2,
influenced by their expectations about the values of variables outside their control at
period 2 – determined variables such as prices, and environmental variables like the
weather – will condition their choices at period 1. We therefore need a theoretical
framework to analyse the formation of plans and expectations, and their influence
on current choices.
We proceed as follows: suppose there exists a vector of environmental variables
(e1, e2, . . . , en), where each environmental variable is capable of taking on a finite
number of values in period 2. Let Ej denote the set of values which can be taken
by environmental variable ej ( j 1, 2, . . . , n). For example, e1 could be the average
448 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
temperature over period 2, measured to the nearest degree centigrade, and E1 could
be the set {e1⎪50°C e1 80°C}, which has a finite number of elements (since
the temperature is measured in units of 1°C). Define a state of the world as a specific
combination of the values of the environmental variables, i.e. as a specific value of
the vector (e1, e2, . . . , en). Since each element of the vector can take only a finite
number of values the number S of states of the world is also finite, though possibly
very large. We index the states of the world by a number s 1, 2, . . . , S and use
the index to label the value of the choice variables or determined variables in each
state of the world. Thus, for example, we can use ys to denote the level of income
the individual gets in state s.
Three fundamental properties of the set of states of the world should be clear:
(a) The set is exhaustive, in that it contains all the states of the world which could
possibly obtain at period 2.
(b) Members of the set are mutually exclusive, in that the occurrence of any one rules
out the occurrence of any other.
(c) The states of the world are outside the control of any decision-taker, so that the
occurrence of any one of them cannot be influenced by the choice of any
economic agent, or indeed by any coalition of agents.
The definition and properties of ‘states of the world’ are basic to all subsequent
analysis. They can be regarded as an attempt to eliminate the elements of doubt,
apprehension, and muddle which are part of the every-day meaning of the word
‘uncertainty’, and to give the situation a precise formalization, for purposes of the
theory. Three further assumptions which can be made are:
(a) All decision-takers have in their minds the same sets of states of the world – they
classify the possible combinations of environmental variables in the same way.
(b) When period 2 arrives, all decision-takers will be able to recognize which state
of the world exists, and will all agree on it.
(c) At period 1, each decision-taker is able to assign a probability to the event that
a particular state of the world will occur at period 2. The probabilities may differ
for different decision-takers, but all probability assignments satisfy the basic
probability laws. The probability associated with the sth state by decision-taker
i, denoted π s
i , lies on the interval 1 π s
i 0, with π s
i 1 implying that i regards
state s as certain to occur, and π s
i 0 implying that he regards state s as certain
not to occur. The probability of one or another of several states occurring is the
sum of their probabilities (with the probability of their simultaneous occurrence
being zero), and, in particular, one of the S states must occur, i.e. ΣS
s 1 π s
i 1.
Each of these assumptions is quite strong, and plays an important part in what follows.
The first is necessary if we are to portray decision-takers as making agreements
in state-contingent terms: in order for one to agree with another that ‘if state 1 occurs
I will do A, in return for your doing B if state 2 occurs’, it is necessary that they
should understand each other’s references to states.
The second assumption is also required for the formation and discharge of agreements
framed in state-contingent terms. If parties to an agreement would differ about
which state of the world exists ex post, they are unlikely to agree ex ante on some
exchange which is contingent on states of the world. The assumption also rules out
problems which might arise from differences in the information which different
C. CHOICE UNDER UNCERTAINTY 449
decision-takers may possess. Suppose, for example, that individual I cannot tell
whether it is state 1 or state 2 which actually prevails at period 2, while individual
J does know. Then I is unlikely to conclude an agreement with J under which,
say, I gains and J loses if state 1 occurs, while J gains and I loses if state 2 occurs,
because of course I could be exploited by J.
C. Choice under uncertainty
We now consider the question of optimal choice under uncertainty. First, we need
to define the objects of choice, and then we can consider the question of the
decision-taker’s preference ordering over these choice objects. We present what is
usually called the von Neumann–Morgenstern Theory of Expected Utility.
Initially, we assume that there is a single good, which is measured in units of
account, and which can be thought of as ‘income’. Let ys (s 1, 2, . . . , S) denote
an amount of income which the decision-taker will have if and only if state s occurs
(in this section we shall be concerned only with a single decision-taker and so do
not need to burden ourselves with a notation which distinguishes among decisiontakers).
Assume that the individual assigns a probability πs to state of the world s,
and denote the vector of probabilities by π [π1, π2, . . . , πS], while y [ y1, y2, . . . ,
yS] is the corresponding vector of state-contingent incomes. Define a prospect, P, as a
given income vector with an associated probability vector,
P (π, y) (π1, . . . , πS, y1, . . . , yS) [C.1]
Changing the probability vector π, or the income vector y (or both) produces a
different prospect. Another term for a prospect would be a probability distribution
of incomes.
The choice objects of our theory are prospects such as P. Any decision has as its only
and entire consequence some prospect P, and so choice between alternative actions
or decisions is equivalent to choice between alternative prospects. A preference
ordering over decisions can only be derived from a preference ordering over their
associated prospects.
For example, consider the decision of a market gardener to insure or not against
loss of income through sickness or poor weather such as severe frost. Decision A is
the decision not to insure, decision B is to insure. Associated with A is a prospect,
PA (π, y A) where y A is an income vector, the components of which vary across
states of the world. In the subset of states in which he is sick, income will take on
one value; in the subset of states in which there is frost, income takes on another
value; in the subset in which he is sick and there is frost, there will be a third value;
and when he is not sick and there is no frost, there will be a fourth (and presumably
the highest) value. Associated with B is a certain prospect (assuming that compensation
for loss of income through sickness or frost is complete) PB (π, yB), where each
element of yB is equal to what income would be in the absence of sickness and
frost, minus the insurance premium, which must be paid in all states of the world.
