Journal of Risk and Uncertainty, 9:239-256 (1994)
© 1994 Kluwer Academie Publishers
Observing Different Orders of Risk Aversion
GRAHAM LOOMES
Department of Economies, University of York, YorkYOl 5DD, UK
UZI SEGAL
Department of Economics, University of Toronto, Ontario, M5S lAI Cattada
Abstract
A decision maker's attitude towards risk is said to be of order /,; = 1, 2, if for every given risk e with expected
value zero, the risk premium the decision maker is willing to pay to avoid the risk te goes with / to zero at the
same order as;'. This article presents an experiment testing the order of decision makers' attitudes toward risk.
Its major result is that both attitudes exist, eaeb in signifieant proportions. Moreover, two elasses of first-order
behavior are defined. The rank-dependent model (Quiggin, 1982) belongs to one, the disappointment aversion
model (Gul, 1991) to the other. We show that only the first of these two classes appears among our subjeets.
Key words: risk attitude, decision theory, experimental economics
Introduction
One ofthe most interesting results of Machina's (1982) generalized nonexpected utility
analysis is that under some smoothness assumptions, many analytical results of expected
utility, especially comparative statics analysis and other results concerning optimization,
carry over to nonlinear preferences (see also Machina, 1989). It turns out, however, that
not all nonexpected utility models share these properties. In particular, smoothness of
preferences with respect to outcomes, guaranteed in the expected utility model by the
differentiability of the utility function, is not satisfied (at least not always) by some of
these models.
Let e be a nontrivial random variable such that E[e] = 0. Define T7(f) to be the risk
premium the decision maker is willing to pay out of the nonstochastic wealth level w to
avoid the random variable te. Formally, the decision maker is indifferent between w -\- te
and w - j:(t). Segal and Spivak (1990) define the attitude towards risk associated with a
preference relation as being of first- or second-order if for every w and such e and for
sufficiently small /, the risk premium 'n-(?) is of the same order of magnitude as t or z^,
respectively. Pratt (1964) proved that in expected utility theory (with a differentiable
utility function), as t goes to zero, 'u(t) is proportional to t'^. There are nonexpected utility
models—for example, Quiggin's (1982) rank-dependent and Gul's (1991) disappointment aversion, where, as t goes to zero, the risk premium is proportional t o ; (see also
Montesano, 1988).
240
GRAHAM LOOMES/UZI SEGAL
This seemingly technical definition has some strong economic implications. For example, under second-order risk aversion, some risk averse decision makers will always
prefer random prices to the certainty of their average. On the other hand, for a decision
maker who is first-order risk averse, random prices should be considered inferior to their
sure average price, provided that these prices vary in a bounded neighborhood (Segal
and Spivak, 1992). Also, some conclusions concerning efficient allocations of risk do not
hold in the presence of first-order risk aversion. In particular, suppose that the economy
is sufficiently large and has two types of decision makers, one type satisfying first-order
risk aversion, the other type satisfying second-order risk aversion. Then the Pareto efficient allocations of the risk associated with a public project and the Pareto efficient
allocations of risk in an insurance market may require that only the second-order individuals will share the risk. However, if everyone satisfies second-order risk aversion, then
in both cases everyone should carry some risk (Segal and Spivak, 1992). This last observation is particularly relevant, since, as we show below, both types appear to exist in
significant proportions.
There are some other important implications in the area of insurance and finance. For
example, second-order risk aversion implies that decision makers will buy full insurance
if and only if there is no marginal loading (i.e., if the insurance premium, exactly equals
the probability of loss). By contrast, first-order attitude implies that decision makers will
buy full insurance even in the presence of some marginal loading, provided it is not too
high (Segal and Spivak, 1990). In addition, Epstein (1992) argues that first-order risk
aversion permits more flexibility in functional forms. He sbows that constant relative risk
aversion under second-order attitude implies either enormous risk premiums for large
gambles or almost zero premiums for moderate risks. By contrast, first-order risk aversion
permits more realistic premiums. Epstein and Zin (1991) show that models of first-order
risk aversion explain stock and bond returns data much better than do models of secondorder risk aversion. For other applications of this concept, see Epstein and Zin (1990).
Since orders of risk aversion have such potentially widespread implications for economic analysis, it is natural to try to find out whether different orders exist. This article
reports an experiment which attempts to observe the order of decision makers' attitudes
to risk. Our major finding is that both first-order and second-order attitudes exist, and in
significant proportions. Moreover, the design of our experiment also allows us to discriminate between two different classes of first-order risk behavior.
The article is organized as follows. Section 1 sets out the theoretical predictions to be
tested by the experiment. Sections 2 and 3 describe the experiment and its results, while
section 4 contains some concluding remarks. Data related to the experiment are collected in tables 1 through 6.
1. Orders of risk aversion
Let L be the set of simple lotteries over finite wealth levels in D = [0, M], a compact
interval in St. The lottery (xi,p\;... ;x,,,p,,) yieldsx, with probability/?,, i = l,...,n.
