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Journal of Risk and Uncertainty, 9:239-256 (1994)© 1994 Kluwer Academie Publishers

Observing Different Orders of Risk Aversion

GRAHAM LOOMESDepartment of Economies, University of York, YorkYOl 5DD, UK

UZI SEGALDepartment 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 expectedvalue zero, the risk premium the decision maker is willing to pay to avoid the risk te goes with / to zero at thesame 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-orderbehavior are defined. The rank-dependent model (Quiggin, 1982) belongs to one, the disappointment aversionmodel (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 utilityanalysis is that under some smoothness assumptions, many analytical results of expectedutility, especially comparative statics analysis and other results concerning optimization,carry over to nonlinear preferences (see also Machina, 1989). It turns out, however, thatnot all nonexpected utility models share these properties. In particular, smoothness ofpreferences with respect to outcomes, guaranteed in the expected utility model by thedifferentiability of the utility function, is not satisfied (at least not always) by some ofthese models.

Let e be a nontrivial random variable such that E[e] = 0. Define T7(f) to be the riskpremium the decision maker is willing to pay out of the nonstochastic wealth level w toavoid the random variable te. Formally, the decision maker is indifferent between w -\- teand w - j:(t). Segal and Spivak (1990) define the attitude towards risk associated with apreference relation as being of first- or second-order if for every w and such e and forsufficiently 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 differentiableutility function), as t goes to zero, 'u(t) is proportional to t'^. There are nonexpected utilitymodels—for example, Quiggin's (1982) rank-dependent and Gul's (1991) disappoint-ment aversion, where, as t goes to zero, the risk premium is proportional t o ; (see alsoMontesano, 1988).

240 GRAHAM LOOMES/UZI SEGAL

This seemingly technical definition has some strong economic implications. For exam-ple, under second-order risk aversion, some risk averse decision makers will alwaysprefer random prices to the certainty of their average. On the other hand, for a decisionmaker who is first-order risk averse, random prices should be considered inferior to theirsure average price, provided that these prices vary in a bounded neighborhood (Segaland Spivak, 1992). Also, some conclusions concerning efficient allocations of risk do nothold in the presence of first-order risk aversion. In particular, suppose that the economyis sufficiently large and has two types of decision makers, one type satisfying first-orderrisk aversion, the other type satisfying second-order risk aversion. Then the Pareto effi-cient allocations of the risk associated with a public project and the Pareto efficientallocations of risk in an insurance market may require that only the second-order indi-viduals will share the risk. However, if everyone satisfies second-order risk aversion, thenin both cases everyone should carry some risk (Segal and Spivak, 1992). This last obser-vation is particularly relevant, since, as we show below, both types appear to exist insignificant proportions.

There are some other important implications in the area of insurance and finance. Forexample, second-order risk aversion implies that decision makers will buy full insuranceif and only if there is no marginal loading (i.e., if the insurance premium, exactly equalsthe probability of loss). By contrast, first-order attitude implies that decision makers willbuy full insurance even in the presence of some marginal loading, provided it is not toohigh (Segal and Spivak, 1990). In addition, Epstein (1992) argues that first-order riskaversion permits more flexibility in functional forms. He sbows that constant relative riskaversion under second-order attitude implies either enormous risk premiums for largegambles or almost zero premiums for moderate risks. By contrast, first-order risk aversionpermits more realistic premiums. Epstein and Zin (1991) show that models of first-orderrisk aversion explain stock and bond returns data much better than do models of second-order risk aversion. For other applications of this concept, see Epstein and Zin (1990).

Since orders of risk aversion have such potentially widespread implications for eco-nomic analysis, it is natural to try to find out whether different orders exist. This articlereports an experiment which attempts to observe the order of decision makers' attitudesto risk. Our major finding is that both first-order and second-order attitudes exist, and insignificant proportions. Moreover, the design of our experiment also allows us to discrim-inate between two different classes of first-order risk behavior.

The article is organized as follows. Section 1 sets out the theoretical predictions to betested by the experiment. Sections 2 and 3 describe the experiment and its results, whilesection 4 contains some concluding remarks. Data related to the experiment are col-lected 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 compactinterval in St. The lottery (xi,p\;... ;x,,,p,,) yieldsx, with probability/?,, i = l,...,n. ForxG 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 distributionfunction ofX given by f"; '(x) = Pr(A'^ x).