The choice between A and B, i.e. the decision whether or not to insure, depends on
whether PA is or is not preferred to PB. To analyse choice under uncertainty therefore
requires us to construct a theory of the preference ordering over prospects.
If certain assumptions (axioms) concerning a decision-taker’s preferences are
satisfied, then we are able to represent those preferences – the criterion by which
450 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
he takes his choices – in a simple and appealing way. A test of the appropriateness
of the assumptions would be to show that we can correctly predict choices not
yet observed, on the basis of observation of choice already made. It should be
emphasized that our theory is a device for permitting such predictions, rather
than for describing whatever thought process a decision-taker goes through when
making choices. The objects of choice consist of a set of prospects, which we can
denote by {P1, P2, . . . , Pn}. The five axioms are described next.
Axiom 1: Ordering of prospects
Given any two prospects, the decision-taker prefers one to the other, or is indifferent
between them, and these relations of preference and indifference are transitive.
In the notation of Chapter 2, for any two prospects Pj, Pk, exactly one of the statements
Pj Pk, Pj Pk, Pj Pk, is true, while
Pj Pk and Pk Pl ⇒ Pj Pl [C.2]
and similarly for the indifference relation . This axiom means that the preference
ordering over prospects has the same desirable properties of completeness and consistency
which were attributed to the preferences ordering over bundles of goods
in Chapter 2.
Before stating the second axiom, we need to introduce the concept of a standard
prospect. Given the set of prospects under consideration we can take all the income
values which appear in them, regardless of the state and the prospect to which they
belong, as defining a set of values of the variable, income. Since there is a finite
number of states and prospects, there is a finite number of such income values (at
most, nS of them). There will be a greatest and a smallest income value. Denote these
values by yu and yL respectively. It follows that all income values lie on the interval
[ yL, yu], and we can construct the theory so as to apply to this entire interval on the
real line. Define a standard prospect, P0, as a prospect involving only the two outcomes
yu and yL, with probabilities v and 1 v respectively, where 1 v 0. A
specific standard prospect, P0
1, can be written as
P0
1 (v1, yu, yL) [C.3]
(where, for convenience, we do not bother to write the second probability 1 v1).
We obtain a second standard prospect, P0
11, by changing v1, the probability of getting
the better outcome, to v11, so that
P0
11 (v11, yu, yL) [C.4]
We can then state the second axiom.
Axiom 2: Preference increasing with probability
Given any two standard prospects P0
1 and P0
11,
P0
1 P0
11 ⇔ v1 v11 [C.5]
P0
1 P0
11 ⇔ v1 v11 [C.6]
The decision-taker always prefers the standard prospect which gives the better
chance of getting the higher-valued outcome, while two standard prospects with the
same chance of getting the better outcome would be regarded as equivalent.
C. CHOICE UNDER UNCERTAINTY 451
Axiom 3: Equivalent standard prospects
Given any certain income value y1 such that yu y1 yL, there exists one and only
one value v1 such that
y1 P0
1 (v1, yu, yL) [C.7]
where P0
1 will be called the equivalent standard prospect for y1.
We can take a value of income in the given interval and always find a probability
of getting the better outcome in the standard prospect such that the decision-taker
would be indifferent between getting the income for certain, and having the standard
prospect. This is true for two values of income, namely yu and yL. We must have
yu Pu
0 (1, yu, yL) [C.8]
yL PL
0 (0, yu, yL) [C.9]
since Pu
0 and PL
0 correspond to the certain receipt of yu and yL respectively. Axiom 3
asserts that we could choose any income between yu and yL and always find a unique
v value to define an equivalent standard prospect. But then [C.7] defines a function
from the income domain yu y1 yL to the range of probability values 1 v 0.
We could write the function as v( y1). On the plausible assumption that more income
is preferred to less, axiom 2 implies that the value of v must increase as y1 increases,
in order to maintain indifference between y1 and the standard prospect. Hence v(y1)
is an increasing function. Given yu and yL, axiom 3 implies that the function is
uniquely defined. However, the values of yu and yL are essentially arbitrary and can
be changed without affecting the basic nature of the theory. For example, if we
define a new standard prospect, P0
1, by choosing a better outcome y1
u yu, then we
could apply axiom 3 to find the v values in P0
1 corresponding to each y value, and
we would expect them to be different – in particular, we would now have u(yu) 1.
Similarly, we could choose y1
L, to define a new standard prospect, and again we
would expect to obtain a different relationship between v and y. Thus the function
v(y) cannot be unique in a general sense, but only relative to a specific choice of
outcomes for the standard prospect. We return to this point later, when we take up
again the properties of the function v(y).
To describe the fourth axiom we need to introduce yet another type of prospect,
known as a compound prospect. In general, a compound prospect Pc is one which has,
for at least one of its outcomes, another prospect, rather than a single value of
income. A commonplace example of a compound prospect is a so-called ‘accumulator’
bet which one may place on horse-racing. If one places a ‘double’, one puts
a stake t on a horse in race 1, and specifies that, if it wins, the gross payout t W1
will be used as a stake on a horse in race 2. The possible outcomes of the prospect
are therefore to lose the original stake, t, with probability π1, or to gain a further
gamble, on the second race, with probability 1 π1. This second gamble, or prospect,
has the possible outcomes of losing the stake with probability π2, and winning
the net payout on the ‘double’, W2. Hence, the ‘double’, as a compound prospect,
can be written
Pd
c [π1, y t, (π2, y t, y W2)] (π1, y t, P1) [C.10]
where P1 (π2, y W1 t W1, y W2) is the gamble on the second race, W1
is the payout on winning the first race and y is income without the bet. (As with
the notation for the standard prospect, whenever a prospect involves only two
452 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
outcomes, only the probability of the first outcome will be written, since the probability
of the second is simply 1 minus the probability of the first.)