Forx
G D, let 8^ be the degenerate lottery (x, 1). For random variable e =
(e\,p\;...;e,,,p,,)
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
241
a n d / e 0i,\ette = (tei,pi;...; re,,,/p,,). For X e L, let F ^ be the cumulative distribution
function ofX given by f";^'(x) = Pr(A'^ x).
On L there exists a complete, transitive, continuous, and monotonic (with respect
to first-order stochastic dominance) preference relation >. As usual, > and ~
denote strict preference and indifference, respectively. A function V: L -^ Si represents the preference relation > if for every X and Y in L, V{X) 5 V{Y) if and only if
X > Y. For A' e L, let the certainty equivalent of A", denoted CE(X), be implicitly
defined by (CE(AO, 1) ~ X.
The risk premium -iT'*'(e) which a decision maker is willing to pay to avoid adding the
random variable e to the nonstochastic wealth level w is defined by w - TT"'(e) = CE(M'
+ e). If E[e] = 0, then by definition, the risk premium is positive whenever the decision
maker is risk averse. For the sake of simplicify, we often omit w from our notation.
Consider now the risk premium •^(t; e) which the decision maker is willing to pay to
avoid adding the random variable te to the current wealth level w. Of course, TT(O; e) = 0.
Moreover, by continuify, lim,_^o T7(;; e) = 0. We assume that ^^{t; e) is a differentiable
function of;, except maybe at; = 0.' Following Segal and Spivak (1990), define orders of
risk aversion as the order of the first non-zero derivative of TT with respect to f at ? = 0 +
for e such that E[e] = 0. Formally,
Definition 1. The decision maker's attitude towards risk at w is of order one if for every
e such that E[e] = 0, d-^{t; e)/af|,=o+ ^ 0. It is of order two if for every such e,
^
^
o* ^ 0.
Orders of risk aversion have a nice geometric interpretation. Consider the set of
lotteries {(x,/?;_y,l - p):x,y G D}foragivenp G [0,1]. Define (p(x,3') = V(x,p;y,\ p) to be the utilify from this lottery and assume, without loss of generalify, that ip(x,x) =
X. The preference relation > induces an indifference curves map in 9^-. The slope ofthe
indifference curve through (x,y) is -<f)i(x,}')/92(jf,3')Suppose now thatp = q = \. Obviously, in this case, the function cp is symmetric, i.e.,
<3;,{x,y) = (p(y,x). Let the decision maker's current wealth level be w, and consider the
random variable e = (1, j ; -1, \). The risk premium 'n-(;; e) satisfies
ip(w
— 17, W —
IT)
= (p(w + t , W — t ) ^ W
— 77 = (p(w + t , W — t ) .
Differentiate both sides with respect to t to obtain
— == ( P 2 ( W ' + t,W - t) - ( p i ( w + t , W -
t ) ^
If the decision maker is second-order risk averse, then, by definition, C)TT/(9;| , = o = 0, and
therefore ipi(iv, w) = 92 (^, w)- Since the slope ofthe indifference curve through (w, w)
is - (pi/'P2, it follows that this slope is - 1 .
242
GRAHAM LOOMES/UZl SEGAL
Suppose, on the other hand, that the preference relation satisfies risk aversion of order
one. Then, dTr/dt\,=o+ > 0. It follows that the limit from the left of the slope of the
indifference curve through (w, w) is less than - 1, and since the function ip is symmetric,
the limit from the right ofthe slope ofthe indifference curve through this point is greater
than - 1 . In other words, indifference curves, as in figure 1, have a kink along the main
diagonal (for similar pictures, see Segal and Spivak, 1990; Gul, 1991; Epstein, 1992).
We now introduce a concept of orders of conditional risk aversion. Let A' and e be as
before and define ;('|, + e to be the lottery (x,,/?,;. . . ; X , _ | , / ? , _ I ; A : , + e|,p,(7,;.. .;A:, +
(^m,Pi<:ini',^i+\,Pi+\',--',x ,,,p,,). The lottery A'l, + e is obtained from A" by adding the
"noise" e to the outcomex,. The conditional risk premium is defined as that amount of
money which the decision maker is willing to pay out of x, in order to avoid this noise.
Formally,
We define orders of conditional risk aversion similarly to before. Let E[e] = 0. We say
that the preference relation satisfies first-order conditional risk aversion if for every
nondegenerate X, e and /, the first-order derivative of the conditional risk premium of
X\ i + te with respect to f at f = 0 + is positive. It satisfies second-order conditional risk
aversion if all these derivatives equal zero, but the second-order derivatives are positive
(see also Segal and Spivak, 1988).
Next, we briefiy consider some models. Throughout, we assume that the decision
maker is risk averse. In this text, we simply summarize results. Proofs can be found in the
references listed and in the appendix.
Expected utility with a differentiable utility function u exhibits both second-order
behavior and second-order conditional risk aversion. The same is true for a number of
nonexpected utility models. For example, Segal and Spivak (1990) prove that Chew's
\
\
X
/
/
Figure I. A nondifferentiable Indifference curve.