On L there exists a complete, transitive, continuous, and monotonic (with respectto first-order stochastic dominance) preference relation >. As usual, > and ~denote strict preference and indifference, respectively. A function V: L -^ Si repre-sents the preference relation > if for every X and Y in L, V{X) 5 V{Y) if and only ifX > Y. For A' e L, let the certainty equivalent of A", denoted CE(X), be implicitlydefined by (CE(AO, 1) ~ X.

The risk premium -iT'*'(e) which a decision maker is willing to pay to avoid adding therandom 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 decisionmaker 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 toavoid 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 differentiablefunction of;, except maybe at; = 0.' Following Segal and Spivak (1990), define orders ofrisk 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 everye 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 oflotteries {(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 oftheindifference 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 therandom variable e = (1, j ; -1, \). The risk premium 'n-(;; e) satisfies

ip(w — 17, W — I T ) = (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, andtherefore 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 orderone. Then, dTr/dt\,=o+ > 0. It follows that the limit from the left of the slope of theindifference 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 greaterthan - 1 . In other words, indifference curves, as in figure 1, have a kink along the maindiagonal (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 asbefore 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 ofmoney 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 saythat the preference relation satisfies first-order conditional risk aversion if for everynondegenerate X, e and /, the first-order derivative of the conditional risk premium ofX\ i + te with respect to f at f = 0 + is positive. It satisfies second-order conditional riskaversion 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 decisionmaker is risk averse. In this text, we simply summarize results. Proofs can be found in thereferences listed and in the appendix.

Expected utility with a differentiable utility function u exhibits both second-orderbehavior and second-order conditional risk aversion. The same is true for a number ofnonexpected utility models. For example, Segal and Spivak (1990) prove that Chew's

//

\

\

X,--//

Figure I. A nondifferentiable Indifference curve.


(1983) weighted utility satisfies second-order risk aversion. The proof that it satisfiessecond-order conditional risk aversion is similar.

By contrast, the rank-dependent functional with nonlinear probability transformationfunction (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 second-order conditional risk aversion (see appendix). Expected utility with (continuous but)nondifferentiable utility represents first-order (and first-order conditional) risk aversionaround any point of nondifferentiability,

2. Experimental design

Consider the following problem. There are three mutually exclusive and exhaustivestatesof the world, 51, 52, and S3, with probabilitiesp + e,p - s, and 1 - 2/7, respec-tively, such that p G [0, 0,5] and e G [0, p]. The deeision maker has £7 to allocatebetween the first two events while winning £/v if 3 happens. In other words, the decisionmaker will participate in the lottery (w + x,p + e;w + T - x,p - E;W + K,\ - 2p). Wedo not impose the constraint thatJC G [0, 7], This represents a simple problem of optimalportfolio selection. If p = 0,5 and we restriet x to be between 772 and T, then thisproblem has a clear interpretation in terms of insurance. Suppose that w = 0 (that is, thewealth level is T) and that the decision maker faces the possibility of a total loss in case ^3happens. 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 premium772, 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 thedecision maker set X = 7/2?

We start with the case where/? = 0,5 or/C = 7/2, Then by settings = 7/2, the deeisionmaker can guarantee the sure gain of 7/2, Suppose that p = 0,5, and eonsider againfigure 1, The deeision maker is constrained to choose a point on the linex + y = T (notshown 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 theindifference curve at this point is - 1), However, if e > 0, then the slope of the indiffer-ence curve through (7/2, 7/2) is less than - 1 , and the best point along the budget line isto the right ofthe main diagonal. On the other hand, if indifference curves have kinks alongthe main diagonal (as is the ease under first-order risk aversion), then for a sufficiently smalle > 0, the best point along the budget line is still atx = y. This intuitive analysis leads toPrediction 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 ofX is 7/2 if and only if E =0,


2. If the decision maker's attitude towards risk is of order one, then there exists E* > 0such 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 toPrediction 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 theoptimal 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 thereexists 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 towardsrisk 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 willa decision maker setx = ^ in this problem? There is, however, another issue which wewish 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 staticsquestion, but we are interested here in general properties that follow by first-orderstochastic 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, byfirst-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 dominanceaxiom, and assume that for some/?, K, and s* > 0, x(e*) = 7/2. Then for the samep andA:,X(E) = 772 for all E E [0, E*].

The aim of the experiment was to use variants of the simple optimal portfolio selectionproblem 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 individ-ual decision makers. One can, of course, argue, following Mas Collel (1974), that noexperiment can prove nondifferentiability of preferences. However, our aim is not toprove differentiability or nondifferentiabilify of indifference curves, but to check whatmodels fit observed behavior better.