How should a rational decision-taker evaluate a compound prospect? Take our
accumulator bet, the ‘double’, as an example. The punter may lose in one of two
mutually exclusive ways: losing on the first race, with probability π1; and losing on
the second race, with probability π2, given that he has won on the first race, with
probability 1 π1. Applying standard probability laws, the overall probability that
he will lose on the second race is π2(1 π1) (probability of joint event ‘win on first
race and lose on second’. Where the separate probabilities are independent), and so
the probability that he will lose on either the first or second race is π1 π2(1 π1)
K. The probability that he will win the net payout W2 is the probability of
winning the second race and winning the first, i.e. (1 π1)(1 π2) (1 K). If
the punter loses, his income loss is t, the loss of the stake money. Looking at the
net income difference brought about by the bet, he ends up either with having lost
the stake t, with probability K, or having won the payout W2, with probability 1 K.
We can define the simple prospect Pd, which summarizes the overall net income
represented by the compound prospect Pd
c , as
Pd (K, y t, y W2) [C.11]
We could argue that Pd is equivalent to Pd
c, in that a rational gambler, working out the
final possible income positions, and their associated probabilities, would conclude that
the compound prospect ‘boils down’ to this simple prospect. However this is not quite
the same as saying that any decision-taker would be indifferent between the compound
prospect Pd
c and its rational equivalent Pd. We might feel that a rational individual
ought to be indifferent between the two, but it is not hard to find punters who
prefer to go for a double even when exactly the same payouts are available with separate
bets on the two races (Question 4, Exercise 17C, asks you to discuss this further).
Now let us generalize from this example and consider the compound prospect
which has standard prospects as its outcomes. Such compound prospects have
P1c
[π 1, (v1, yu, yL), (v2, yu, yL), . . . , (vS, yu, yL)]
(π1, P1
0, P2
0, . . . , PS0
) [C.12]
where π1 is a vector of probabilities, π1 [π1
1, π 1
2, . . . , π1
S], and vs (s 1, 2, . . . , S) is
the probability of getting the better outcome yu in the sth standard prospect. Thus,
π1s
is the probability of getting the sth standard prospect which has probability vs of
getting yu. The compound prospect P1c
has S different ways of getting either yu or yL.
The probability of getting yu overall is
G π 1
1v1 π 1
2v2 . . . π1s
vs . . . π1
SvS π1s
vs [C.13]
yu may be won by winning prospect Ps0
with probability π1s
and then winning yu with
probability vs, so that the probability of winning yu in this particular way is π1s
vs.
Hence the probability of winning yu in one of these S mutually exclusive ways is
G ΣS
s 1π1s
vs. The overall probability of winning yL is 1 G. Thus, we can define as
the rational equivalent of P1c
in [C.12] the standard prospect
P1
0 (G, yu, yL) [C.14]
since P1
0 yields yu with probability G, and yL with probability 1 G. We can now state
the fourth axiom.
s
S
= Σ
1
C. CHOICE UNDER UNCERTAINTY 453
Axiom 4: Rational equivalence
Given any compound prospect P1c
, having as outcomes only standard prospects (as
in [C.12]), and given its rational equivalent P1
0 (as in [C.14]), then P1c
P1
0.
Axiom 4 asserts that the decision-taker does indeed rationally evaluate the probabilities
of ultimately obtaining the two outcomes, and is not affected by the twostage
nature of the gamble – we could perhaps say that he does not suffer from ‘risk
illusion’. Clearly, the axiom incorporates a strong assumption about the rationality
and computational ability of the individual decision-taker.
Axiom 3 stated that for any income value y1 we can find an equivalent standard
prospect P1
0 by suitable choice of a value v1. Take now any one of the prospects
P1, P2, . . . Pn, which form the original set of objects of choice for the decision-taker,
with:
Pj (π, yj ) j 1, 2, . . . , n [C.15]
where π is the vector of probabilities and yj [yj1
, yj2
, . . . , y j
S ] is the vector of statecontingent
incomes. Applying axiom 3, we can find for each y j
s (s 1, 2, . . . , S) the
equivalent standard prospect P0
js such that
ys
j P0
js (vjs, yu, yL) [C.16]
where, consistent with our earlier notation, we have
vjs v(ys
j ) j 1, 2, . . . , n s 1, 2, . . . , S [C.17]
Now consider the compound prospect, Pj
c, which is formed from Pj by replacing each
component of the income vector by its equivalent standard prospect, i.e.
Pj
c (π, (P0 j1, P0 j2, . . . , P0
jS )) [C.18]
where each P0
js (s 1, 2, . . . , S) satisfies [C.16]. Thus, whereas the outcomes in Pj are
amounts of income, the outcomes in Pj
c are the equivalent standard prospects. Then
the fifth and final axiom can be stated.
Axiom 5: Context independence
Pj Pj
c all j 1, 2, . . . , n
In words: the decision-taker is indifferent between a given prospect and a compound
prospect formed by replacing each value of income by its equivalent standard
prospect. For example, suppose that the decision-taker is in turn indifferent between
(a) £70 for certain, and a 50–50 chance of £200 or £10, and (b) £100 for certain, and
a 75–25 chance of £200 or £10. Axiom 5 asserts that he would then be indifferent
between a 50–50 chance of £70 or £100, on the one hand, and a 50–50 chance of
obtaining one of two further gambles: (a) a 50–50 chance of £200 or £10, and (b) a
75–25 chance of £200 or £10, on the other. The fact that values of income, and their
equivalent standard prospects, may be included in prospects, does not change their
basic relation of indifference (which is what the term ‘context independence’ tries
to convey). We could represent this example as
([0.5 0.5], [£70 £100]) ([0.5 0.5][(0.5, £200, £10)(0.75, £200, £10)]) [C.19]
Thus equivalent standard prospects can be substituted for incomes, without changing
the place of a prospect in the preference ordering.