,
/
/
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
243
(1983) weighted utility satisfies second-order risk aversion. The proof that it satisfies
second-order conditional risk aversion is similar.
By contrast, the rank-dependent functional with nonlinear probability transformation
function (see Quiggin, 1982; Yaari, 1987; Segal, 1989) and cumulative prospect theory
(Tversky and Kahneman, 1992) represent first-order risk aversion (Segal and Spivak,
1990), Similarly, they represent first-order conditional risk aversion. However, Gul's
(1991) disappointment aversion model satisfies first-order risk aversion, but secondorder conditional risk aversion (see appendix). Expected utility with (continuous but)
nondifferentiable utility represents first-order (and first-order conditional) risk aversion
around any point of nondifferentiability,
2. Experimental design
Consider the following problem. There are three mutually exclusive and exhaustive
statesof the world, 51, 52, and S3, with probabilitiesp + e,p - s, and 1 - 2/7, respectively, such that p G [0, 0,5] and e G [0, p]. The deeision maker has £7 to allocate
between the first two events while winning £/v if ^3 happens. In other words, the decision
maker will participate in the lottery (w + x,p + e;w + T - x,p - E;W + K,\ - 2p). We
do not impose the constraint thatJC G [0, 7], This represents a simple problem of optimal
portfolio selection. If p = 0,5 and we restriet x to be between 772 and T, then this
problem has a clear interpretation in terms of insurance. Suppose that w = 0 (that is, the
wealth level is T) and that the decision maker faces the possibility of a total loss in case ^3
happens. The decision maker can buy a partial insurance, the priee of which 50 pence
(50p) for eaeh pound insured. For example, the total payment for a full insurance is 7/2,
in which case the deeision maker will have £ r i n both cases less the insuranee premium
772, It follows immediately by first-order stoehastic dominance that if e > 0, then x ^
7/2, We are interested in this seetion in the question: Under what conditions will the
decision maker set X = 7/2?
We start with the case where/? = 0,5 or/C = 7/2, Then by settings = 7/2, the deeision
maker can guarantee the sure gain of 7/2, Suppose that p = 0,5, and eonsider again
figure 1, The deeision maker is constrained to choose a point on the linex + y = T (not
shown in the picture). Suppose indifference eurves are everywhere differentiable. If e =
0, thenx = y = 7/2 is the best point along thex + y = T line (reeall that the slope of the
indifference curve at this point is - 1), However, if e > 0, then the slope of the indifference curve through (7/2, 7/2) is less than - 1 , and the best point along the budget line is
to the right ofthe main diagonal. On the other hand, if indifference curves have kinks along
the main diagonal (as is the ease under first-order risk aversion), then for a sufficiently small
e > 0, the best point along the budget line is still atx = y. This intuitive analysis leads to
Prediction 1, whieh follows from Proposition 1 in Segal and Spivak (1990),
Prediction 1. Suppose that/? = 0,5 or that K = 7/2, Then
1, If the decision maker's attitude towards risk is of order two, then the optimal value of
X is 7/2 if and only if E =0,
244
GRAHAM LOOMES/UZI SEGAL
2. If the decision maker's attitude towards risk is of order one, then there exists E* > 0
such that for e G [0, e*] the optimal value ofx is r/2.
Ifp < 0.5 and K ^ T/2, then there is no way for the decision maker to avoid some risk,
and the natural concept is therefore conditional orders of risk aversion. Similarly to
Prediction 1, we now have
Prediction 2.
Suppose that/? < 0.5 and that K ^ T/2. Then
1. If the decision maker's attitude towards conditional risk is of order two, then the
optimal value ofx is T/2 if an only if e = 0.
2. If the decision maker's attitude towards conditional risk is of order one, then there
exists 8* > 0 such that for e £ [0, E*] the optimal value ofx is T/2.
This optimization problem can thus be used in testing what kind of attitude towards
risk decision makers have. We describe an experiment based on these results below.
Assume w = 0, and consider again the optimization problem maXj. V(x,p + e;T - x,
p - z;K,\ ~ 2p). Predictions 1 and 2 deal with the question: Under what conditions will
a decision maker setx = ^ in this problem? There is, however, another issue which we
wish to address, namely the connection between the value of e and the optimal value ofx.
Of course, different functional forms may give specific answers to this comparative statics
question, but we are interested here in general properties that follow by first-order
stochastic dominance. Denote the solution of the above optimization problem by x(8).
Suppose that for some p, X(E*) = T/2, but for some s < E*, X(E) > 772. Then, by
first-order stochastic dominance and by the definition ofx(E), it follows that (x(z),p + E*;
T - x(e),p
- B*; K, I - 2p) > {x(e),p + B;T - x(e),p
- e; /f, 1 - 2p) > (T/2, 2p; K,
1 - 2/7). ThereforeX(E*) ^ 7/2, a contradiction. This leads to
Prediction 3. Let the preference relation satisfy the first-order stochastic dominance
axiom, and assume that for some/?, K, and s* > 0, x(e*) = 7/2. Then for the samep and
A:,X(E) = 772 for all E E [0, E*].
