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OCR for page 134
7
Choice Under Uncertainty:
Problems Solved and Unsolved
MARK ]. MACHINA
Fifteen years ago, the theory of choice under uncertainty could be
considered one of the 'Success stories" of economic analysis: it rested
on solid axiomatic foundations;) it had seen important breakthroughs in
the analytics of risk and risk aversion and their applications to economic
issues;2 and it stood ready to provide the theoretical underpinnings for
the newly emerging "information revolution" in economics.3 1bday, choice
under uncertainty is a field in flux: the standard theory and, implicitly, its
public policy implications are being challenged on several grounds from
both within and outside the field of economics. The nature of these
challenges, and of economists' responses to them, is the topic of this paper.
The following section provides a brief but self-contained description
of the economist's canonical model of individual choice under uncertainty,
the expected Slid model of preferences over lotteries. I shall describe this
model from two different perspectives. The first perspective is the most
familiar and has traditionally been the most useful for addressing standard
economic questions. However, the second, more modern perspective will
be the most useful for illustrating some of the problems that have beset
this model, as well as some of the proposed responses.
Each of the following sections is devoted to one of these problems.
All are important; some are more completely "solved" than others. In each
Mark J. Machina is professor in the Department of Economics at the University of California,
San Diego.
1 See, for example, van Neumann and Morgenstern (1947), Marschak (1950), and Savage (1954~.
2See, for example, Arrow (1963, 1974), Pratt (1964) and Rothschild and Stiglitz (1970, 1971~.
For surveys of applications, see Lippman and McCall (1981) and Hey (1979~.
3See, for example, Akerlof (1970) and Spence and Zeckhauser (1971~. For overviews of the
subsequent development of this area, see Stiglitz (1975, 1985~.
134
OCR for page 135
Al4RK ~ ~4 CHINA
135
case, I begin with a specific example or description of the phenomenon in
question. I then review the empirical evidence regarding the uniformity
and extent of the phenomenon. Finally, I shall report on how these findings
have changed, or are likely to change, or should change, the way economists
view and model private and public decisions under uncertainty. On this
last topic, the disclaimer that "my opinions are my own" has more than the
usual significance.
THE EXPECTED UTILITY MODEL
The Classical Perspective: Cardinal Utility and Attitudes Toward Risk
In light of current trends toward generalizing this model, it is useful
to note that the expected utility hypothesis was itself first proposed as an
alternative to an earlier, more restrictive theory of risk-bearing. During
the development of modern probability theory in the 17th century, such
mathematicians as Blaise Pascal and Pierre de Fermat assumed that the
attractiveness of a gamble offering the payoffs (at, ..., an) with probabilities
(Pi, ~Pn) was given by its expected value x (i.e., the weighted average of
the payoffs where each payoff is multiplied by its associated probability, so
that x = Up + . .. + Alps). The fact that individuals consider more than
just expected value, however, was dramatically illustrated by an example
posed by Nicholas Bernoulli in 1728 and now known as the St. Petersburg
Paradox:
Suppose someone offers to toss a fair coin repeatedly until it comes up
heads, and to pay you $1 if this happens on the first toss, $2 if it takes
two tosses to land a head, $4 if it takes three tosses, $8 if it takes four
tosses, and so on. What is the largest sure payment you would be willing
to forgo in order to undertake a single play of this game?
Because this gamble offers a 1/2 chance of winning $1, a 1/4 chance of
winning $2, and so forth, its expected value is (1/2~$1 + (1/4~$2 + (1/8~$4
+ . . . = $1/2 + $1/2 + $1/2 + .. . = boo; thus, it should be preferred
to any finite sure gain. However, it is clear that few individuals would forgo
more than a moderate amount for a one-shot play. Although the unlimited
financial backing needed to actually make this offer is somewhat unrealistic,
it is not essential for making the point: agreeing to limit the game to at
most a million tosses will still lead to a striking discrepancy between a
typical individual's valuation of the modified gamble and its expected value
of $500,000.