454 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
We now show that the five axioms lead to an appealing way of representing the
decision-taker’s preference ordering. Axiom 5 implies that given any set of prospects
{P1, P2, . . . , Pn} we can replace each by the corresponding compound prospects
P1c
, P2c
, . . . , Pnc
, where each Pj
c ( j 1, 2, . . . , n) satisfies [C.18]. In other words, we can
express each of the ‘primary’ prospects as a compound prospect involving only various
chances of obtaining standard prospects. This is an important step, since it puts
the individual prospects on to a common basis of comparison – they become simply
different ways of winning one or other of the same two outcomes. Moreover, axiom 4 tells
us that each of the Pjc
will be indifferent to its rational equivalent, i.e. a standard
prospect involving only outcomes yu and yL, with probabilities derived in a straightforward
way from those appearing in the Pjc
. Thus, we can write
Pj Pjc
Pj0
[C.20]
or, more fully,
(π, yj ) (π, [P0 j1, P0 j2, . . . , P0
jS]) (Gj, yu, yL) [C.21]
where P0
js (vjs, yu, yL), and
[C.22]
Now, from axiom 2, we have that P0 j Pk
0, if and only if Gj Gk, and that P0 j Pk
0
if and only if Gj Gk. It follows that the rational equivalent standard prospects P1
0,
P2
0, . . . , Pn
0 can be completely ordered by the values of G which appear in them. The
preferred rational equivalent standard prospect will be that with the highest G value.
Thus the decision-taker chooses among the prospects P0 j in such a way as to maximize
the value of G. He acts as if his intention is to make G as large as possible. But, from
axiom 1, the preference ordering over all prospects (including standard prospects) is
transitive, so that
Pj P0 j and Pk Pk
0 and P0 j Pk
0 ⇒ Pj Pk [C.23]
Thus, because of [C.20], the preference ordering over the rational equivalent standard
prospects P0 j , represented by the values of the Gj, is identical to the preference
ordering over the initial prospects, Pj ( j 1, 2, . . . , n). It follows that choice among
the initial prospects can be modelled as if the decision-taker maximizes G.
The axioms provide a procedure for predicting the choices among prospects of a
decision-taker to whom they apply. We might proceed as follows: by a large number
(in principle an infinity) of paired comparisons between certain income values
on the interval [yu, yL], and standard prospects P0, we could find the function v(y).
We could then take two prospects, say P1 and P2, and, by inserting the values of the
incomes y1s
, y2
s (s 1, 2, . . . , S) into this function, we obtain the values v1s
v(y1s
),
v2s
v( y2s
). Then we can calculate G1 Σs πsv1s
, and G2 Σs πsv2s
and predict that the
prospects with the higher of these two values will be chosen.
In theoretical analysis we do not know the specific functions v(y). Rather, we have
to use certain general properties of this function. In the next section, we consider
the most important properties in some detail. We conclude this section with a note
on terminology. It is usual to call the function v( y) a utility function, since it is a
real-valued numerical representation of a preference ordering. It should be clear
from the way in which this function is derived that ‘utility’ is not to be interpreted
as a quantity of satisfaction, well-being or other psychic sensation but simply as a
name for the numbers which result when we carry out a series of paired comparisons.
G j
s
js
s
S
s s
j
s
S
π v π v( y )
= =
Σ Σ
1 1
C. CHOICE UNDER UNCERTAINTY 455
We refer to the value Gj ΣSs
1 πsvj
s as the expected utility of prospect Pj, and we can
interpret the axioms to mean that the decision-taker chooses among projects as if
to maximize expected utility. The theory based on the axioms is often called the
Expected Utility Theory of Choice under Uncertainty.
We have set out the axioms underlying the Expected Utility Theory. In the next
section, we discuss some properties which can be attributed to the utility function
v(y).
EXERCISE 17C
1. Ms A has a utility function given by v √y, where y is income. She is asked to enter a
business venture, which involves a 50–50 chance of an income of £900, or £400, and so the
expected value of income from the venture is £650.
(a) If asked to pay a ‘fair price’ of £650 in order to take part in the venture, would she accept?
(b) What is the largest sum of money she would be prepared to pay to take part in the venture?
(Hint: find the certainty equivalent of the prospect defined by the venture, i.e. the income
which, if received for certain, has a utility equal to the expected utility of the prospect.
Don’t worry, for the moment, that the v-values in this question do not lie between 0 and 1.
This is discussed in the next section.)
2. Now suppose Ms A has the utility function (a) v ay where a 0, or (b) v y2. How would
your answers to question 1 change?
3. St Petersburg Paradox. Suppose we define the following gamble: we toss a coin, and if it
lands heads, you receive £2, and if it lands tails, we toss it again; if on the second toss it
lands heads, you receive £4 ( £22), and if it lands tails, we toss it again; if on the third toss
it lands heads, you receive £8 ( £23), and if it lands tails, we toss it again . . . and so on ad
infinitum. Thus on the nth toss, a head wins you £2n, while a tail leads to a further toss.
Now, assuming the coin is fair, the probability of a head on the first toss is 1–2 ; the probability
of a head on the second toss (the sequence, a tail then a head) is (1–2 )2; and the probability of
a sequence of n 1 tails and then a head on the nth toss is ( 1–2 )n. Therefore, the expected
value of the game is:
E ( 1–2 )£2 ( 1–2 )2£4 ( 1–2 )2£8 . . . ( 1–2 )n£2n . . .