The aim of the experiment was to use variants of the simple optimal portfolio selection
problem described at the beginning of this section to observe the extent of second-order,
first-order unconditional, and first-order conditional risk aversion in a sample of individual decision makers. One can, of course, argue, following Mas Collel (1974), that no
experiment can prove nondifferentiability of preferences. However, our aim is not to
prove differentiability or nondifferentiabilify of indifference curves, but to check what
models fit observed behavior better.
Figure 2 gives two examples of the form in which such decision problems were presented in the experiment. Columns represent states ofthe world whose respective probabilities, expressed as chances out of a hundred, are shown at the base of each column.
The mechanism for determining which state of the world occurs is a single random draw
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
1
245
60.61
60
:
42.4.3
100
40
80.81
!
8.65
!
20
I
42
!
38
100
Figttre 2. Two examples of the form in which decision problems were presented in the experiment.
from a box containing one hundred cloakroom tickets numbered 1 to 100 inclusive. So, in
the lower of the two examples shown in figure 2, S\ occurs if the randomly drawn ticket
bears a number between 1 and 42 inclusive, ^2 occurs if the ticket is between 43 and 80,
and S3 occurs otherwise.
In terms of the notation used in the previous section, the first example in figure 2 is
the case where/? = 0.5 and e =0.1; the second example involves/? = 0.4, e = 0.02
andK = 8.65.
Participants in the experiment were randomized between four subsamples. Each
subsample was presented with a series of 16 variants of the formats shown in figure 2.
For eight ofthe 16 problems, the total sum to be distributed between S\ and 52 was
£22.40. For the other eight problems, T = £17.30. In cases where there was a third
state of the world, K was set either at 0 or at T/2 (the second case in figure 2 is an
example of the latter.) The values ofp and E varied extensively, with 0.1 ^ p ^ 0.5
and 0.005 ^ e ^ 0.3. Full details of the parameters used in all four subsamples are
given in tables 1-4.
Each subsample was presented with five, six, or seven problems where either there
were only two states of the world or else the payoff in S3 was T/2. In these cases,
individuals could, if they wished, opt for certainty by settings = T/2. Alternatively, by
settings > T/2, they could raise the expected value of the portfolio by accepting some
level of risk. In these cases, second-order risk averse individuals will always set x > T/2.
By contrast, individuals who are first-order risk averse will setx = 772 if e is less than or
equal to their critical value E*. Since the value of e* may vary from one individual to
another and from one value ofp to another, the number of observations of x = T/2 in
these cases will tend to decrease as E increases. However, if there are individuals who are
first-order risk averse, we should expect to observe them setting x = T/2 in problems
where e is sufficiently small.
Those whose attitude to risk is w«conditional first-order (e.g. disappointment aversion) should be expected to set x = T/2 only in (some of) the five, six, or seven cases
where p = 0.5 or K = T/2. However, those who exhibit first-order conditional risk
246
GRAHAM LOOMESAJZl SEGAL
aversion may also be expected to setx = T/2 in a number of cases where/7 < 0.5, K = 0
and e is sufficiently small.
The experiment discussed in this article was conducted in two stages, in February
and in June 1991. In February a letter was distributed widely among students and
nonacademic stafi" at the University of York describing the general nature of the
experiment, giving examples of two- and three-state problems, explaining the incentive system, and inviting potential participants to sign up for whichever session was
most convenient for them. As a result, a total of 196 individuals participated in
groups of up to 16 at a time. On arrival at their session, individuals were randomly
allocated to one of the four subsamples and seated at a separate computer terminal,
not adjacent to anyone in the same subsample. Each session began with two practice
rounds (common to all subsamples) during which the instructions were reviewed and
any questions were answered. The two practice problems and the 16 incentive-linked
problems all followed the same pattern. The first screen displayed five alternatives,
all with the same values oip, e, and K (if/? <0.5), but with T divided difi:erently
between S\ and S2. For example, the first screen for the second example given in
figure 2 offered the five possible allocations:
v4: {15.00, 2.30}
5 : {13.50, 3.80}
C: {12.00, 5.30}
D: {10.50, 6.80}
£: {9.00, 8.30}.
(The pairs of numbers were filled into the empty boxes in figure 2).
Subjects were asked to consider the alternatives and choose the one they most preferred. After they had confirmed their choice, a second screen presented four more
alternatives, which differed from the first five only in thatx and T - x were now clustered
more closely around the distribution chosen from the first screen. For example, someone
who chose alternative C in the first screen, would now be presented with the following
pairs of \x, T - x}\
^ : {12.75, 4.55}
fi: {12.25, 5.05}
C: {11.75, 5.55}
D: {11.25, 6.05}.