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136
CHOICE UNDER UNCERTAINTY
The resolution of this paradox was proposed independently by Gabriel
Cramer and Nicholas's cousin Daniel Bernoulli.4 Arguing that a gain of
$2,000 was not necessarily "worth" twice as much as a gain of $1,000, they
hypothesized that individuals possess what is now termed a von Neumann-
Morgenstem utility of wealth Scion U f ). Rather than evaluating gambles
on the basis of their expected value ~ = Up + . . . + ~nPn, individuals will
evaluate them on the basis of their expected utility u = U(x~)p~ + ... +
U(Xn)Pn This value is calculated by weighting the utility of each possible
outcome by its associated probability, and it can therefore incorporate the
fact that successive increments to wealth may yield successively diminishing
increments to utility. Thus, if utility took the logarithmic form U(x) = Infix)
(which exhibits this property of diminishing increments) and the individual's
wealth at the start of the game were, let us say, $50,000, the sure- gain that
would yield just as much utility as taking this gamble (i.e., the individual's
certainty equivalent of the gamble), would be about $9, even though the
gamble has an infinite expected value.5
Although it shares the name "utility," this function Ural is quite distinct
from the ordinal utility function of standard consumer theory. Although the
latter can be subjected to any monotonic transformation, a von Neumann-
Morgenstern utility function is cardinal in that it can only be subjected to
transformations that change the origin point or the scale (or both) of the
vertical axis, but do not affect the "shape" of the function. The ability to
choose the origin and scale factor is often exploited to normalize the utility
function-for example, to set U(O) = 0 and U(M) = 1 for some large value
M.
~ see how this shape determines risk attitudes, let us consider Figures
la and lb. The monoton~city of the curves in each figure reflects the
property of stochastic dominance preference, by which one lottery Is said
to stochastical) dominate another if it can be obtained from it by shifting
probability from lower to higher outcome levels.6 Stochastic dominance
preference is thus the probabilistic extension of the attitude that "more Is
better."
Consider a gamble offering a 2/3 chance of a wealth level of a' and
a 1/3 chance of a wealth levels of x". The amount x = (2/3)x' + (1/3)x"
in the figures gives the expected value of this gamble; Ua = (2/3)Ua~x') +
4Bernoulli (1738~. For a historical overview of the St. Petersburg paradox and its impact, see
Samuelson (1977~.
5Algebraically, the certainty equivalent of the Petersburg gamble is given by the value ~ that
solves U(W+~) = (1/2)U(W+1) + (1/4)U(W+2) + (1/~)U(W+4) + ..., where W denotes the
individual's initial wealth (i.e., wealth going into the gamble).
61bus, for example, a 2/3:1/3 chance of $100 or $20 and a 1/2:1/2 chance of $100 or $30 both
stochastically dominate a 1/2:1/2 chance of $100 or $20.
OCR for page 137
CLARK J. MA CHINA
A
Ua(x )
c
a
cn
c
a
~Uafx')
z
c
o
ua
B
Ub(X )
a)
~n
c
-
o
c
~Ub(X)
Z Ub(X )
137
7
. /~
/'
{,
f
1
1
1
Ua(~)
. , I I
X' X
ub
x
Wealth
1 Ub(~)
,"'/ ~
'~' / I
~1
~1
''1' 1
~^-_' 1
'' ~1
___ ~1 1
_, 1 1
1 1 1
1 1
~1' ~
J I l
1 1 1
1 1 1
X X X
Weatth
FIGURE 1 Utility functions of risk. A: Concave utility function of a risk averter. B:
Convex utility function of a risk lover.