Figure 2 gives two examples of the form in which such decision problems were pre-sented in the experiment. Columns represent states ofthe world whose respective prob-abilities, 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


1 60.61 100

60 : 40

42.4.3 80.81 100

! 8.65I

42 ! 38 ! 20

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, inthe lower of the two examples shown in figure 2, S\ occurs if the randomly drawn ticketbears 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 isthe case where/? = 0.5 and e =0.1; the second example involves/? = 0.4, e = 0.02andK = 8.65.

Participants in the experiment were randomized between four subsamples. Eachsubsample 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 thirdstate of the world, K was set either at 0 or at T/2 (the second case in figure 2 is anexample of the latter.) The values ofp and E varied extensively, with 0.1 ^ p ^ 0.5and 0.005 ^ e ^ 0.3. Full details of the parameters used in all four subsamples aregiven in tables 1-4.

Each subsample was presented with five, six, or seven problems where either therewere 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, bysettings > T/2, they could raise the expected value of the portfolio by accepting somelevel 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 orequal to their critical value E*. Since the value of e* may vary from one individual toanother and from one value ofp to another, the number of observations of x = T/2 inthese cases will tend to decrease as E increases. However, if there are individuals who arefirst-order risk averse, we should expect to observe them setting x = T/2 in problemswhere e is sufficiently small.

Those whose attitude to risk is w«conditional first-order (e.g. disappointment aver-sion) should be expected to set x = T/2 only in (some of) the five, six, or seven caseswhere 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 = 0and e is sufficiently small.

The experiment discussed in this article was conducted in two stages, in Februaryand in June 1991. In February a letter was distributed widely among students andnonacademic stafi" at the University of York describing the general nature of theexperiment, giving examples of two- and three-state problems, explaining the incen-tive system, and inviting potential participants to sign up for whichever session wasmost convenient for them. As a result, a total of 196 individuals participated ingroups of up to 16 at a time. On arrival at their session, individuals were randomlyallocated 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 practicerounds (common to all subsamples) during which the instructions were reviewed andany questions were answered. The two practice problems and the 16 incentive-linkedproblems 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:erentlybetween S\ and S2. For example, the first screen for the second example given infigure 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 pre-

ferred. After they had confirmed their choice, a second screen presented four morealternatives, which differed from the first five only in thatx and T - x were now clusteredmore closely around the distribution chosen from the first screen. For example, someonewho chose alternative C in the first screen, would now be presented with the followingpairs 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 ap-peared. This displayed six alternatives, clustered around and including the option se-lected from the second screen, and now with the values ofx differing by only 5p from onealternative 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 refiecton the portfolio which they had arrived at, and make any final adjustments to the


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 theright arrow key moved money in the opposite direction. When completely satisfiedwith the distribution, subjects confirmed their decisions, and the computer moved tothe next decision problem, repeating the same four-screen sequence,^

The idea behind this procedure was to use a series of choices to enable subjects toiterate towards their most preferred portfolio in each case. It might seem more straight-forward simply to present the (fourth) screen with arbitrary {x, T - x) pairs alreadyinserted 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 endow-ment," may in some way bias the results, especially since subjects have to do differentamounts of work to achieve different final portfolios. The procedure that we adoptedwas arguably more neutral and gave people time to absorb the parameters of the problem and "home in" on a final decision. Moreover, since the first three screens requiredthe same number of keystrokes whichever portfolio was eventually chosen, no portfolioinvolved 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 ofsubjects 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. Inmore than 90% of cases, the responses that they gave were in whole pounds. Under thesecircumstances, it would be very difficult to tell how often setting x = T/2 is due tofirst-order risk aversion, and how often it is primarily due to rounding. In the design ofthe present experiment, we selected values of 7 which made T/2 neither more nor lessround than adjacent values of J:.

It may at this stage be worth noting other respects in which the present experimentdiffers from the one reported in Loomes (1991), That earlier experiment was prima-rily concerned with testing the independence axiom of expected utility theory ratherthan 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 forthe task. For example, in cases where/? = 0,5, E was never lower than 0,1; and in allcases where there was a third state of the world, K was set at 0, In the earlierexperiment, only 12 sets of parameters were used, and E never fell below 0,05 in anyof 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 greatlyextend the range of parameters, especially the number of cases where 0 < E ^ 0,05, ifwe wished to try to discriminate between difTerent orders of risk aversion.