After the most preferred second screen option had been chosen, a third screen appeared. This displayed six alternatives, clustered around and including the option selected from the second screen, and now with the values ofx differing by only 5p from one
alternative to the next. Having chosen one of the six alternatives, a fourth (and final)
screen displayed only that chosen option. This gave subjects an opportunity to refiect
on the portfolio which they had arrived at, and make any final adjustments to the
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
247
distribution of T if they wished. This could be done by using two of the cursor keys:
pressing the left arrow key moved money from ^2 to 51 in 5p increments; pressing the
right arrow key moved money in the opposite direction. When completely satisfied
with the distribution, subjects confirmed their decisions, and the computer moved to
the next decision problem, repeating the same four-screen sequence,^
The idea behind this procedure was to use a series of choices to enable subjects to
iterate towards their most preferred portfolio in each case. It might seem more straightforward simply to present the (fourth) screen with arbitrary {x, T - x) pairs already
inserted and then ask subjects to use the cursor keys to achieve their optimal portfolio.
The potential danger with such a procedure is that the starting point, the "initial endowment," may in some way bias the results, especially since subjects have to do different
amounts of work to achieve different final portfolios. The procedure that we adopted
was arguably more neutral and gave people time to absorb the parameters of the prob
lem and "home in" on a final decision. Moreover, since the first three screens required
the same number of keystrokes whichever portfolio was eventually chosen, no portfolio
involved more time or effort to select than any other.
By setting T at either £22,40 or £17,30, we attempted to counteract a tendency of
subjects to give responses in "round" numbers. In the experiments reported in Loomes
(1991), subjects were given the task of dividing £20,00 between two states of the world. In
more than 90% of cases, the responses that they gave were in whole pounds. Under these
circumstances, it would be very difficult to tell how often setting x = T/2 is due to
first-order risk aversion, and how often it is primarily due to rounding. In the design of
the present experiment, we selected values of 7 which made T/2 neither more nor less
round than adjacent values of J:.
It may at this stage be worth noting other respects in which the present experiment
differs from the one reported in Loomes (1991), That earlier experiment was primarily concerned with testing the independence axiom of expected utility theory rather
than orders of risk aversion, and the data could not be used for this latter purpose.
Besides the problem of rounding, the parameters were generally not appropriate for
the task. For example, in cases where/? = 0,5, E was never lower than 0,1; and in all
cases where there was a third state of the world, K was set at 0, In the earlier
experiment, only 12 sets of parameters were used, and E never fell below 0,05 in any
of them. So, although the earlier experiment yielded a number of responses where x
= 772, it was clear that we needed to reduce the propensity for rounding, and greatly
extend the range of parameters, especially the number of cases where 0 < E ^ 0,05, if
we wished to try to discriminate between difTerent orders of risk aversion.
Subjects were asked to work through the 16 decisions at their own speed, answering
each one as accurately as they could, on the understanding that when they had completed all 16, one and only one would be selected at random to be played out for real.
The procedure was that when subjects indicated that they had finished, a supervisor
came to their terminal and asked them to roll a 20-sided die until it came to rest with a
number between 1 and 16 face up. The subject's decision in that problem was retrieved
from the computer's memory and displayed on the screen. The subject then picked a
248
GRAHAM LOOMES/UZI SEGAL
ticket at random from a box of 100 cloakroom tickets, and was paid whatever amount was
due under the relevant state of the world.-'
Throughout the February experiment, subjects did not know that they would be invited to make the 16 decisions again at a later date. But, in June, they were each sent a
letter inviting them to participate again. This letter was accompanied by a sheet which
displayed—in the format shown in figure 2—all 16 decision problems which had been
presented to them in February.
On this occasion, subjects were asked to decide on the distribution of T in each case in
advance, write their most preferred values ofx and T - x into the appropriate columns
on the sheet, and then come to the laboratory at any convenient time during two fivehour periods to enter their decisions into the computer. This time they were not asked to
go through the three-screen iterative choice procedure, but to enter their decisions
directly via the "fourth" screens.
Our reason for inviting subjects to make the 16 decisions again was to check whether
the opportunity to see all 16 decision problems together and to consider all their responses at their leisure would markedly change the observed patterns of behavior, either
at an individual or at an aggregate level.
3. Results
Altogether, 136 of the original 196 subjects took part again in June. In reporting the
results, we shall focus on the responses of these 136 individuals.'' Tables 1-4 present
the aggregate patterns. (The individual data are available upon request.) In each
table, the sixteen decision problems are listed in ascending order of E. In cases where e is
the same in two or more questions, these are listed in descending order of p. In each
subsample, there is a pair of problems with the same values of both E and p: in these
cases, the problem where K = T/2 is listed above the one where K = Q. Problems where
either/? = 0.5 ox K = T/2 are marked with an asterisk. The columns headed February
and June show the numbers of individuals who setx = T/2 on each occasion, with the
numbers in brackets reporting cases where individuals setx < T/2. All four tables exhibit
patterns in both February and June which are consistent with an appreciable degree of
first-order risk aversion. There is clear evidence of some differences in responses between February and June. Judging by the frequencies of violations of stochastic
dominance^—which fell from 45 cases committed by 29 individuals down to 25 cases
committed by 13 individuals—the opportunity to consider decisions more carefully
helped to reduce aberrant responses. It also served to markedly increase the observability of first-order risk aversion in all four subsamples.