1 1 1 . . .
which is infinite, assuming nothing prevents us playing the game for ever. It has been
noticed, however, that the maximum amount one would pay to take part in the game is,
for most people, finite. Discuss possible explanations of this, and, in particular, what it
might imply about the utility function v(y).
4. Consider an accumulator bet on horse racing, known as a treble: the punter specifies three
horses in successive races, and places a stake on the horse in the first race; if it wins, the
winnings become the stake on the horse in the second race; if this wins, the winnings
become the stake on the horse in the third race. Write down the ‘treble’ in the notation for
prospects, assuming the punter assigns probabilities to the events of each horse winning its
race. Now derive the rational equivalent of this prospect. How many outcomes does it have?
Suppose that the punter also has the option of making three separate bets on each of the
races, the bet on the second race being laid after the result of the first is known, and the
bet on the third race being made after the result of the second is known. Describe this
option as a prospect, and compare it with the treble. Discuss possible reasons for a punter
preferring the treble.
5. Choose someone with enough patience, and try to construct their utility function for
incomes over the range £0–£100 a week, in the manner suggested by the section. Note the
problems you encounter, and relate them to the axioms set out above.
456 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
D. Properties of the utility function
The axioms set out in the previous section imply certain properties of the utility
function v(y). It increases with income, y; it is uniquely defined relative to the values
yu and yL; and it is bounded above by the value 1 and below by the value 0. Moreover,
as we shall see, the fact that the decision-taker can be assumed to act as if he
maximized expected utility implies a further important property of the function. But
there are certain properties of the function which we would like it to have but which
are not implied by the axioms set out so far, and so further assumptions have to be
made to endow it with them. We shall consider these assumptions first and then
look at the properties already implied by the axioms of the previous section.
For the kinds of analysis we wish to carry out, it is useful if the utility function is
differentiable at least twice in its entire domain, that is, if the derivatives v′( y) and
v″( y) exist for all y in the interval [yL, yu]. Thus we make the assumption that the
utility function is at least twice differentiable, at all income levels in the given
domain. Call v′(y) the marginal utility of income. v″(y) is the rate at which marginal
utility of income changes with income. Note that the differentiability assumption
implies that the utility function is continuous.
The second assumption concerns the attitude of the decision-taker towards risk.
Suppose he is confronted with a prospect P (π, y1, y2). The expected value of the
outcomes is D π y1 (1 π)y2. Define the certainty equivalent of the prospect, yc , as
that value of income which satisfies
yc P [D.1]
or equivalently
v( yc) G π v( y1) (1 π)v( y2) [D.2]
yc is the amount of income which, if received for certain, would be regarded by the
decision-taker as just as good as the prospect P. [D.1] says that yc is indifferent to P,
and [D.2] that its utility must equal the expected utility of P, which, in the light of
the analysis in the previous section, is an equivalent statement.
Consider the three possible relationships between the certainty equivalent, yc, and
the expected value of the outcomes, D.
(a) yc D. The decision-taker values the prospect at its expected value. For example,
offered a bet that if a fair coin lands heads he will receive £6, and if it lands tails
he must pay £4, he would certainly accept the bet. When his original income is y
the choice is between the prospect P1 (0.5, y £6, y £4), and the prospect
P2 (1, y, y) since to refuse the bet is to accept the certainty of no income gain.
The expected value of the prospect P2 is D2 y, while that of P1 is D1 £1 y. But
since certainty equivalents equal expected values, and v(y 1) v(y), he must
prefer P1 and so will take the bet. Indeed, he would be prepared to pay anything
up to £1 for the opportunity to engage in the bet, since the expected value of his
winnings, £1 minus what he pays, will still be positive. In general terms in this case
π v(y1) (1 π)v(y2) G v(yc) v(D) [D.3]
where D π y1 (1 π)y2. A preference ordering over alternative prospects can
then be based entirely on the expected values of the outcomes of the prospects,
with higher expected value always being preferred to lower.
D. PROPERTIES OF THE UTILITY FUNCTION 457
(b) yc D. The decision-taker values the prospect at less than its expected value. For
example, given the above example of the prospect P1 (0.5, y £6, y £4)
we can no longer be sure that it will be accepted in preference to the prospect
P2 (1, y, y), and the decision-taker would certainly pay less than £1 for the
opportunity to take it. In general terms,
π v(y1) (1 π)v(y2) G v(yc) v(D) [D.4]
In this case a preference ordering over alternative prospects could not be based
on the expected values of outcomes, since they overstate the values of the
prospects. To predict the rankings we would need to know the utility function
or the certainty equivalents.
(c) yc D. The decision-taker values the prospect at more than its expected value.
In our previous example, he would certainly accept the prospect P1, since
yc D y, while he would actually be prepared to pay more than D y £1
for the opportunity to take the gamble. In general terms,
π v(y1) (1 π)v(y2) G v(yc) v(D) [D.5]
Again, a preference ordering over prospects could not be based on the expected
values of outcomes, since these now understate the values of the prospects. To
predict the ranking we would again need to know the utility function or the
certainty equivalents.
The three cases provide a way of classifying attitudes to risk, based on a comparison
of the certainty equivalent and expected value. In the first case, where prospects are
valued at their expected values, the decision-taker is risk neutral; in the second,
where prospects are valued at less than their expected values, he is risk-averse; and in
the third case he is risk-attracted. As we shall see later, there are strong arguments for
regarding risk-aversion as typical.
We consider now the implications of these three cases for the utility function
v(y). First recall (Appendix B) the definitions of convex and concave functions.
