Jehle reny 2.4 uncertainty
2.4 UNCERTAINTY
Until now, we have assumed that decision makers act in a world of absolute certainty. The
consumer knows the prices of all commodities and knows that any feasible consumption
bundle can be obtained with certainty. Clearly, economic agents in the real world cannot
always operate under such pleasant conditions. Many economic decisions contain some
element of uncertainty. When buying a car, for example, the consumer must consider the future price of petrol, expenditure on repairs, and the resale value of the car several years
later – none of which is known with certainty at the time of the decision. Decisions like this
involve uncertainty about the outcome of the choice that is made. Whereas the decision
maker may know the probabilities of different possible outcomes, the final result of the
decision cannot be known until it occurs.
At first glance, uncertainty may seem an intractable problem, yet economic theory
has much to contribute. The principal analytical approach to uncertainty is based on the
pathbreaking work of von Neumann and Morgenstern (1944).
2.4.1 PREFERENCES
Earlier in the text, the consumer was assumed to have a preference relation over all consumption
bundles x in a consumption set X. To allow for uncertainty we need only shift
perspective slightly. We will maintain the notion of a preference relation but, instead of
consumption bundles, the individual will be assumed to have a preference relation over
gambles.
To formalise this, let A = {a1, . . . , an} denote a finite set of outcomes. The ai’s
might well be consumption bundles, amounts of money (positive or negative), or anything
at all. The main point is that the ai’s themselves involve no uncertainty. On the other hand,
we shall use the set A as the basis for creating gambles.
For example, let A = {1,−1}, where 1 is the outcome ‘win one dollar’, and −1 is
the outcome ‘lose one dollar’. Suppose that you have entered into the following bet with
a friend. If the toss of a fair coin comes up heads, she pays you one dollar, and you pay
her one dollar if it comes up tails. From your point of view, this gamble will result in one
of the two outcomes in A: 1 (win a dollar) or −1 (lose a dollar), and each of these occurs
with a probability of one-half because the coin is fair.
More generally, a simple gamble assigns a probability, pi, to each of the outcomes
ai, in A. Of course, because the pi’s are probabilities, they must be non-negative, and
because the gamble must result in some outcome in A, the pi’s must sum to one.We denote
this simple gamble by (p1 ◦ a1, . . . , pn ◦ an). We define the set of simple gambles GS as
follows.
When one or more of the pi’s is zero, we shall drop those components from the
expression when it is convenient to do so. For example, the simple gamble (α ◦ a1, 0 ◦
a2, . . . , 0 ◦ an−1, (1 − α) ◦ an) would be written as (α ◦ a1, (1 − α) ◦ an). Note that GS
contains A because for each i, (1 ◦ ai), the gamble yielding ai with probability one, is in
GS. To simplify the notation further, we shall write ai instead of (1 ◦ ai) to denote this
gamble yielding outcome ai with certainty.
Returning to our coin-tossing example where A = {1,−1}, each individual, then,
was faced with the simple gamble (12
◦ 1, 12
◦ −1). Of course, not all gambles are simple.
For example, it is quite common for state lotteries to give as prizes tickets for the next
lottery! Gambles whose prizes are themselves gambles are called compound gambles.
Note that there is no limit to the level of compounding that a compound gamble
might involve. Indeed, the example of the state lottery is a particularly extreme case in
point. Because each state lottery ticket might result in another lottery ticket as a prize,
each ticket involves infinitely many levels of compounding. That is, by continuing to win
lottery tickets as prizes, it can take any number of plays of the state lottery before the
outcome of your original ticket is realised.
For simplicity only, we shall rule out infinitely layered compound gambles like the
state lottery. The compound gambles we shall consider must result in an outcome in A after
finitely many randomisations.
Let G then, denote the set of all gambles, both simple and compound. Although it is
possible to give a more formal description of the set of compound gambles, and therefore
of G, for our purposes this is not necessary. Quite simply, a gamble can be viewed as a
lottery ticket, which itself might result in one of a number of other (perhaps quite distinct)
lottery tickets, and so on. But ultimately, after finitely many lotteries have been played,
some outcome in A must result. So, if g is any gamble in G, then g = (p1 ◦ g1, . . . , pk ◦ gk),
for some k ≥ 1 and some gambles gi ∈ G, where the gi’s might be compound gambles,
simple gambles, or outcomes. Of course, the pi’s must be non-negative and they must sum
to one.2
The objects of choice in decision making under uncertainty are gambles. Analogous
to the case of consumer theory, we shall suppose that the decision maker has preferences,
, over the set of gambles, G. We shall proceed by positing a number of axioms, called
axioms of choice under uncertainty, for the decision maker’s preference relation, . As
before, ∼ and denote the indifference and strict preference relations induced by . The
first few axioms will look very familiar and so require no discussion.
AXIOM 1: Completeness. For any two gambles, g and g in G, either g g , or g g.
AXIOM 2: Transitivity. For any three gambles g, g , g in G, if g g and g g , then
g g .
Because each ai in A is represented in G as a degenerate gamble, Axioms G1
and G2 imply in particular that the finitely many elements of A are ordered by . (See Exercise 2.16.) So let us assume without loss of generality that the elements of A have
been indexed so that a1 a2 · · · an.
It seems plausible then that no gamble is better than that giving a1 with certainty, and
no gamble is worse than that giving an with certainty (although we are not directly assuming
this). That is, for any gamble g, it seems plausible that (α ◦ a1, (1 − α) ◦ an) g,
when α = 1, and g (α ◦ a1, (1 − α) ◦ an) when α = 0. The next axiom says that if
indifference does not hold at either extreme, then it must hold for some intermediate
value of α.
AXIOM 3: Continuity. For any gamble g in G, there is some probability, α ∈ [0, 1], such
that g ∼ (α ◦ a1, (1 − α) ◦ an).