Subjects were asked to work through the 16 decisions at their own speed, answeringeach one as accurately as they could, on the understanding that when they had com-pleted 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 supervisorcame to their terminal and asked them to roll a 20-sided die until it came to rest with anumber between 1 and 16 face up. The subject's decision in that problem was retrievedfrom the computer's memory and displayed on the screen. The subject then picked a


ticket at random from a box of 100 cloakroom tickets, and was paid whatever amount wasdue under the relevant state of the world.-'

Throughout the February experiment, subjects did not know that they would be in-vited to make the 16 decisions again at a later date. But, in June, they were each sent aletter inviting them to participate again. This letter was accompanied by a sheet whichdisplayed—in the format shown in figure 2—all 16 decision problems which had beenpresented to them in February.

On this occasion, subjects were asked to decide on the distribution of T in each case inadvance, write their most preferred values ofx and T - x into the appropriate columnson the sheet, and then come to the laboratory at any convenient time during two five-hour periods to enter their decisions into the computer. This time they were not asked togo through the three-screen iterative choice procedure, but to enter their decisionsdirectly via the "fourth" screens.

Our reason for inviting subjects to make the 16 decisions again was to check whetherthe opportunity to see all 16 decision problems together and to consider all their re-sponses at their leisure would markedly change the observed patterns of behavior, eitherat an individual or at an aggregate level.

3. Results

Altogether, 136 of the original 196 subjects took part again in June. In reporting theresults, we shall focus on the responses of these 136 individuals.'' Tables 1-4 presentthe aggregate patterns. (The individual data are available upon request.) In eachtable, the sixteen decision problems are listed in ascending order of E. In cases where e isthe same in two or more questions, these are listed in descending order of p. In eachsubsample, there is a pair of problems with the same values of both E and p: in thesecases, the problem where K = T/2 is listed above the one where K = Q. Problems whereeither/? = 0.5 ox K = T/2 are marked with an asterisk. The columns headed Februaryand June show the numbers of individuals who setx = T/2 on each occasion, with thenumbers in brackets reporting cases where individuals setx < T/2. All four tables exhibitpatterns in both February and June which are consistent with an appreciable degree offirst-order risk aversion. There is clear evidence of some differences in responses be-tween February and June. Judging by the frequencies of violations of stochasticdominance^—which fell from 45 cases committed by 29 individuals down to 25 casescommitted by 13 individuals—the opportunity to consider decisions more carefullyhelped to reduce aberrant responses. It also served to markedly increase the observabil-ity of first-order risk aversion in all four subsamples.

Since respondents were randomized between subsamples, a convenient way of con-veying the overall pattern is to combine the data as in table 5, which shows the averagepercentages of cases where x was set equal to 772 for different levels of E. TO ensurerepresentation from all four subsamples within each cell, responses to certain pairs ofadjacent E'S have been pooled. It is clear that for 0.005 $ E 0.025, the February pattern


Tabte I. Subsample 1 responses (n = 33)

ProblemNumber

106*4*

1612*8*239

14*7

131151*

15

e

.005

.01

.01

.01

.02

.02

.075

.075

.075

.10

.10

.125

.125

.20

.30

.30

P

.475

.50

.20

.20

.40

.10

.375

.225

.125

.50

.15

.375

.275

.44

.50

.45

February

8(2)64(2)5(2)3(1)

1(1)2

1(1)21(2)111211

June

12(2)8

12(1)12(1)5(2)

7(1)11

3

1(1)3(1)2(1)1111

One respondent set.*: = 4 for all 16 questions in both February and June.

Table 2. Subsample 2 responses (;j = 32)

ProblemNumber

8*161114*4*

3*26

1375*

10121

159*

E

.005

.005

.015

.015

.02

.025

.05

.05

.05

.075

.10

.10

.125

.15

.15

.25

P

.125

.125

.475

.375

.50

.125

.20

.13

.10

.175

.50

.40

.325

.35

.30

.50

February

9(1)66(1)5(2)5(2)3

2(1)4(1)2

2(2)233(1)3(1)32

June

14

12(2)4(1)

106(1)4(1)424

333312

1

One respondent setA- = y for all 16 questions in both February and June.