OCR for page 138
138
CHOICE UNDER UNCERTAINTY
( 1/3) Ua (x" ) and Ub = ( 2/3) Ub (x' ) + ( 1/3) Ub (x") give its expected utility
for the utility functions UP ) and Ub( ). For the concave (i.e., bowed
upward) utility function Ua(~), we have Ua(x) > ha, which implies that this
individual would prefer a sure gain of x twhich would yield utility Ua(x)]
to the gamble. Because someone with a concave utility function will in
fact always rather receive the expected value of a gamble than receive the
gamble itself, concave utility functions are termed risk averse. For the convex
(bowed downward) utility function Ubt ), we have ub > USA). Because this
preference for bearing the risk rather than receiving the expected value
will also extend to all gambles, Ub(~) is termed risk-loving. In their famous
article, Friedman and Savage (1948) showed how a utility function that was
concave at low-wealth levels and convex at high-wealth levels could explain
the behavior of individuals who both incur risk by purchasing lottery tickets
as well as avoid risk by purchasing insurance.7 Algebraically, Arrow (1963,
1974), Pratt (1964) and others have shown that the degree of concavity of
a utility function, as measured by the curvature index-U"(x)/U'~x), can
lead to predictions of how risk attitudes, and hence behavior, will vary with
wealth or across individuals in a variety of- situations.8
Because a knowledge of UP ~ would allow the prediction of preferences
(and hence behavior) in any risly situation, experimenters and applied
decision analysts are frequently interested in eliciting or recovering their
subjects' (or clients') von Neumann-Morgenstern utility functions. One
means of doing this is the.fiactile method. This approach begins by adopting
the normalization U(0) = 0 and U(M) = 1 for some positive amount M
and fixing a "mixture probability" ~say, p = 1/2. The next step involves
obtaining the individual's certainty equivalent (~ of a gamble yielding a
1/2 chance of M and a 1/2 chance of 0, which will have the property that
U(~) = 1/2.9 Finding the certainty equivalent of a gamble yielding a 1/2
chance of (, and a In chance of 0 yields the value (2 satm6mg U(62) = 1/4.
7 How risk attitudes actually differ over gains versus losses is itself an unsolved problem: evidence
consistent with or contradictory to the Friedman-Savage observation of risk seeking over gains
and risk aversion over losses can be found in Williams (1966), Kahneman and Tversky (1979),
Fishburn and Kochenberger (1979), Grether and Plott (19 79), Hershey and Schoemaker (1980a),
Payne, Laughhunn, and Crum (1980, 1981), Hershey, Kunreuther, and Schoemaker (1982), and
the references cited in these articles. Finally, Feather (1959) and Slovic (1969a) found evidence
that subjects' risk attitudes over gains and losses systematically changed when hypothetical situ-
ations were replaced by situations involving real money.
For example, if Uc( ) and Ua( ) satisfy -U"(x)/U'(x) > -U`''(~)/U<~(~) for all x [i.e., if Uc( ) is
at least as risk averse as U`, ( A, an individual with utility function UC ( ) would always be willing to
pay at least as much as an individual with utility function Us ( ) for (complete) insurance against
any risk. See also the related analyses of Ross (1981) and Kihlstrom, Romer, and Williams (1981~.
9Because the utility of it, will equal the expected utility of the gamble, it follows that UP ) =
(1/2)U(M) + (1/2)U(0), which under the normalization U(o) = o and U(M) = 1 will equal l/2.
OCR for page 139
MARK ~ CHINA
139
Finding the certainty equivalent of a gamble yielding a 1/2 chance of M
and a 1/2 chance of (1 yields the value <3 satiating U(63) = 3/4.10 By
repeating this procedure (i.e., 1/8, 3/8, 5/8, 7/8, 1/16, 3/16, etc.), the utility
function can (in the limit) be completely assessed.
1b see how the expected utility model can be applied to risk policy, let
us consider a disastrous event that is expected to occur with probability p
and involve a loss L (L can be measured in either dollars or lives). In many
cases, there will be some scope for influencing the magnitudes of either p
or L, often at the expense of the other. For example, replacing one large
planned nuclear power plant with two smaller, geographically separated
plants may (to a first approximation) double the possibility that a nuclear
accident will occur. However, the same action may lower the magnitude of
the loss (however it is measured) if an accident occurs.