Axiom G3 has implications that at first glance might appear unreasonable. For example,
suppose that A = {$1000, $10, ‘death’}. For most of us, these outcomes are strictly
ordered as follows: $1000 $10 ‘death’. Now consider the simple gamble giving $10
with certainty. According to G3, there must be some probability α rendering the gamble
(α◦ $1000, (1 − α) ◦ ‘death’) equally attractive as $10. Thus, if there is no probability α
at which you would find $10 with certainty and the gamble (α◦ $1000, (1 − α) ◦ ‘death’)
equally attractive, then your preferences over gambles do not satisfy G3.
Is, then, Axiom G3 an unduly strong restriction to impose on preferences? Do not
be too hasty in reaching a conclusion. If you would drive across town to collect $1000 –
an action involving some positive, if tiny, probability of death – rather than accept a $10
payment to stay at home, you would be declaring your preference for the gamble over the
small sum with certainty. Presumably, we could increase the probability of a fatal traffic
accident until you were just indifferent between the two choices. When that is the case, we
will have found the indifference probability whose existence G3 assumes.
The next axiom expresses the idea that if two simple gambles each potentially yield
only the best and worst outcomes, then that which yields the best outcome with the higher
probability is preferred.
AXIOM 4: Monotonicity. For all probabilities α, β ∈ [0, 1],
(α ◦ a1, (1 − α) ◦ an) (β ◦ a1, (1 − β) ◦ an)
if and only if α ≥ β.
Note that monotonicity implies a1 an, and so the case in which the decision maker
is indifferent among all the outcomes in A is ruled out.
Although most people will usually prefer gambles that give better outcomes higher
probability, as monotonicity requires, it need not always be so. For example, to a safari
hunter, death may be the worst outcome of an outing, yet the possibility of death adds to
the excitement of the venture. An outing with a small probability of death would then be
preferred to one with zero probability, a clear violation of monotonicity.
The next axiom states that the decision maker is indifferent between one gamble and
another if he is indifferent between their realisations, and their realisations occur with the
same probabilities.
G, and if hi ∼ gi for every i, then h ∼ g.
Together with G1, Axiom G5 implies that when the agent is indifferent between
two gambles he must be indifferent between all convex combinations of them. That is,
if g ∼ h, then because by G1 g ∼ g, Axiom G5 implies (α ◦ g, (1 − α) ◦ h) ∼ (α ◦ g,
(1 − α) ◦ g) = g.
Our next, and final, axiom states that when considering a particular gamble, the
decision maker cares only about the effective probabilities that gamble assigns to each
outcome in A. This warrants a bit of discussion.
For example, suppose that A = {a1, a2}. Consider the compound gamble yielding
outcome a1 with probability α, and yielding a lottery ticket with probability 1 − α, where
the lottery ticket itself is a simple gamble. It yields the outcome a1 with probability β and
the outcome a2 with probability 1 − β.
Now, taken all together, what is the effective probability that the outcome in fact
will be a1? Well, a1 can result in two mutually exclusive ways, namely, as an immediate
result of the compound gamble, or as a result of the lottery ticket. The probability of the
first is clearly α. The probability of the second is (1 − α)β, because to obtain a1 via the
lottery ticket, a1 must not have been the immediate result of the compound gamble and
it must have been the result of the lottery ticket. So, all together, the probability that the
outcome is a1 is the sum, namely, α + (1 − α)β, because the two different ways that a1
can arise are mutually exclusive. Similarly, the effective probability that the outcome is a2,
is (1 − α)(1 − β).
To say that the decision maker cares only about the effective probabilities on the
ai’s when considering the preceding compound gamble is to say that the decision maker
is indifferent between the compound gamble and the simple gamble (α + (1 − α)β ◦
a1, (1 − α)(1 − β) ◦ a2) that it induces.
Clearly, one can derive the (unique) effective probabilities on the ai’s induced by any
compound gamble in a similar way.We shall not spell out the procedure explicitly here, as
it is, at least conceptually, straightforward.
For any gamble g ∈ G, if pi denotes the effective probability assigned to ai by g, then
we say that g induces the simple gamble (p1 ◦ a1, . . . , pn ◦ an) ∈ GS. We emphasise that
every g ∈ G induces a unique simple gamble. Our final axiom is then as follows.3
AXIOM 6: Reduction to Simple Gambles. For any gamble g ∈ G, if (p1 ◦ a1, . . . , pn ◦ an)
is the simple gamble induced by g, then (p1 ◦ a1, . . . , pn ◦ an) ∼ g.
Note that by G6 (and transitivity G2), an individual’s preferences over all
gambles – compound or otherwise – are completely determined by his preferences over
simple gambles.
As plausible as G6 may seem, it does restrict the domain of our analysis. In particular,
this would not be an appropriate assumption to maintain if one wished to model the
behaviour of vacationers in Las Vegas. They would probably not be indifferent between playing the slot machines many times during their stay and taking the single once and
for all gamble defined by the effective probabilities over winnings and losses. On the other
hand, many decisions under uncertainty are undertaken outside of Las Vegas, and for many
of these, Axiom G6 is reasonable.
2.4.2 VON NEUMANN-MORGENSTERN UTILITY
Now that we have characterised the axioms preferences over gambles must obey, we once
again ask whether we can represent such preferences with a continuous, real-valued function.
The answer to that question is yes, which should come as no surprise. We know from
our study of preferences under certainty that, here, Axioms G1, G2, and some kind of continuity
assumption should be sufficient to ensure the existence of a continuous function
representing . On the other hand, we have made assumptions in addition to G1, G2, and
continuity. One might then expect to derive a utility representation that is more than just
continuous. Indeed, we shall show that not only can we obtain a continuous utility function
representing on G, we can obtain one that is linear in the effective probabilities on the
outcomes.