250 GRAHAM LOOMESAJZl SEGAL

Table 3. Subsample 3 responses {ii = 39)

ProblemNumber

6*

16*12

814

2*4*

10971*53

11*15*13

E

.005

.01

.01

.01

.02

.02

.05

.05

.10

.15

.20

.20

.20

.25

.30

.30

P

.225

.45

.45

.10

.20

.14

.50

.35

.25

.25

.50

.40

.30

.50

.50

.45

February

10(1)4(1)6(1)

4(3)4(4)

1(1)2220(1)11

1(2)111

June

14

1313(1)8

56

531

2122211

Table 4. Subsample 4 responses (n = 32)

ProblemNumber

16*10

4*8

12*2

14

15

9*1115

76

3*13

E

.005

.005

.015

.015

.025

.025

.05

.05

.05

.10

.10

.10

.125

.125

.25

.25

P

.325

.325

.225

.135

.375

.225

.45

.25

.15

.50

.30

.20

.225

.175

.50

.45

February

9(1)6(1)6(2)52

1(1)3(1)4

1(1)10000

0(1)0

June

1821(1)11(1)8(2)4851

1(1)1

20(1)2(1)

1(1)00(1)


Table 5. Percentages of cases where J: =

E

February

June

.005

23.9

45.9

.01/.015

15.0

28.7

_ T~ 2

.02/.025

7.4

16.7

.05/.075

6.8

8.2

.10

3.7

5.4

.125/.15

4.3

5.3

.20/.25/.30

2.8

2.9

Table 6. First-order behavior in pairs of problems with same {p, E)

February (June)

Subs. 1:4* & 16Subs. 2: 8* & 16Subs. 3: 16*&12Subs. 4: 16*& 10

Only when

2(2)6(4)0(1)5(1)

Both

2(10)3(10)4(12)4(17)

Only whenK = 0

3(2)3(2)2(1)2(4)

is reproduced in June, but on approximately twice the scale. There is rather less differ-ence for 0.05 ^ e ^ 0.15, and virtually no difference for e ^ 0.2 where (apart from the twoindividuals who set x = T/2 for all 16 problems on both occasions) there are very fewobservations and little discernible pattern about them. In short, aggregate behavior isconsistent with the existence of first-order risk aversion, and with our hypothesis (basedon the assumption that E* is liable to vary among individuals) that the number of obser-vations ofx = T/2 will tend to decrease as e increases. In our experiment, such behaviordoes not seem attributable to "rounding," or to casual unconsidered responses, and we takeour results to suggest that, for at least some decision makers, first-order risk aversion is acharacteristic 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. InFebruary there were a total of 40 cases where individuals chose x = T/2 in at least onequestion in those sets, of which only five violated Prediction 3. In June there were fourviolations out of 67 cases.

Secondly, although first-order behavior was exhibited by many in our sample, it is alsoevident that a substantial minority of respondents weversetx ^ T/2, either in February orin June. Inspection of the individual data shows this to be the case for 33 (24.3%) of thesample of 136. Third, recall that disappointment aversion entails unconditional, but notconditional, first-order risk aversion, i.e., entails setting x = T/2 only in (some of) theproblems marked with an asterisk. In our experiment, there was little evidence of suchbehavior. Perhaps the most straightforward indication of this comes from examining thepairs of problems involving the same values oip and e, but with K = T/2 in one case andK = Oin the other. Table 6 shows the data by focusing attention only on those responseswhere 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 conformwith disappointment aversion never outnumber those that do not; and in June, whengiven 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 toaccommodate first-order conditional risk aversion, and should not be restricted to theunconditional case.

4. Conclusion

Rather than focus on some specific axioms of decision theory, we have concentrated in thisarticle on a more general characteristic, namely, the order of risk attitude. In addition to theexisting distinction between unconditional first-order and second-order risk aversion, weextended the analysis to encompass order of conditional risk aversion and showed howthis extended analysis enabled us to distinguish among several different decision models.

We then reported an experiment designed to discover whether different attitudescould 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 afour-month interval in between. On the second occasion, each participant had the op-portunity to contemplate his/her decisions at leisure, the idea being to check whetherbehavior which departs from conventional expected utility theory diminishes when indi-viduals have ample opportunity to consider their responses, make calculations, etc.While the frequency of violations of stochastic dominance was effectively halved on thesecond occasion, evidence of first-order risk aversion became more pronounced; at thesame time, it was also clear that a substantial minority of respondents conformed withsecond-order risk aversion on both occasions.