Given some function f (y), defined on a convex set Y, the function is concave if and
only if
f (D) kf(y1) (1 k)f (y2) 0 k 1 y1, y2 Y [D.6]
where D ky1 (1 k)y2. A linear function satisfies [D.6] as an equality, while a
strictly concave function satisfies it as a strict inequality. But in equations [D.3],
[D.4] and [D.5], if we replace k by the probability π (defined on precisely the same
interval), and replace f by v, we see that case (a), risk neutrality, corresponds to a
linear utility function (at least over the range [y1, y2]), while case (b), risk aversion,
corresponds to a strictly concave utility function (over the range [y1, y2]), Moreover,
the function f (y) is strictly convex if f (y) is strictly concave, and so case (c), risk
attraction, corresponds to a strictly convex utility function. Figure 17.1 illustrates
these propositions. In (a) of the figure, the utility function v(y) is strictly concave.
Corresponding to the income levels y1, y2 are the utility values v(y1), v(y2), at points
a and b respectively. Given a value of π, the expected value D π y1 (1 π)y2 will
lie somewhere between y1 and y2. Consider the straight line ab, whose end points
are at the values v(y1) and v(y2) respectively. The expected utility of the prospect
P (π, y1, y2) is at point c, on the line ab directly above the expected value of
458 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
the prospect, D π y1 (1 π)y2. Thus, since the slope of the straight line ab is
[v(y2) v(y1)]/(y2 y1), letting f(y) denote its height at y, we have
[D.7]
In particular the height of the line at c above D π y1 (1 π)y2 is
f(y) v(y1)
v(y1)
v(y1) [v(y2) v(y1)](1 π )
π v(y1) (1 π )v(y2) G [D.8]
which gives G as point c in the figure.
We can now find the certainty equivalent of the prospect P, the income which
satisfies
v(yc) G [D.9]
It is given by yc in the figure since, at point d, the utility of y, received for certain, is
equal to the expected utility of the prospect. Thus, we see the equivalence between
the inequality yc D, and the strict concavity of the utility function. Each can
represent the assumption of risk aversion.
In (b) of the figure, v(y) is strictly convex. D again shows the expected value of
income, for a particular choice of π, and the expected utility G is given by g in the
[ ( ) ( )]( )( )
v y v y y y
y y
2 1 2 1
2 1
1

π
[ ( ) ( )][ ( ) ]
v y v y y y y
y y
2 1 1 2 1
2 1
1

π π
f y v y
v y v y y y
y y
( ) ( )
[ ( ) ( )]( )


1
2 1 1
2 1
Figure 17.1
D. PROPERTIES OF THE UTILITY FUNCTION 459
figure. Now, however, the certainty equivalent is yc D, since v(yc) G at h. Thus,
risk attraction is equivalent to a strictly convex utility function, or to a certainty
equivalent in excess of the expected value of income.
Risk neutrality could be illustrated in either part of the figure. Take (a), and suppose
that the utility function is the line drawn between a and b. Then G v(D) (whereas
in the strictly concave case v(D) G), and D is the certainty equivalent of the prospect
P. Hence, risk neutrality corresponds to the case of a linear utility function.
Risk premium
The risk averse individual in Fig. 17.1(a) will prefer to have a certain income of D
rather than the risky prospect P (π, y1, y2), where D is the mean of the risky incomes
y1 and y2. The risky prospect is costly to the risk averse individual in that it reduces
expected utility compared with the certain prospect of D. A monetary measure of
the cost of risk can be obtained by asking the individual how much of his certain
income he would be willing to give up rather than face the risky prospect. (Recall
the discussion of the compensating variation in section 3C.) This sum of money r
is the risk premium or the cost of risk and is defined by
v(D r) π v(y1) (1 π )v(y2) G [D.10]
since the individual is indifferent between the risky prospect P with expected
income D and the certain income D r.
Comparing [D.10] which defines the risk premium, with [D.9] which defines the
certainty equivalent income yc we see that
v(yc) G v(D r)
and we can equivalently define the risk premium for the prospect P by
r D yc [D.11]
In Fig. 17.1(a) the risk premium is the horizontal distance cd and shows how much
the individual would be willing to pay rather than face a risky prospect with an
expected income equal to his certain income D.
If the individual is risk attracted, as in Fig. 17.1(b), he would have a larger expected
utility with the risky prospect than with the certain income D. The risk premium
would be negative because he would be willing to pay to have P rather than the
certainty of D. Hence in Fig. 17.1(b), r D yc 0.
A fair bet is one which leaves expected income unchanged, as for example if a
gambler wins £5 if a coin turns up heads and loses £5 if it turns up tails. If an individual
has a certain income of D then the prospect P (π, y1, y2) which yields an
expected income of D is a fair bet since π(y1 D) (1 π)(y2 D) 0. We can therefore
equivalently define a risk averter as an individual who would refuse a fair bet
since he will prefer D to P, i.e. v(D) G. Similarly, risk preferers are made better off
by fair bets and risk neutral individuals are indifferent to such bets.
There is yet another way to describe the decision-taker’s attitude to risk. The differentiability
assumption allows us to express convexity and concavity in terms of
derivatives of the utility function. Thus marginal utility v′(y) is the slope at a point
of the curve v(y) in Fig. 17.1, while the second derivative, v″(y), is determined by
the curvature. Thus, strict concavity over the range of y values implies that, at every
460 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
Nothing in the axioms of section B implies a particular shape for the utility
function or attitude to risk for individuals in general. The usual assumption is that
decision-takers are risk averse and so have strictly concave utility functions. Part of
the reason for the assumption is empirical: most people behave in ways consistent
with risk aversion rather than risk neutrality or risk preference. Offered a 50–50 chance
of winning, say, £5000, or losing £5000, most people would reject the gamble.
Another common example of risk averse behaviour is the purchase of insurance at a
premium which is higher than would be actuarially fair (see section 19B).