To be more precise, suppose that u: G→R is a utility function representing on G.4
So, for every g ∈ G, u(g) denotes the utility number assigned to the gamble g. In particular,
for every i, u assigns the number u(ai) to the degenerate gamble (1 ◦ ai), in which the
outcome ai occurs with certainty. We will often refer to u(ai) as simply the utility of the
outcome ai. We are now prepared to describe the linearity property mentioned above.
DEFINITION 2.3 Expected Utility Property
The utility function u: G→R has the expected utility property if, for every g ∈ G,
u(g) =
n
i=1
piu(ai),
where (p1 ◦ a1, . . . , pn ◦ an) is the simple gamble induced by g.
Thus, to say that u has the expected utility property is to say that it assigns to each
gamble the expected value of the utilities that might result, where each utility that might
result is assigned its effective probability.5 Of course, the effective probability that g yields
utility u(ai) is simply the effective probability that it yields outcome ai, namely, pi.
Note that if u has the expected utility property, and if gs = (p1 ◦ a1, . . . , pn ◦ an) is
a simple gamble, then because the simple gamble induced by gs is gs itself, we must have
u(p1 ◦ a1, . . . , pn ◦ an) =
n
i=1
piu(ai), ∀ probability vectors (p1, . . . , pn).
Consequently, the function u is completely determined on all of G by the values it assumes
on the finite set of outcomes, A.
If an individual’s preferences are represented by a utility function with the expected
utility property, and if that person always chooses his most preferred alternative available,
then that individual will choose one gamble over another if and only if the expected utility
of the one exceeds that of the other. Consequently, such an individual is an expected utility
maximiser.
Any such function as this will have some obvious analytical advantages because
the utility of any gamble will be expressible as a linear sum involving only the utility of
outcomes and their associated probabilities. Yet this is clearly a great deal to require of the
function representing , and it is unlike anything we required of ordinary utility functions
under certainty before. To help keep in mind the important distinctions between the two,
we refer to utility functions possessing the expected utility property as von Neumann-
Morgenstern (VNM) utility functions.
We now present a fundamental theorem in the theory of choice under uncertainty.
THEOREM 2.7 Existence of a VNM Utility Function on G
Just before the statement of Theorem 2.8, we stated that the class of VNM utility
representations of a single preference relation is characterised by the constancy of ratios
of utility differences. This in fact follows from Theorem 2.8, as you are asked to show in
an exercise.
Theorem 2.8 tells us that VNM utility functions are not completely unique, nor are
they entirely ordinal.We can still find an infinite number of them that will rank gambles in
precisely the same order and also possess the expected utility property. But unlike ordinary
utility functions from which we demand only an order-preserving numerical scaling, here
we must limit ourselves to transformations that multiply by a positive number and/or add
a constant term if we are to preserve the expected utility property as well. Yet the less
than complete ordinality of the VNM utility function must not tempt us into attaching
undue significance to the absolute level of a gamble’s utility, or to the difference in utility
between one gamble and another. With what little we have required of the agent’s binary
comparisons between gambles in the underlying preference relation, we still cannot use
VNM utility functions for interpersonal comparisons of well-being, nor can we measure
the ‘intensity’ with which one gamble is preferred to another.
2.4.3 RISK AVERSION
In Example 2.4 we argued that the VNM utility function we created there reflected some
desire to avoid risk. Now we are prepared to define and describe risk aversion more formally.
For that, we shall confine our attention to gambles whose outcomes consist of
different amounts of wealth. In addition, it will be helpful to take as the set of outcomes, A,
all non-negative wealth levels. Thus, A = R+. Even though the set of outcomes now contains
infinitely many elements, we continue to consider gambles giving only finitely many
outcomes a strictly positive effective probability. In particular, a simple gamble takes the
form (p1 ◦ w1, . . . , pn ◦ wn), where n is some positive integer, the wi’s are non-negative
wealth levels, and the non-negative probabilities, p1, . . . , pn, sum to 1.6 Finally, we shall
assume that the individual’s VNM utility function, u(·), is differentiable with u
(w) > 0
for all wealth levels w.
We now investigate the relationship between a VNM utility function and the agent’s
attitude towards risk. The expected value of the simple gamble g offering wi with probability
pi is given by E(g) = ni
=1 piwi. Now suppose the agent is given a choice between
accepting the gamble g on the one hand or receiving with certainty the expected value
of g on the other. If u(·) is the agent’s VNM utility function, we can evaluate these two The first of these is the VNM utility of the gamble, and the second is the VNM utility of
the gamble’s expected value. If preferences satisfy Axioms G1 to G6, we know the agent
prefers the alternative with the higher expected utility. When someone would rather receive
the expected value of a gamble with certainty than face the risk inherent in the gamble
itself, we say they are risk averse. Of course, people may exhibit a complete disregard of
risk, or even an attraction to risk, and still be consistent with Axioms G1 through G6. We
catalogue these various possibilities, and define terms precisely, in what follows.
As remarked after Definition 2.3, a VNM utility function on G is completely determined
by the values it assumes on the set of outcomes, A. Consequently, the characteristics
of an individual’s VNM utility function over the set of simple gambles alone provides a
complete description of the individual’s preferences over all gambles. Because of this, it is
enough to focus on the behaviour of u on GS to capture an individual’s attitudes towards
risk. This, and the preceding discussion, motivate the following definition.
DEFINITION 2.4 Risk Aversion, Risk Neutrality, and Risk Loving
Let u(·) be an individual’s VNM utility function for gambles over non-negative levels of
wealth. Then for the simple gamble g = (p1 ◦ w1, . . . , pn ◦ wn), the individual is said to be
1. risk averse at g if u(E(g)) > u(g),
2. risk neutral at g if u(E(g)) = u(g),
3. risk loving at g if u(E(g)) < u(g).