Since respondents were randomized between subsamples, a convenient way of conveying the overall pattern is to combine the data as in table 5, which shows the average
percentages of cases where x was set equal to 772 for different levels of E. TO ensure
representation from all four subsamples within each cell, responses to certain pairs of
adjacent E'S have been pooled. It is clear that for 0.005 $ E ^ 0.025, the February pattern
249
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
Tabte I. Subsample 1 responses (n = 33)
Problem
Number
10
6*
4*
16
12*
8*
2
3
9
14*
7
13
11
5
1*
15
e
P
February
June
.005
.01
.01
.01
.02
.02
.075
.075
.075
.10
.10
.125
.125
.20
.30
.30
.475
.50
.20
.20
.40
.10
.375
.225
.125
.50
.15
.375
.275
.44
.50
.45
8(2)
6
4(2)
5(2)
3(1)
12(2)
8
12(1)
12(1)
5(2)
1(1)
2
7(1)
1
1
3
1(1)
2
1(2)
1
1
1
2
1
1
1(1)
3(1)
2(1)
1
1
1
1
One respondent set.*: = 4 for all 16 questions in both February and June.
Table 2. Subsample 2 responses (;j = 32)
Problem
Number
8*
16
11
14*
4*
3*
2
6
13
7
5*
10
12
1
15
9*
E
P
February
June
.005
.005
.015
.015
.02
.025
.05
.05
.05
.075
.10
.10
.125
.15
.15
.25
.125
.125
.475
.375
.50
.125
.20
.13
.10
.175
.50
.40
.325
.35
.30
.50
9(1)
6
6(1)
5(2)
5(2)
3
14
12(2)
4(1)
10
6(1)
4(1)
4
2
4
3
3
3
3
1
2
1
2(1)
4(1)
2
2(2)
2
3
3(1)
3(1)
3
2
One respondent setA- = y for all 16 questions in both February and June.
250
GRAHAM LOOMESAJZl SEGAL
Table 3. Subsample 3 responses {ii = 39)
Problem
Number
E
P
February
June
6*
16*
12
8
14
2*
4*
10
9
7
1*
5
3
11*
15*
13
.005
.01
.01
.01
.02
.02
.05
.05
.10
.15
.20
.20
.20
.25
.30
.30
.225
.45
.45
.10
.20
.14
.50
.35
.25
.25
.50
.40
.30
.50
.50
.45
10(1)
4(1)
6(1)
4(3)
4(4)
1(1)
2
2
2
0(1)
1
1
1(2)
1
1
1
14
13
13(1)
8
5
6
5
3
1
2
1
2
2
2
1
1
Table 4. Subsample 4 responses (n = 32)
Problem
Number
E
P
February
June
16*
10
4*
8
12*
2
14
1
5
9*
11
15
7
6
3*
13
.005
.005
.015
.015
.025
.025
.05
.05
.05
.10
.10
.10
.125
.125
.25
.25
.325
.325
.225
.135
.375
.225
.45
.25
.15
.50
.30
.20
.225
.175
.50
.45
9(1)
6(1)
6(2)
5
2
18
21(1)
11(1)
8(2)
4
8
5
1
1(1)
3(1)
4
1(1)
1
0
0
0
0
0(1)
0
1(1)
1
2
0(1)
2(1)
1(1)
0
0(1)
251
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
_
Table 5. Percentages of cases where J: =
T
~ 2
.005
.01/.015
.02/.025
.05/.075
.10
.125/.15
.20/.25/.30
February
23.9
15.0
7.4
6.8
3.7
4.3
2.8
June
45.9
28.7
16.7
8.2
5.4
5.3
2.9
E
Table 6. First-order behavior in pairs of problems with same {p, E)
Only when
February (June)
Subs. 1:4* & 16
Subs. 2: 8* & 16
Subs. 3: 16*&12
Subs. 4: 16*& 10
2(2)
6(4)
0(1)
5(1)
Both
Only when
K=0
2(10)
3(10)
4(12)
4(17)
3(2)
3(2)
2(1)
2(4)
is reproduced in June, but on approximately twice the scale. There is rather less difference for 0.05 ^ e ^ 0.15, and virtually no difference for e ^ 0.2 where (apart from the two
individuals who set x = T/2 for all 16 problems on both occasions) there are very few
observations and little discernible pattern about them. In short, aggregate behavior is
consistent with the existence of first-order risk aversion, and with our hypothesis (based
on the assumption that E* is liable to vary among individuals) that the number of observations ofx = T/2 will tend to decrease as e increases. In our experiment, such behavior
does not seem attributable to "rounding," or to casual unconsidered responses, and we take
our results to suggest that, for at least some decision makers, first-order risk aversion is a
characteristic of their preferenees.