The key tool used in evaluating whether such adjustments should be
undertaken is the individual's (or society's) marginal rate of substitution
MRSp,L, which specifies the rate at which an individual (or society) would
be willing to trade off a (small) change in p against an offsetting change
in L. If the potential adjustment involves better terms than this minimum
acceptable rate, it will obviously be preferred; if it involves worse terms,
it will not be preferred. Although the exact value of this marginal rate of
substitution will depend upon the individual's (or society's) utility function
UP ), the expected utility model does offer some general guidance regardless
of the shape of the utility function: namely, for a given loss magnitude L,
a doubling (tripling, halving, etc.) of the loss probability p should double
(triple, half, etc.) the rate at which one would be willing to trade reductions
in p against increases in L.~i
The discussion so far has paralleled the economic literature of the
1960s and 1970s by emphasizing the flexibility of the expected utility model
in comparison with the Pascal-Fermat expected value approach. The need
to analyze and respond to growing empirical challenges, however, has
led economists in the l980s to concentrate on the behavioral restrictions
implied by the expected utility hypothesis. These restrictions are the subject
of the next section.
iOAs in the previous note, U(~2) = (1/2)U(~) + (1/2)U(0) and U(63) = (~/2)U(M) +
(~/2)U((, ), which from the normalization U(o) = o, U(M) = ~ and the fact that Up) = i/2
will equal t/4 and 3/4, respectively.
~ iBecause expected utility in this example is given by u = (I - p)U(W) + pU(W - L) (where W
is initial wealth or lives), an application of the standard economic formula for the marginal rate
of substitution (e.g., see Henderson and Quandt [1980:10-113) yields MRSp,~ = -(Bu/BL)/
(Bu/0p) = -pU'(W - L)/[U(W) - U(W - L)] which, for fixed L, varies proportionately with
P.
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140
CHOICE UNDER UNCERTAINTY
A Modern Perspective:
Linearity in the Probabilities as a Testable Hypothesis
As a theory of Individual behavior, the expected utility model shares
many of the underlying assumptions of standard economic consumer theory.
In each case, it is assumed that the objects of choice, either commodity
bundles or lotteries, can be unambiguously and objectively described and
that situations that ultimately imply the same set of availabilities (e.g., the
same budget set) will lead to the same choice. In each case, it is also
assumed that the individual is able to perform the mathematical operations
necessary to actually determine the set of availabilities for example, to
add up the quantities in different sized containers or to calculate the
probabilities of compound or conditional events. Finally, in each case, it is
assumed that preferences are transitive, so that if an individual prefers one
object (either a commodity bundle or a risky prospect) to a second, and
prefers this second object to a third, he or she will prefer the first object to
the third. The validity of these assumptions for choice under uncertainty is
examined in later sections.
The strongest and most specific implication of the expected utility
hypothesis stems from the form of the expected utility maximand or pr~er-
encefilnction U(x~)p1 + . .. + U(xn~pn. Although this preference function
generalizes the expected value form HIPS + ... + caps by dropping the
property of linearity in the payoff levels (i.e., the xi's), it retains the other
key property of this form, namely, linearly in the probabilities.
Graphically, the property of linearity in the probabilities may be illus-
trated by considering the set of all lotteries or prospects over some set of
fixed outcome levels ~1 < x2 < ~3, which can be represented by the set
of all probability triples of the form P = (\Pi,P2,P3) where Pi = probed)
and pi + P2 + p3 = 1.~2 Making the substitution P2 = 1 - pi-p3, this
set of lotteries can be represented by the points in the unit triangle in
the (P1,P3) plane, as in Figure 2.~3 Because upward movements in the
triangle increase p3 at the expense Of P2 (i.e., shift probability from the
outcome x2 up to ~3) and leftward movements reduce P1 to the benefit of
P2 (i.e., shift probability from ~1 up to Aid these movements (and, more
generally, all northwest movements) lead to stochastically dominating lot-
teries and would accordingly be preferred. For the purposes of illustrating
many of the following discussions it will be useful to plot the ~ndiv~dual's
indifference curves in this diagram; that Is, the curves in the diagram that
12Thus, if x' = $20, ~2 = $30, and X3 = $100, the three prospects in footnote 6 would be repre
sented by the points (p! ,p3 ) = (1/3,2/3), (p,, ,p3 ) = (O. ~ /2) and ~l ,p3) = (1/2, 1/2), respectively.