If for every non-degenerate7 simple gamble, g, the individual is, for example, risk averse
at g, then the individual is said simply to be risk averse (or risk averse on G for emphasis).
Similarly, an individual can be defined to be risk neutral and risk loving (on G).
Each of these attitudes toward risk is equivalent to a particular property of the VNM
utility function. In the exercises, you are asked to show that the agent is risk averse, risk
neutral, or risk loving over some subset of gambles if and only if his VNM utility function
is strictly concave, linear, or strictly convex, respectively, over the appropriate domain of
wealth. averse.
In Fig. 2.6, the individual prefers E(g) with certainty to the gamble g itself. But there
will be some amount of wealth we could offer with certainty that would make him indifferent
between accepting that wealth with certainty and facing the gamble g. We call this
amount of wealth the certainty equivalent of the gamble g. When a person is risk averse
and strictly prefers more money to less, it is easy to show that the certainty equivalent is
less than the expected value of the gamble, and you are asked to do this in the exercises.
In effect, a risk-averse person will ‘pay’ some positive amount of wealth to avoid the gamble’s
inherent risk. This willingness to pay to avoid risk is measured by the risk premium.
The certainty equivalent and the risk premium, both illustrated in Fig. 2.6, are defined in
what follows.
DEFINITION 2.5 Certainty Equivalent and Risk Premium
The certainty equivalent of any simple gamble g over wealth levels is an amount of wealth,
CE, offered with certainty, such that u(g) ≡ u(CE). The risk premium is an amount of
wealth, P, such that u(g) ≡ u(E(g) − P). Clearly, P ≡ E(g)−CE.
Many times, we not only want to know whether someone is risk averse, but also
how risk averse they are. Ideally, we would like a summary measure that allows us both
to compare the degree of risk aversion across individuals and to gauge how the degree of
risk aversion for a single individual might vary with the level of their wealth. Because risk
aversion and concavity of the VNM utility function in wealth are equivalent, the seemingly
most natural candidate for such a measure would be the second derivative, u
(w), a basic
measure of a function’s ‘curvature’. We might think that the greater the absolute value of
this derivative, the greater the degree of risk aversion.
But this will not do. Although the sign of the second derivative does tell us whether
the individual is risk averse, risk loving, or risk neutral, its size is entirely arbitrary.
Theorem 2.8 showed that VNM utility functions are unique up to affine transformations.
This means that for any given preferences, we can obtain virtually any size second derivative
we wish through multiplication of u(·) by a properly chosen positive constant. With
this and other considerations in mind, Arrow (1970) and Pratt (1964) have proposed the
following measure of risk aversion.
DEFINITION 2.6 The Arrow-Pratt Measure of Absolute Risk Aversion
The Arrow-Pratt measure of absolute risk aversion is given by Ra(w) is positive, negative, or zero as the agent is risk averse, risk loving, or risk
neutral, respectively. In addition, any positive affine transformation of utility will leave the
measure unchanged: adding a constant affects neither the numerator nor the denominator;
multiplication by a positive constant affects both numerator and denominator but leaves
their ratio unchanged.
To demonstrate the effectiveness of the Arrow-Pratt measure of risk aversion, we
now show that consumers with larger Arrow-Pratt measures are indeed more risk averse in
a behaviourally significant respect: they have lower certainty equivalents and are willing
to accept fewer gambles.
To see this, suppose there are two consumers, 1 and 2, and that consumer 1 has VNM
utility function u(w), and consumer 2’s VNM utility function is v(w). Wealth, w, can take
on any non-negative number. Let us now suppose that at every wealth level, w, consumer
1’s Arrow-Pratt measure of risk aversion is larger than consumer 2’s. That is, where the inequality, called Jensen’s inequality, follows because h is strictly concave,
and the final two equalities follow from (2.15) and (2.13), respectively. Consequently,
u(wˆ 1) < u(wˆ 2), so that because u is strictly increasing, wˆ 1 < wˆ 2 as desired.
We may conclude that consumer 1’s certainty equivalent for any given gamble is
lower than 2’s. And from this it easily follows that if consumers 1 and 2 have the same
initial wealth, then consumer 2 (the one with the globally lower Arrow-Pratt measure)
will accept any gamble that consumer 1 will accept. (Convince yourself of this.) That is,
consumer 1 is willing to accept fewer gambles than consumer 2.
Finally, note that in passing, we have also shown that (2.12) implies that consumer
1’s VNM utility function is more concave than consumer 2’s in the sense that (once again
putting x = v(w) in (2.13))
u(w) = h(v(w)) for all w ≥ 0, (2.16)
where, as you recall, h is a strictly concave function. Thus, according to (2.16), u is a
‘concavification’ of v. This is yet another (equivalent) expression of the idea that consumer
1 is more risk averse than consumer 2.
Ra(w) is only a local measure of risk aversion, so it need not be the same at every
level of wealth. Indeed, one expects that attitudes toward risk, and so the Arrow-Pratt measure,
will ordinarily vary with wealth, and vary in ‘sensible’ ways. Arrow has proposed a
simple classification of VNM utility functions (or utility function segments) according to
how Ra(w) varies with wealth. Quite straightforwardly, we say that a VNM utility function
displays constant, decreasing, or increasing absolute risk aversion over some domain
of wealth if, over that interval, Ra(w) remains constant, decreases, or increases with an
increase in wealth, respectively.
Decreasing absolute risk aversion (DARA) is generally a sensible restriction to
impose. Under constant absolute risk aversion, there would be no greater willingness
to accept a small gamble at higher levels of wealth, and under increasing absolute risk
aversion, we have rather perverse behaviour: the greater the wealth, the more averse
one becomes to accepting the same small gamble. DARA imposes the more plausible
restriction that the individual be less averse to taking small risks at higher levels of wealth.