Figure 17.1 raises the question: is it valid to ascribe a particular shape to the
utility function v(y) representing the preferences of a given individual who satisfies
the axioms of section B; or, equivalently, to place restrictions on the sign of the
second derivative v″(y)? Recall the discussion of the ordinal utility function in
Chapter 2. We argued there that the utility function was unique only up to a positive
monotonic transformation. It could not be specified as ‘convex’ or ‘concave’,
since functions of both kinds could be permissible representations of the preferences
satisfying the axioms stated in the chapter. Indeed, this provided the motivation
for the introduction of the concepts of quasi-concavity and quasi-convexity. If v(y)
were an ordinal utility function, we could not place a sign restriction on its second
derivative. For example, if v √y were a permissible ordinal utility function, with
v″(y) 1–4 y 3/2 0, so would be g v2, or g v4, which have zero and positive
second derivatives, respectively. The implicit assumption in the preceding discussion
of attitudes to risk that it is meaningful to place sign restrictions on v″(y) hints
that the utility function v(y) is not an ordinal utility function.
The utility function v(y) is not a unique representation of the decision-taker’s
preferences. By changing one or both of the outcomes in the standard prospect
we would obtain, for each certain income y, a different probability v at which the
decision-taker would be indifferent between the certain income and the standard
prospect. Figure 17.2 illustrates. In (a) of the figure, we assume first that v(y) is
measured against a standard prospect containing yL and yu, and then we replace yL
by y′L yL. We then obtain the new utility function, denoted g(y), for which we must
point, v″(y) 0, i.e. there is diminishing marginal utility of income. Similarly, strict
convexity implies v″(y) 0 or increasing marginal utility of income, and linearity
implies v″(y) 0 or constant marginal utility of income. A decision-taker is risk
averse, risk attracted or risk neutral, as his marginal utility of income is decreasing,
increasing or constant.
The different but equivalent characterizations of attitudes to risk are summarized
in Table 17.1.
Table 17.1
Risk aversion Risk neutrality Risk preference
Certainty equivalence yc ( yc ( yc (
Risk premium r 0 r 0 r 0
Fair bets Reject Indifferent Accept
Utility function Strictly concave Linear Strictly convex
v″ 0 v″ 0 v″ 0
D. PROPERTIES OF THE UTILITY FUNCTION 461
have g(yL′) 0, g(yu) 1 and g(y) v(y) for yL y yu (explain why). Thus g(y) and
v(y) must bear the general relationship to each other shown in the figure. In (b) of
the figure, we assume v(y) is as before, while g(y) is obtained by replacing yu with yu ′
in the standard prospect. Here, we must have g(yL) 0, g(yu ′) 1 and g(y) v(y) for
yL y yu (explain why), and so the general relationship between the two functions
is as shown in the figure.
Although the utility function representing preferences satisfying the axioms of
expected utility cannot be unique, the relationship among all possible utility functions,
of which v(y) and g(y) are examples, is much more restricted than among
the ordinal utility functions which represent consumer preferences under certainty.
We can show that any two utility functions v(y) and g(y) for a particular decisiontaker
whose preferences conform to our axioms must satisfy the linear relationship:
g(y) a bv(y) b 0 [D.12]
As the level of certain income varies, the utility numbers obtained by use of one
standard prospect must vary linearly with those obtained by use of a different standard
prospect. Figure 17.3 illustrates. A given income level, y, implies a pair of values
(v, g) (such as at α), and, as the income value varies, through y1 to y3, the pair of
utility values varies along the line through β to γ. The restriction on the class of
permissible utility functions is strong. The utility function v(y) is unique up to a
positive linear transformation (cf. the ordinal utility function’s property of uniqueness
up to a positive monotonic transformation). The restriction justifies attaching
significance to the sign of v″(y), since, if g(y) must satisfy [D.12], g″(y) must have the
same sign (see Question 4, Exercise 17D).
The property of uniqueness up to a linear transformation is implied by the
axioms. Take three income values y1 y2 y3, and let π be the probability that
y2 (π, y1, y3) [D.13]
The decision-taker is indifferent between y2 for certain and a prospect with y1 and y3.
(The axioms imply that such a π always exists.) Then
Figure 17.2
462 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
v(y2) π v(y1) (1 π)v(y3) [D.14]
g(y2) π g(y1) (1 π)g(y3) [D.15]
for any two permissible utility functions v(y) and g(y). Given that the decision-taker
conforms to our axioms, [D.14] and [D.15] are equivalent ways of expressing [D.13].
Writing [D.14] and [D.15] in vector notation gives
[v2, g2] π[v1, g1] (1 π )[v3, g3] 0 π 1 [D.16]
where v2 v( y2), g2 g( y2), and so on. But [D.16] states that the pair of utility
values (v2, g2) lie on the line joining the pair (v1, g1) with the pair (v3, g3), since it is
equivalent to the standard expression for a convex combination of two vectors.
Hence any two permissible functions v(y) and g(y) must be linearly related.
Uniqueness up to a positive linear transformation implies that the utility function
v(y) is a cardinal rather than an ordinal measure of utility. The von Neumann–
Morgenstern axioms provide a measure of utility comparable to measures of
temperature – the Fahrenheit and Centigrade scales are similarly linearly related. If
the theory holds, utility is cardinally measurable. This does not mean that we have
succeeded in ‘measuring utility’ as if utility were a physical magnitude in the same
sense as weight or height. Nothing has been said about the ‘amount’ of intrinsic
pleasure which the decision-taker receives from various amounts of income. The
basis of the theory is still the ordering relation of preference or indifference, and
the decision-taker’s ability to compare and rank is the only aspect of his psychology
in which we are interested. What we have shown is that, if his preferences satisfy
a number of conditions of rationality and consistency, then he can be represented
as acting as if he maximized the expected value of a numerical function which
must therefore be cardinal. The label ‘utility’ attached, perhaps unfortunately, to
the function implies nothing about the measurability of sensations of pleasure or
satisfaction.