The patterns of behavior exhibited by our sample at an individual and at an aggregatelevel suggest that first-order risk aversion, conditional as well as unconditional, may be asignificant phenomenon, and that the implications of first-order attitudes for many areasof economic activity deserve serious consideration. Such steps were taken by Epstein andZin (1990,1991) and by Segal and Spivak (1992).

Appendix

This appendix discusses some of the claims presented at the end of section 1. As statedthere, we assume throughout that the decision maker is risk averse. Expected utility witha 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 withthe 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-orderconditional risk aversion.

The rank-dependent functional (see Quiggin, 1982; Yaari, 1987; Segal, 1989) is de-noted RD(A^ and is given by


= £u{x)df{Fx{x)) (1)

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 probabil-ity transformation function / is not linear. This functional represents risk aversion if andonly if M and / are both concave (see Chew, Karni, and Safra, 1987). In that case, a properrank-dependent functional represents first-order risk aversion. Consider the lottery 8^ + ttwith 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 rank-dependent functional (with concave u and / ) represents first-order conditional riskaversion.

Gul (1991) suggested another model for decision making under uncertainty, calleddisappointment aversion. We show below that this functional satisfies first-order riskaversion, but second-order conditional risk aversion.

Let X = (x\, p\;...; Xf,, p,,), and assume that x\ ^ . . . ^ x,,. An elation-disappointment decomposition (EDD) of A' is a triple (a, Y, Z) such that the possibleoutcomes o f y = (y\,qh ... ;>»,„,(?,„) are those outcomes^ of Xsatisfyingx ^ CE(A^,thepossible outcomes of Z = {zi,r\;... ;z^,r^) are the outcomes^ of A'such tha tx^ CE(A^,and Fx = aFy + (1 - a)Fz. Let 7 : [0,1] -^ &lhe given by y(a) = a/[\ + (1 - a)|3] forsome p G ( - 1 , 00). According to the disappointment aversion theory, the preferencerelation ?= 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 preferencerelation > over the lotteries can be represented by a proper disappointment aversionfunctional. Assume further that the risk premium 'n-(/; e) is difTerentiable from the rightat f = 0 +. Then the deeision maker's attitude towards risk is of order 1.

Proof: Lete = (e\,p];... ;£„,/?„)withE[e] = Owhereei < . . . <en,andlet^(Obethecertainty 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


2. There exists /'o and two sequences {^j, such that for) = 1 , 2 ,(a)lim^.^^ 4 = O;and(b) For every/c,4(/'^) < w + tîêi^^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

MOdt

= 0+ ' = '0+1

(Recall that E[e] = 0 and p >0). Similar calculations for case 2 show that jr(t) is notdifferentiable a t ; = 0 + , a contradiction to our assumption that T:{f, e) is differen-tiable from the right at f = 0 +. Finally, if ^{ti^) = w + tên) for some ('o and a sequencetk -^ 0, then either the risk premium is nondifferentiable as in case 2, or its derivativeis positive as in case 1. •

Consider now the issue of conditional orders of risk aversion. Let X and e be as in thedefinition of conditional orders of risk aversion, and suppose that the certainty equiva-lent of A" is (strictly) betweenx,,, andx,,,+1. For a sufficiently small t, adding te tox, leavesthe 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 Theanalysis of this case is therefore similar to that of expected utility, and the disappoint-ment aversion model satisfies second-order conditional risk aversion, although it is ofunconditional order one.'


Acknowledgments

We gratefully acknowledge the financial support of the Economic and Social ResearchCouncil (U.K), Award No. R000231194 and of the Social Sciences and HumanitiesResearch 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 Segaland Spivak (1990, 1992). Note that since a monotonic function is almost everywhere differentiable, non-differentiability 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 neartbe 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 aroundtbe same area of the screen were those (few) who opted for an equal or near equal division of tbe money inall cases. But since those people mostly exhibited the same pattern of answers under a quite differentprocedure (discussed later in this section), tbere is no reason to think tbat their responses were an artifaetdue 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 tookpart 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 wasgenerally very small.

6. Epstein (1992) has already indicated, without a formal proof, tbat the disappointment aversion functionalimplies first-order risk aversion.

7. The only case where the first-order derivative is not zero is when tbe certainty equivalent of A" equals somex,|| and the noise is added to tbis outcome. We ignore tbis case bere because for every/;, the set of lotteriesof length n having this property has measure zero in the set of all lotteries of length /;. Also, in tbeexperiment reported above, the noise was added to tbe certainty equivalent only when X = 8,.

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