Until now, we have assumed that decision makers act in a world of absolute certainty. The
consumer knows the prices of all commodities and knows that any feasible consumption
bundle can be obtained with certainty. Clearly, economic agents in the real world cannot
always operate under such pleasant conditions. Many economic decisions contain some
element of uncertainty. When buying a car, for example, the consumer must consider the future price of petrol, expenditure on repairs, and the resale value of the car several years
later – none of which is known with certainty at the time of the decision. Decisions like this
involve uncertainty about the outcome of the choice that is made. Whereas the decision
maker may know the probabilities of different possible outcomes, the final result of the
decision cannot be known until it occurs.
At first glance, uncertainty may seem an intractable problem, yet economic theory
has much to contribute. The principal analytical approach to uncertainty is based on the
pathbreaking work of von Neumann and Morgenstern (1944).
2.4.1 PREFERENCES
Earlier in the text, the consumer was assumed to have a preference relation over all consumption
bundles x in a consumption set X. To allow for uncertainty we need only shift
perspective slightly. We will maintain the notion of a preference relation but, instead of
consumption bundles, the individual will be assumed to have a preference relation over
gambles.
To formalise this, let A = {a1, . . . , an} denote a finite set of outcomes. The ai’s
might well be consumption bundles, amounts of money (positive or negative), or anything
at all. The main point is that the ai’s themselves involve no uncertainty. On the other hand,
we shall use the set A as the basis for creating gambles.
For example, let A = {1,−1}, where 1 is the outcome ‘win one dollar’, and −1 is
the outcome ‘lose one dollar’. Suppose that you have entered into the following bet with
a friend. If the toss of a fair coin comes up heads, she pays you one dollar, and you pay
her one dollar if it comes up tails. From your point of view, this gamble will result in one
of the two outcomes in A: 1 (win a dollar) or −1 (lose a dollar), and each of these occurs
with a probability of one-half because the coin is fair.
More generally, a simple gamble assigns a probability, pi, to each of the outcomes
ai, in A. Of course, because the pi’s are probabilities, they must be non-negative, and
because the gamble must result in some outcome in A, the pi’s must sum to one.We denote
this simple gamble by (p1 ◦ a1, . . . , pn ◦ an). We define the set of simple gambles GS as
follows.
When one or more of the pi’s is zero, we shall drop those components from the
expression when it is convenient to do so. For example, the simple gamble (α ◦ a1, 0 ◦
a2, . . . , 0 ◦ an−1, (1 − α) ◦ an) would be written as (α ◦ a1, (1 − α) ◦ an). Note that GS
contains A because for each i, (1 ◦ ai), the gamble yielding ai with probability one, is in
GS. To simplify the notation further, we shall write ai instead of (1 ◦ ai) to denote this
gamble yielding outcome ai with certainty.
Returning to our coin-tossing example where A = {1,−1}, each individual, then,
was faced with the simple gamble (12
◦ 1, 12
◦ −1). Of course, not all gambles are simple.
For example, it is quite common for state lotteries to give as prizes tickets for the next
lottery! Gambles whose prizes are themselves gambles are called compound gambles.
Note that there is no limit to the level of compounding that a compound gamble
might involve. Indeed, the example of the state lottery is a particularly extreme case in
point. Because each state lottery ticket might result in another lottery ticket as a prize,
each ticket involves infinitely many levels of compounding. That is, by continuing to win
lottery tickets as prizes, it can take any number of plays of the state lottery before the
outcome of your original ticket is realised.
For simplicity only, we shall rule out infinitely layered compound gambles like the
state lottery. The compound gambles we shall consider must result in an outcome in A after
finitely many randomisations.
Let G then, denote the set of all gambles, both simple and compound. Although it is
possible to give a more formal description of the set of compound gambles, and therefore
of G, for our purposes this is not necessary. Quite simply, a gamble can be viewed as a
lottery ticket, which itself might result in one of a number of other (perhaps quite distinct)
lottery tickets, and so on. But ultimately, after finitely many lotteries have been played,
some outcome in A must result. So, if g is any gamble in G, then g = (p1 ◦ g1, . . . , pk ◦ gk),
for some k ≥ 1 and some gambles gi ∈ G, where the gi’s might be compound gambles,
simple gambles, or outcomes. Of course, the pi’s must be non-negative and they must sum
to one.2
The objects of choice in decision making under uncertainty are gambles. Analogous
to the case of consumer theory, we shall suppose that the decision maker has preferences,
, over the set of gambles, G. We shall proceed by positing a number of axioms, called
axioms of choice under uncertainty, for the decision maker’s preference relation, . As
before, ∼ and denote the indifference and strict preference relations induced by . The
first few axioms will look very familiar and so require no discussion.
AXIOM 1: Completeness. For any two gambles, g and g in G, either g g , or g g.
AXIOM 2: Transitivity. For any three gambles g, g , g in G, if g g and g g , then
g g .
Because each ai in A is represented in G as a degenerate gamble, Axioms G1
and G2 imply in particular that the finitely many elements of A are ordered by . (See Exercise 2.16.) So let us assume without loss of generality that the elements of A have
been indexed so that a1 a2 · · · an.
It seems plausible then that no gamble is better than that giving a1 with certainty, and
no gamble is worse than that giving an with certainty (although we are not directly assuming
this). That is, for any gamble g, it seems plausible that (α ◦ a1, (1 − α) ◦ an) g,
when α = 1, and g (α ◦ a1, (1 − α) ◦ an) when α = 0. The next axiom says that if
indifference does not hold at either extreme, then it must hold for some intermediate
value of α.
AXIOM 3: Continuity. For any gamble g in G, there is some probability, α ∈ [0, 1], such
that g ∼ (α ◦ a1, (1 − α) ◦ an).