Several other points of interest emerge from the data. Across the four subsamples,
there were a number of sets of questions whieh allowed us to check Prediction 3. In
February there were a total of 40 cases where individuals chose x = T/2 in at least one
question in those sets, of which only five violated Prediction 3. In June there were four
violations out of 67 cases.
Secondly, although first-order behavior was exhibited by many in our sample, it is also
evident that a substantial minority of respondents weversetx ^ T/2, either in February or
in June. Inspection of the individual data shows this to be the case for 33 (24.3%) of the
sample of 136. Third, recall that disappointment aversion entails unconditional, but not
conditional, first-order risk aversion, i.e., entails setting x = T/2 only in (some of) the
problems marked with an asterisk. In our experiment, there was little evidence of such
behavior. Perhaps the most straightforward indication of this comes from examining the
pairs of problems involving the same values oip and e, but with K = T/2 in one case and
K = Oin the other. Table 6 shows the data by focusing attention only on those responses
where some sort of first-order risk aversion (conditional or unconditional) is exhibited.
Even in February, with less time to refiect upon their decisions, the cases which conform
with disappointment aversion never outnumber those that do not; and in June, when
given the opportunity to consider their deeisions in advance, the great majority who set x
252
GRAHAM LOOMESAJZI SEGAL
= 7/2 do SO for both problems in each pair, in disagreement with disappointment theory.
This strongly suggests that models which entail first-order risk aversion should be able to
accommodate first-order conditional risk aversion, and should not be restricted to the
unconditional case.
4. Conclusion
Rather than focus on some specific axioms of decision theory, we have concentrated in this
article on a more general characteristic, namely, the order of risk attitude. In addition to the
existing distinction between unconditional first-order and second-order risk aversion, we
extended the analysis to encompass order of conditional risk aversion and showed how
this extended analysis enabled us to distinguish among several different decision models.
We then reported an experiment designed to discover whether different attitudes
could be detected. We asked a substantial number of individuals to participate twice,
presenting each person with the same set of decision problems on both occasions, with a
four-month interval in between. On the second occasion, each participant had the opportunity to contemplate his/her decisions at leisure, the idea being to check whether
behavior which departs from conventional expected utility theory diminishes when individuals have ample opportunity to consider their responses, make calculations, etc.
While the frequency of violations of stochastic dominance was effectively halved on the
second occasion, evidence of first-order risk aversion became more pronounced; at the
same time, it was also clear that a substantial minority of respondents conformed with
second-order risk aversion on both occasions.
The patterns of behavior exhibited by our sample at an individual and at an aggregate
level suggest that first-order risk aversion, conditional as well as unconditional, may be a
significant phenomenon, and that the implications of first-order attitudes for many areas
of economic activity deserve serious consideration. Such steps were taken by Epstein and
Zin (1990,1991) and by Segal and Spivak (1992).
Appendix
This appendix discusses some of the claims presented at the end of section 1. As stated
there, we assume throughout that the decision maker is risk averse. Expected utility with
a differentiable utility function u exhibits second-order behavior. Let V(X) =
Jou(x)dFx(x). Risk aversion implies that u is concave. Consider the lottery 8^ + te with
the expected utility /DM(W -I- te)dFe(e). If E[e] = 0, that is, if JoedFi.(e) = 0, then dtrQ;
e)/af|, = o = -lDeu'(w)dFi(e) = 0 hut d^'rr(t; e)/dt^\,=ii+= -Soe^u"(w)dFe(e) > 0.
Similarly, expected utility (with a differentiable utility function ii) satisfies second-order
conditional risk aversion.
The rank-dependent functional (see Quiggin, 1982; Yaari, 1987; Segal, 1989) is denoted RD(A^ and is given by
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
253
=£
(1)
u{x)df{Fx{x))
for some strictly increasing and continuous function/, satisfying/(O) = Oand/(1) = 1.
Call this functional/jroper rank-dependent if it is not expected utility, i.e., if the probability transformation function / is not linear. This functional represents risk aversion if and
only if M and / are both concave (see Chew, Karni, and Safra, 1987). In that case, a proper
rank-dependent functional represents first-order risk aversion. Consider the lottery 8^ + tt
with E[e] = 0. Then RD(8H, + te) = JDM(W + te)df{Fi(e)). Hence dTv{t; e)/*|,=()+ =
-J[)Xu'(w)df{F^(e)) > 0 (see Segal and Spivak, 1990). Similarly, a strict rankdependent functional (with concave u and / ) represents first-order conditional risk
aversion.
Gul (1991) suggested another model for decision making under uncertainty, called
disappointment aversion. We show below that this functional satisfies first-order risk
aversion, but second-order conditional risk aversion.
Let X = (x\, p\;...;
Xf,, p,,), and assume that x\ ^ . . . ^ x,,. An elationdisappointment decomposition (EDD) of A' is a triple (a, Y, Z) such that the possible
outcomes o f y = (y\,qh ... ;>»,„,(?,„) are those outcomes^ of Xsatisfyingx ^ CE(A^,the
possible outcomes of Z = {zi,r\;... ;z^,r^) are the outcomes^ of A'such t h a t x ^ CE(A^,
and Fx = aFy + (1 - a)Fz. Let 7 : [0,1] -^ &lhe given by y(a) = a/[\ + (1 - a)|3] for
some p G ( - 1 , 00). According to the disappointment aversion theory, the preference
relation ?= can be represented by the functional
7(a) X q,u(yi) + (1 - ^(a)) t nu^z,)
(2)
for some EDD of X It represents risk aversion if the function u is concave and p > 0.