13Although it is fair to describe the renewal of interest in this approach as "modern," modified
versions of this triangle diagram can be found as far back as Marschak (1950) and Markowitz
(1959:Chap 11~.
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11~4RK ~ CHINA
11
1
/
pi = prod (x,)
FIGURE 2 Expected utility indifference curares in the triangle diagram.
141
connect points of equal expected utility.~4 Because each such curve will
consist of the set of all ~i, p3) points that solve an equation of the form
u = U(~)p~ ~ Up-pi-p3) ~ U(X3)p3 = k for some constant k, and
because the probabilities pi and p3 enter linearly (i.e., as multiplicative co-
efficients) into this equation, the indifference curves will consist of parallel
straight lines, with more preferred indifference curves lying to the north-
west. This means that, to know an expected utility maximizer's preferences
over the entire triangle, it suffices to know the slope of a single indifference
curve.
~ see how this diagram can be used to illustrate attitudes toward
risk, let us consider Figures 3a and 3b. The dashed lines in the figures
are not indifference curves but rather iso-expected value lines; that is, lines
connecting points with the same expected value that are hence given by the
solutions to equations of the form ~ = Alps +x2~1-pi-p3~+~3p3 = k for
some constant k. Because northeast movements along these lines do not
change the expected value of the prospect but do increase the probabilities
i4A useful analogy to the concept of indifference curves is the "constant-altitude" curves on a
topographic map, each of which connect points of the same altitude. Just as these curares can be
used to determine whether a given movement on the map will lead to a greater or lower altitude,
indifference curves can be used to determine whether a given movement in the triangle will lead
to greater or lower expected utility.
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142
CHOICE UNDER UNCERTAINTY
A
P3
B
P3
1
o
1
f /
o lo
P1
Non,.
1~
/ '/
I'd
\
2~
FIGURE 3 A: Relatively steep indifference curves of a risk averter. B: Relatively flat
indifference curves of a risk lover.
OCR for page 143
M4RK J: AL4 CHINA
143
of the extreme outcomes x~ and x3 at the expense of the middle outcome
x2, they are simple examples of mean preserving spreads or "pure" increases
in risker When the utility function Up ~ is concave (i.e., risk averse), its
indifference curves can be shown to be steeper than the iso-expected value
lines (Figure 3a),~6 and such increases in risk will lead to less preferred
indifference curves. When Up ~ is convex (risk loving), its indifference
curves will be flatter than the iso-expected value lines (Figure 3b), and
these increases in risk will lead to more preferred indifference curves.
Finally, if one compares two different utility functions, the one that is
more risk averse (in the above Arrow-Pratt sense) will possess the steeper
indifference cu~ves.~7
Behaviorally, the property of linearity in the probabilities can be viewed
as a restriction on the individual's preferences over probability mixtures of
lotteries. If P* = (P~,...,Pn) and P = (P., APE) are two lotteries over a
common outcome set {at, ..., anti the cat: (1-cry probability mixture of P*
and P is the lottery clip* + (1-a)P = kept +(1-alps, . . ., rip* +(1-a)Pn).
This may be thought of as that prospect that yields the same ultimate
probabilities over fx~...~3cn} as the two-stage lottery that offers an c':
(1- a) chance of winning P* or P. respectively. It can be shown that
expected utility maximizers will exhibit the following property, known as
the independence Axiom:
If the lottery Pa is preferred (respectively indifferent) to the lottery
P. then the mixture cop* + (1-~)P** will be preferred (respectively
indifferent) to the mixture c'P + (1-c'jP** for all c' > 0 and P**.