Axiom G3 has implications that at first glance might appear unreasonable. For example,
suppose that A = {$1000, $10, ‘death’}. For most of us, these outcomes are strictly
ordered as follows: $1000 $10 ‘death’. Now consider the simple gamble giving $10
with certainty. According to G3, there must be some probability α rendering the gamble
(α◦ $1000, (1 − α) ◦ ‘death’) equally attractive as $10. Thus, if there is no probability α
at which you would find $10 with certainty and the gamble (α◦ $1000, (1 − α) ◦ ‘death’)
equally attractive, then your preferences over gambles do not satisfy G3.
Is, then, Axiom G3 an unduly strong restriction to impose on preferences? Do not
be too hasty in reaching a conclusion. If you would drive across town to collect $1000 –
an action involving some positive, if tiny, probability of death – rather than accept a $10
payment to stay at home, you would be declaring your preference for the gamble over the
small sum with certainty. Presumably, we could increase the probability of a fatal traffic
accident until you were just indifferent between the two choices. When that is the case, we
will have found the indifference probability whose existence G3 assumes.
The next axiom expresses the idea that if two simple gambles each potentially yield
only the best and worst outcomes, then that which yields the best outcome with the higher
probability is preferred.
AXIOM 4: Monotonicity. For all probabilities α, β ∈ [0, 1],
(α ◦ a1, (1 − α) ◦ an) (β ◦ a1, (1 − β) ◦ an)
if and only if α ≥ β.
Note that monotonicity implies a1 an, and so the case in which the decision maker
is indifferent among all the outcomes in A is ruled out.
Although most people will usually prefer gambles that give better outcomes higher
probability, as monotonicity requires, it need not always be so. For example, to a safari
hunter, death may be the worst outcome of an outing, yet the possibility of death adds to
the excitement of the venture. An outing with a small probability of death would then be
preferred to one with zero probability, a clear violation of monotonicity.
The next axiom states that the decision maker is indifferent between one gamble and
another if he is indifferent between their realisations, and their realisations occur with the
same probabilities.
G, and if hi ∼ gi for every i, then h ∼ g.
Together with G1, Axiom G5 implies that when the agent is indifferent between
two gambles he must be indifferent between all convex combinations of them. That is,
if g ∼ h, then because by G1 g ∼ g, Axiom G5 implies (α ◦ g, (1 − α) ◦ h) ∼ (α ◦ g,
(1 − α) ◦ g) = g.
Our next, and final, axiom states that when considering a particular gamble, the
decision maker cares only about the effective probabilities that gamble assigns to each
outcome in A. This warrants a bit of discussion.
For example, suppose that A = {a1, a2}. Consider the compound gamble yielding
outcome a1 with probability α, and yielding a lottery ticket with probability 1 − α, where
the lottery ticket itself is a simple gamble. It yields the outcome a1 with probability β and
the outcome a2 with probability 1 − β.
Now, taken all together, what is the effective probability that the outcome in fact
will be a1? Well, a1 can result in two mutually exclusive ways, namely, as an immediate
result of the compound gamble, or as a result of the lottery ticket. The probability of the
first is clearly α. The probability of the second is (1 − α)β, because to obtain a1 via the
lottery ticket, a1 must not have been the immediate result of the compound gamble and
it must have been the result of the lottery ticket. So, all together, the probability that the
outcome is a1 is the sum, namely, α + (1 − α)β, because the two different ways that a1
can arise are mutually exclusive. Similarly, the effective probability that the outcome is a2,
is (1 − α)(1 − β).
To say that the decision maker cares only about the effective probabilities on the
ai’s when considering the preceding compound gamble is to say that the decision maker
is indifferent between the compound gamble and the simple gamble (α + (1 − α)β ◦
a1, (1 − α)(1 − β) ◦ a2) that it induces.
Clearly, one can derive the (unique) effective probabilities on the ai’s induced by any
compound gamble in a similar way.We shall not spell out the procedure explicitly here, as
it is, at least conceptually, straightforward.
For any gamble g ∈ G, if pi denotes the effective probability assigned to ai by g, then
we say that g induces the simple gamble (p1 ◦ a1, . . . , pn ◦ an) ∈ GS. We emphasise that
every g ∈ G induces a unique simple gamble. Our final axiom is then as follows.3
AXIOM 6: Reduction to Simple Gambles. For any gamble g ∈ G, if (p1 ◦ a1, . . . , pn ◦ an)
is the simple gamble induced by g, then (p1 ◦ a1, . . . , pn ◦ an) ∼ g.
Note that by G6 (and transitivity G2), an individual’s preferences over all
gambles – compound or otherwise – are completely determined by his preferences over
simple gambles.
As plausible as G6 may seem, it does restrict the domain of our analysis. In particular,
this would not be an appropriate assumption to maintain if one wished to model the
behaviour of vacationers in Las Vegas. They would probably not be indifferent between playing the slot machines many times during their stay and taking the single once and
for all gamble defined by the effective probabilities over winnings and losses. On the other
hand, many decisions under uncertainty are undertaken outside of Las Vegas, and for many
of these, Axiom G6 is reasonable.
2.4.2 VON NEUMANN-MORGENSTERN UTILITY
Now that we have characterised the axioms preferences over gambles must obey, we once
again ask whether we can represent such preferences with a continuous, real-valued function.
The answer to that question is yes, which should come as no surprise. We know from
our study of preferences under certainty that, here, Axioms G1, G2, and some kind of continuity
assumption should be sufficient to ensure the existence of a continuous function
representing . On the other hand, we have made assumptions in addition to G1, G2, and
continuity. One might then expect to derive a utility representation that is more than just
continuous. Indeed, we shall show that not only can we obtain a continuous utility function
representing on G, we can obtain one that is linear in the effective probabilities on the
outcomes.