However, if p = 0, the functional at Eq. (2) is reduced to the expected utility form.
Consequently, we call this functional proper disappointment aversion if p > 0.
Lemma 1. Suppose that the decision maker is risk averse and that the preference
relation > over the lotteries can be represented by a proper disappointment aversion
functional. Assume further that the risk premium 'n-(/; e) is difTerentiable from the right
at f = 0 +. Then the deeision maker's attitude towards risk is of order 1.^
Proof: Lete = (e\,p];... ;£„,/?„)withE[e] = Owhereei < . . . <en,andlet^(Obethe
certainty equivalent of the lottery 8^; + te. There are two possible cases:
1. There exists f* > 0 and /Q sueh that \ft £ (0, t*), i,(t) G. [w + /e,,,, w + tei^^+ \]; or
254
GRAHAM LOOMES/UZI SEGAL
2. There exists /'o and two sequences {^j, such that for) = 1 , 2 ,
(a)lim^.^^ 4 = O;and
(b) For every/c,4(/'^) < w + t^i^ei^^hut ^{t\) >w + t\ei^y
Consider case 1, and assume first that for every / £ (0, t*), ^{t) G (w + /e,,,, w +
,,i+1). Then, the EDD is unique, and by Eq. (2) we get that
where a = 2"=,,|+i A-Therefore, «(w -
T7(/))
= V{^w + te) implies
„ pieju'{w + tej)
'• = ' 0 + 1
MO
dt
= 0+
' = '0+1
(Recall that E[e] = 0 and p >0). Similar calculations for case 2 show that jr(t) is not
differentiable a t ; = 0 + , a contradiction to our assumption that T:{f, e) is differentiable from the right at f = 0 +. Finally, if ^{ti^) = w + t^en) for some ('o and a sequence
tk -^ 0, then either the risk premium is nondifferentiable as in case 2, or its derivative
is positive as in case 1.
•
Consider now the issue of conditional orders of risk aversion. Let X and e be as in the
definition of conditional orders of risk aversion, and suppose that the certainty equivalent of A" is (strictly) betweenx,,, andx,,,+1. For a sufficiently small t, adding te tox, leaves
the certainty equivalent between x,,, andx/,|+1 (thus a and therefore ^(a) do not change)
and, moreover, will add outcomes only on one side of the certainty equivalent of X The
analysis of this case is therefore similar to that of expected utility, and the disappointment aversion model satisfies second-order conditional risk aversion, although it is of
unconditional order one.'
OBSERVING DIFFERENT ORDERS OF RISK AVERSION
255
Acknowledgments
We gratefully acknowledge the financial support of the Economic and Social Research
Council (U.K), Award No. R000231194 and of the Social Sciences and Humanities
Research Council of Canada.
Notes
1. For a discussion of the differentiability of the risk premium in expected and nonexpected utility models,
and especially the issue of nondifferentiability of the utility function in the expected utility model, see Segal
and Spivak (1990, 1992). Note that since a monotonic function is almost everywhere differentiable, nondifferentiability is the exception, not the rule, within the expected utility model.
2. We also checked tbe conjecture that some subjects migbt have tended to repeatedly pick alternatives near
tbe middle of tbe screen, somebow figuring that tbis is probably a reasonable thing to do. Tbere is,
bowever, no evidence for such behavior. Tbe only people who consistently picked alternatives from around
tbe same area of the screen were those (few) who opted for an equal or near equal division of tbe money in
all cases. But since those people mostly exhibited the same pattern of answers under a quite different
procedure (discussed later in this section), tbere is no reason to think tbat their responses were an artifaet
due to the screen displays.
3. Following Holt (1986), there has been some question about wbether tbis "random lottery selection"
procedure may distort tbe way individuals report their preferences. Subsequently, Starmer and Sugden
(1991) examined tbis question directly and found no evidenee of any systematic distortion.
4. No significant difference was found between the February responses of tbe 136 individuals who took
part on both occasions and the 60 individuals who did not return in June.
5. Most of tbese cases involvedx being set only slightly lower than 7/2 i.e., the potential cost of such errors was
generally very small.
6. Epstein (1992) has already indicated, without a formal proof, tbat the disappointment aversion functional
implies first-order risk aversion.
7. The only case where the first-order derivative is not zero is when tbe certainty equivalent of A" equals some
x,|| and the noise is added to tbis outcome. We ignore tbis case bere because for every/;, the set of lotteries
of length n having this property has measure zero in the set of all lotteries of length /;. Also, in tbe
experiment reported above, the noise was added to tbe certainty equivalent only when X = 8,.
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