Figure 17.3
D. PROPERTIES OF THE UTILITY FUNCTION 463
Measures of risk aversion
The risk premium r (cost of risk) defined in [D.10] depends on the individual’s
attitude to risk and the prospect she is confronted with. It will be larger the greater
is the risk and the more risk averse the individual. We can make this explicit by
a method of approximating the risk premium which shows how it depends on a
measure of risk and on a measure of the individual’s attitude to risk.
Suppose that an individual has a certain income D. We offer her a bet which pays
off zs in state s where the probability of state s is πs. Assume that the bet is fair:
Ezs Σπszs 0 and all the payoffs zs (negative or positive) are small. Her income in
state s with the bet is ys D zs. Her expected income if she accepts the bet is
unchanged since Σπsys Σπs(D zs) D Σπszs D but her expected utility will
be reduced if she is risk averse. The risk premium for the bet is defined, as in the twostate
case in [D.10], by
v( D r) Σπsv( ys) Σπsv(D zs) [D.17]
We can approximate v( D zs) by a second-order Taylor series expansion about D:
v(ys) v(D zs) v(D) v′(D)zs 1–2 v″(D)z2s
[D.18]
Similarly, we can approximate v(D r) by a first-order expansion
v(D r) v(D) v′(D)r [D.19]
(Provided the zs are small the risk premium r will also be small and the approximations
will not be too inaccurate.) Multiplying both sides of [D.18] by the probabilities
πs and summing over all s gives
Σπsv(ys) Σπsv(D) Σπsv′(D)zs 1–2 Σπsv″(D)z2
s
v(D) Σπs v′(D) Σπszs 1–2 v″(D) Σπsz2
s [D.20]
Since the πs are probabilities, Σπs 1. The bet is fair and so Ezs Σπszs 0. The
variance of the zs is σ 2
z Σπs(zs Ezs)2 Σπsz2
s. Thus [D.20] reduces to
Σπsv(ys) v(D) 1–2 v″(D)σ 2
z [D.21]
From the definition of the risk premium in [D.17] the right-hand sides of [D.19] and
[D.21] must be approximately equal
v(D) v′(D)r v(D) 1–2 v″(D)σ 2
z
and rearranging gives an approximation to the cost of risk (the risk premium) as
r σ 2
z 1–2 A(D)σ 2
z [D.22]
[D.22] shows that the approximate risk premium depends on preferences (the first
and second derivatives of the utility function) and the variance of the bet. The first
term v″(D)/v′(D) A(D) in the approximation is known as the Pratt–Arrow coefficient
of absolute risk aversion and is a useful measure of attitude to risk.
If the individual is risk averse (v″ 0) A is positive. A depends on the individual’s
income and may increase or decrease as income varies. Larger As lead to a larger risk
premium. Although A(y) appears as part of an approximation to the risk premium
its usefulness does not depend on the accuracy of the approximation. As we see in
1
2
′′
′
v
v
( )
( )
D
D
464 CHAPTER 17 • CHOICE UNDER UNCERTAINTY
the next chapter, making assumptions about A(y) enables us to predict how the
decision-taker will respond to changes in her uncertain environment even when
the risks or bets are large and [D.22] would not be a useful approximation.
Risk aversion is related to the concavity of the utility function but using v″ as a
measure of attitudes to risk is only valid if we are interested in very crude distinctions,
i.e. between risk aversion (v″ 0), risk neutrality (v″ 0) and risk preference
(v″ 0). As the discussion of the permissible ways in which we can numerically
represent preferences in risky situations showed, the utility function for a decisionmaker
with given preferences is unique only up to positive linear transformations.
Statements about v″ being negative, positive or zero convey information about
preferences but statements about v″ being larger or smaller at some given level of
income do not. We could make the second derivative of the utility function as large
or small as we like by choosing some other, equivalent representation of preferences.
If v(y) represents preferences so will g(y) a bv(y) if b 0 and so g″(y) bv″(y)
could be made greater or less than v″ by suitable choice of b. Part of the usefulness
of the coefficient of absolute risk aversion is that it is not affected by such linear
transformations. It has the same value whatever acceptable numerical representation
of preferences is used. If g a bv(y) then
[D.23]
The coefficient of absolute risk aversion conveys information about the decisiontaker’s
preferences (i.e. attitude to risk), not about particular numerical representations
of them. Only differences in attitude to risk (as income varies or across
different decision-takers) will affect A.
A(y) was introduced in deriving an approximation to the cost of risk r. r is
measured in monetary units (income) and is the absolute amount of money the
decision-maker would give up rather than face a risky prospect. It is often useful to
have a measure of the risk premium or cost of risk which does not depend on the
monetary units. One obvious measure is the relative cost of risk or relative risk
premium which is the risk premium as a proportion of mean income: r/D. We can
derive an approximation for the relative risk premium which relates it to a measure
of the individual’s attitude to risk and a measure of the riskiness of the prospect.
(We leave the steps in the argument to the exercises since they are very similar to
those for the absolute risk premium.) The relative or proportional risk premium is
approximately
[D.24]
where σz/D is the coefficient of variation for the prospect and R(y) v″(y)y/v′(y) is
the Pratt–Arrow coefficient of relative risk aversion. Note that R(y) is the negative of the
elasticity of marginal utility of income and measures the responsiveness of marginal
utility to changes in income in a way which is independent of the particular utility
function used to represent preferences and of the units in which income is measured.
R(y) is useful in circumstances in which [D.24] would not be a sensible approximation
and it is not affected by the particular utility function used to represent
preferences. Since R Ay assumptions about R frequently place restrictions on A and
vice versa.
r v
v