To be more precise, suppose that u: G→R is a utility function representing on G.4
So, for every g ∈ G, u(g) denotes the utility number assigned to the gamble g. In particular,
for every i, u assigns the number u(ai) to the degenerate gamble (1 ◦ ai), in which the
outcome ai occurs with certainty. We will often refer to u(ai) as simply the utility of the
outcome ai. We are now prepared to describe the linearity property mentioned above.
DEFINITION 2.3 Expected Utility Property
The utility function u: G→R has the expected utility property if, for every g ∈ G,
u(g) =
n
i=1
piu(ai),
where (p1 ◦ a1, . . . , pn ◦ an) is the simple gamble induced by g.
Thus, to say that u has the expected utility property is to say that it assigns to each
gamble the expected value of the utilities that might result, where each utility that might
result is assigned its effective probability.5 Of course, the effective probability that g yields
utility u(ai) is simply the effective probability that it yields outcome ai, namely, pi.
Note that if u has the expected utility property, and if gs = (p1 ◦ a1, . . . , pn ◦ an) is
a simple gamble, then because the simple gamble induced by gs is gs itself, we must have
u(p1 ◦ a1, . . . , pn ◦ an) =
n
i=1
piu(ai), ∀ probability vectors (p1, . . . , pn).
Consequently, the function u is completely determined on all of G by the values it assumes
on the finite set of outcomes, A.
If an individual’s preferences are represented by a utility function with the expected
utility property, and if that person always chooses his most preferred alternative available,
then that individual will choose one gamble over another if and only if the expected utility
of the one exceeds that of the other. Consequently, such an individual is an expected utility
maximiser.
Any such function as this will have some obvious analytical advantages because
the utility of any gamble will be expressible as a linear sum involving only the utility of
outcomes and their associated probabilities. Yet this is clearly a great deal to require of the
function representing , and it is unlike anything we required of ordinary utility functions
under certainty before. To help keep in mind the important distinctions between the two,
we refer to utility functions possessing the expected utility property as von Neumann-
Morgenstern (VNM) utility functions.
We now present a fundamental theorem in the theory of choice under uncertainty.
THEOREM 2.7 Existence of a VNM Utility Function on G
Just before the statement of Theorem 2.8, we stated that the class of VNM utility
representations of a single preference relation is characterised by the constancy of ratios
of utility differences. This in fact follows from Theorem 2.8, as you are asked to show in
an exercise.
Theorem 2.8 tells us that VNM utility functions are not completely unique, nor are
they entirely ordinal.We can still find an infinite number of them that will rank gambles in
precisely the same order and also possess the expected utility property. But unlike ordinary
utility functions from which we demand only an order-preserving numerical scaling, here
we must limit ourselves to transformations that multiply by a positive number and/or add
a constant term if we are to preserve the expected utility property as well. Yet the less
than complete ordinality of the VNM utility function must not tempt us into attaching
undue significance to the absolute level of a gamble’s utility, or to the difference in utility
between one gamble and another. With what little we have required of the agent’s binary
comparisons between gambles in the underlying preference relation, we still cannot use
VNM utility functions for interpersonal comparisons of well-being, nor can we measure
the ‘intensity’ with which one gamble is preferred to another.
2.4.3 RISK AVERSION
In Example 2.4 we argued that the VNM utility function we created there reflected some
desire to avoid risk. Now we are prepared to define and describe risk aversion more formally.
For that, we shall confine our attention to gambles whose outcomes consist of
different amounts of wealth. In addition, it will be helpful to take as the set of outcomes, A,
all non-negative wealth levels. Thus, A = R+. Even though the set of outcomes now contains
infinitely many elements, we continue to consider gambles giving only finitely many
outcomes a strictly positive effective probability. In particular, a simple gamble takes the
form (p1 ◦ w1, . . . , pn ◦ wn), where n is some positive integer, the wi’s are non-negative
wealth levels, and the non-negative probabilities, p1, . . . , pn, sum to 1.6 Finally, we shall
assume that the individual’s VNM utility function, u(·), is differentiable with u
(w) > 0
for all wealth levels w.
We now investigate the relationship between a VNM utility function and the agent’s
attitude towards risk. The expected value of the simple gamble g offering wi with probability
pi is given by E(g) = ni
=1 piwi. Now suppose the agent is given a choice between
accepting the gamble g on the one hand or receiving with certainty the expected value
of g on the other. If u(·) is the agent’s VNM utility function, we can evaluate these two The first of these is the VNM utility of the gamble, and the second is the VNM utility of
the gamble’s expected value. If preferences satisfy Axioms G1 to G6, we know the agent
prefers the alternative with the higher expected utility. When someone would rather receive
the expected value of a gamble with certainty than face the risk inherent in the gamble
itself, we say they are risk averse. Of course, people may exhibit a complete disregard of
risk, or even an attraction to risk, and still be consistent with Axioms G1 through G6. We
catalogue these various possibilities, and define terms precisely, in what follows.
As remarked after Definition 2.3, a VNM utility function on G is completely determined
by the values it assumes on the set of outcomes, A. Consequently, the characteristics
of an individual’s VNM utility function over the set of simple gambles alone provides a
complete description of the individual’s preferences over all gambles. Because of this, it is
enough to focus on the behaviour of u on GS to capture an individual’s attitudes towards
risk. This, and the preceding discussion, motivate the following definition.
DEFINITION 2.4 Risk Aversion, Risk Neutrality, and Risk Loving
Let u(·) be an individual’s VNM utility function for gambles over non-negative levels of
wealth. Then for the simple gamble g = (p1 ◦ w1, . . . , pn ◦ wn), the individual is said to be
1. risk averse at g if u(E(g)) > u(g),
2. risk neutral at g if u(E(g)) = u(g),
3. risk loving at g if u(E(g)) < u(g).