This property, which is in fact equivalent to linearity in the probabilities,
can be interpreted as follows:
In terms of the ultimate probabilities over the outcomes {at, . . ., an),
choosing between the mixtures c'P* + (1-c~)P** and cap + (1-~)P** is
the same as being offered a coin with a probability 1-cat of landing tails,
in which case you will obtain the lottery P**, and being asked before
the flip whether you would rather have Pe or P in the event of a head.
Now either the coin will land tails, in which case your choice won't have
mattered, or else it will land heads, in which case your are "in effect"
5 See, for example, Rothschild and Stiglitz (1970, 1971~.
16This follows because the slope of the indifference curves can be calculated to be [U(~2)-
U(xi)]/[U(x3) - U(~2)], the slope of the iso-expected value lines can be calculated to be [x2 -
=~]/[~3 - 2, and a concave shape for U(~) implies [U(~2) - U(~)]/[~2-XI] > [U(X3) -
U(x2)]/[X3 - 272] wherever r~ < r2 < X3.
17Setting his v, w, x, and y equal to an, ~2, =2, and :1:3, respectively, this follows directly from
theorem 1 of Pratt (1964~.
18See, for example, Marschak (1950) and Samuelson (1952~.
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144
CHOICE UNDER UNCERTAINI'Y
back to a choice between P. or P. and it is only "rational" to make the
same choice as you would before.
Although this is a prescriptive argument, it has played a key role in
economists' adoption of expected utility as a descriptive theory of choice
under uncertainty. The mounting evidence against the model has led
to a growing tension between those who view economic analysis as the
description and prediction of what they consider to be rational behavior and
those who view it as the description and prediction of observed behavior.
Let us turn now to this evidence.
VIOLATIONS OF LINEARITY IN THE PROBABILITIES
The Allais Paradox and "Fanning Out"
One of the earliest and best-known examples of systematic violation
of linearity in the probabilities (or, equivalently, of the independence
axiom) is the well-known Allais paradox.l9 This problem involves obtaining
the individual's preferred option from each of the following two pairs of
gambles (readers who have never seen this Problem may want to circle
their own choices before proceeding):
a~:{l.OO chance of $1,000,000 versus
and
~10 chance of $5,000,000 versus
.10 chance of $5,000,000
a2: .89 chance of $1,000,000
.01 chance of $0
. J .11 chance of $1,000,000
a4 ~ .89 chance of $0
Defining Act, x2, x3) = {$0;$1 miDion;$5 million), these four gambles
are seen to form to a parallelogram in the (pi,p3) triangle (Figures 4a
and 4b). Under the expected utility hypothesis, therefore, a preference for
al in the first pair would indicate that the individual's indifference curves
were relatively steep (as in Figure 4a), which would imply a preference for
a4 in the second pair. In the alternative case of relatively flat indifference
curves, the gambles as and as would be preferred.20 Yet, such researchers
as Allais (1953, 1979a), Morrison (1967), Raiffa (1968), and Slovic and
Tversky (1974) have found that the most common choice has been for al
in the first pair and as in the second, which implies that indifference curves
are not parallel but rather fan out, as in Figure 4b.
19 See, for example, Allais (1952, 1953, 1979a).
20Algebraically, these two cases are equivalent to the expression [.1o U(5,000,000) -
U(1,ooo,ooo) + .01 U(0)], being respectively negative or positive.
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178
CHOICE UNDER UNCERTAINTY
"departures from the strictures of probability theory should be corrected
[by the analyst or the decision maker] but that [systematic] departures from
the structures of expected utility theory should not." This is because the
former involved the determination of the risks associated with alternative
actions or policies, which are in fact matters of accurate representation,
while the latter involve the willingness of individuals, organizations, and
society to bear these risks, which is a matter of preference.
He concludes that analysis must be designed to account for actual
preferences, even those that depart from the tenets of expected utility
theory. Therefore, analysts and decision makers, in assigning values to
policy alternatives, may need to consider departures from expected utility
and weighting schemes to reflect those departures.
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