If for every non-degenerate7 simple gamble, g, the individual is, for example, risk averse
at g, then the individual is said simply to be risk averse (or risk averse on G for emphasis).
Similarly, an individual can be defined to be risk neutral and risk loving (on G).
Each of these attitudes toward risk is equivalent to a particular property of the VNM
utility function. In the exercises, you are asked to show that the agent is risk averse, risk
neutral, or risk loving over some subset of gambles if and only if his VNM utility function
is strictly concave, linear, or strictly convex, respectively, over the appropriate domain of
wealth. averse.
In Fig. 2.6, the individual prefers E(g) with certainty to the gamble g itself. But there
will be some amount of wealth we could offer with certainty that would make him indifferent
between accepting that wealth with certainty and facing the gamble g. We call this
amount of wealth the certainty equivalent of the gamble g. When a person is risk averse
and strictly prefers more money to less, it is easy to show that the certainty equivalent is
less than the expected value of the gamble, and you are asked to do this in the exercises.
In effect, a risk-averse person will ‘pay’ some positive amount of wealth to avoid the gamble’s
inherent risk. This willingness to pay to avoid risk is measured by the risk premium.
The certainty equivalent and the risk premium, both illustrated in Fig. 2.6, are defined in
what follows.
DEFINITION 2.5 Certainty Equivalent and Risk Premium
The certainty equivalent of any simple gamble g over wealth levels is an amount of wealth,
CE, offered with certainty, such that u(g) ≡ u(CE). The risk premium is an amount of
wealth, P, such that u(g) ≡ u(E(g) − P). Clearly, P ≡ E(g)−CE.
Many times, we not only want to know whether someone is risk averse, but also
how risk averse they are. Ideally, we would like a summary measure that allows us both
to compare the degree of risk aversion across individuals and to gauge how the degree of
risk aversion for a single individual might vary with the level of their wealth. Because risk
aversion and concavity of the VNM utility function in wealth are equivalent, the seemingly
most natural candidate for such a measure would be the second derivative, u
(w), a basic
measure of a function’s ‘curvature’. We might think that the greater the absolute value of
this derivative, the greater the degree of risk aversion.
But this will not do. Although the sign of the second derivative does tell us whether
the individual is risk averse, risk loving, or risk neutral, its size is entirely arbitrary.
Theorem 2.8 showed that VNM utility functions are unique up to affine transformations.
This means that for any given preferences, we can obtain virtually any size second derivative
we wish through multiplication of u(·) by a properly chosen positive constant. With
this and other considerations in mind, Arrow (1970) and Pratt (1964) have proposed the
following measure of risk aversion.
DEFINITION 2.6 The Arrow-Pratt Measure of Absolute Risk Aversion
The Arrow-Pratt measure of absolute risk aversion is given by Ra(w) is positive, negative, or zero as the agent is risk averse, risk loving, or risk
neutral, respectively. In addition, any positive affine transformation of utility will leave the
measure unchanged: adding a constant affects neither the numerator nor the denominator;
multiplication by a positive constant affects both numerator and denominator but leaves
their ratio unchanged.
To demonstrate the effectiveness of the Arrow-Pratt measure of risk aversion, we
now show that consumers with larger Arrow-Pratt measures are indeed more risk averse in
a behaviourally significant respect: they have lower certainty equivalents and are willing
to accept fewer gambles.
To see this, suppose there are two consumers, 1 and 2, and that consumer 1 has VNM
utility function u(w), and consumer 2’s VNM utility function is v(w). Wealth, w, can take
on any non-negative number. Let us now suppose that at every wealth level, w, consumer
1’s Arrow-Pratt measure of risk aversion is larger than consumer 2’s. That is, where the inequality, called Jensen’s inequality, follows because h is strictly concave,
and the final two equalities follow from (2.15) and (2.13), respectively. Consequently,
u(wˆ 1) < u(wˆ 2), so that because u is strictly increasing, wˆ 1 < wˆ 2 as desired.
We may conclude that consumer 1’s certainty equivalent for any given gamble is
lower than 2’s. And from this it easily follows that if consumers 1 and 2 have the same
initial wealth, then consumer 2 (the one with the globally lower Arrow-Pratt measure)
will accept any gamble that consumer 1 will accept. (Convince yourself of this.) That is,
consumer 1 is willing to accept fewer gambles than consumer 2.
Finally, note that in passing, we have also shown that (2.12) implies that consumer
1’s VNM utility function is more concave than consumer 2’s in the sense that (once again
putting x = v(w) in (2.13))
u(w) = h(v(w)) for all w ≥ 0, (2.16)
where, as you recall, h is a strictly concave function. Thus, according to (2.16), u is a
‘concavification’ of v. This is yet another (equivalent) expression of the idea that consumer
1 is more risk averse than consumer 2.
Ra(w) is only a local measure of risk aversion, so it need not be the same at every
level of wealth. Indeed, one expects that attitudes toward risk, and so the Arrow-Pratt measure,
will ordinarily vary with wealth, and vary in ‘sensible’ ways. Arrow has proposed a
simple classification of VNM utility functions (or utility function segments) according to
how Ra(w) varies with wealth. Quite straightforwardly, we say that a VNM utility function
displays constant, decreasing, or increasing absolute risk aversion over some domain
of wealth if, over that interval, Ra(w) remains constant, decreases, or increases with an
increase in wealth, respectively.
Decreasing absolute risk aversion (DARA) is generally a sensible restriction to
impose. Under constant absolute risk aversion, there would be no greater willingness
to accept a small gamble at higher levels of wealth, and under increasing absolute risk
aversion, we have rather perverse behaviour: the greater the wealth, the more averse
one becomes to accepting the same small gamble. DARA imposes the more plausible
restriction that the individual be less averse to taking small risks at higher levels of wealth.
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