Experimental comparison of individual behavior under risk and under uncertainty for gains and for losses

ORGANIZATIONAL

BEHAVIOR

AND

HUMAN

DECISION

PROCESSES

39, l-22 (1987)

Experimental Comparison of Individual Behavior under Risk and under Uncertainty for Gains and for Losses MICHELE COHEN,JEAN-YVES JAFFRAY,ANDTANIOS SAID UniversitPs

Paris I, Paris VI, and Paris XIII

This study concerns individual decision making under risk and under nonprobabilized uncertainty. It is based on a sample of 134 college students. These subjects had to perform binary choices offering prospects of fairly large gains or losses and knew that some payments would become effective. Data analysis leads to the following main conclusions: (a) subjects’ behavior on the gain side and their behavior on the loss side are totally unrelated, which fact in particular suffices to disprove the “reflection effect” hypothesis proposed by Kahneman and Tversky; (b) conventional, model independent definitions of risk attitudes are inadequate; (c) on the gain side, subjects generally take the exact probabilities of the events into account, whereas on the loss side, they appear to have recourse only to coarser categories of plausibility; (d) correlatively, on the gain side, subjects discriminate between situations of risk and of uncertainty and are, on the whole, pessimists, whereas on the loss side, they do not differentiate between those two situations. Q 1987 Academic press, hc

INTRODUCTION

Models of decision under uncertainty are commonly constructed from a basic model specific to choice under risk, i.e., probabilized uncertainty. As a typical example, the expected utility (EU) theory (von Neumann & Morgenstern, 1947) provides the core of the subjective expected utility (SEU) theory (Savage, 1954). This two-step approach has proved to be effective for constructing normative models; it is, however, not so promising for constructing descriptive models, unless it turns out that real decision processes under uncertainty indeed involve a “probability-oriented” processing of information, such as subjective probability assessment or lower and upper probability assessment. This would, however, appear questionable, since it would imply that decision makers cope with reducing it to a simpler but unfaa familiar situation -uncertainty-by miliar one -risk (real events seldom have objective probabilities). Actually the reduction processes could as well work the other way around, with decision making under risk following the same decision process as decision making under any particular situation of uncertainty. There have in fact already been several observations which would seem to support this thesis. In experiments on decisions under risk, subPlease send correspondence, including reprint requests, to .I. Y. Jaffray, Laboratoire d’Econom&rie, UniversitC Paris VI, 4 Place Jussieu, Paris 75005, France. 0749-5978187$3.00 CopyrIght 0 1987 by Academx Press, Inc. All rights of reproduction in any form reserved.

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COHEN,

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jects have been shown to be less sensitive to probabilities (overweighing of smaller probabilities, underweighing of larger ones) than predicted by the EU model (Kahneman & Tversky, 1979; Yaari, 1%5). Moreover, probabilities of losses have been found to be given much less import than probabilities of gains (Slavic & Lichtenstein, 1968). Thus, on the one hand, experiments offering subjects the choice between gains with known probabilities and gains with unknown ones (Ellsberg, 1961; MacCrimmon & Larsson, 1979; Slavic & Tversky, 1974) have disproved the subjective probability hypothesis-and hence the reduction of uncertainty to risk-on the gain side; on the other hand, however, the lesser influence of probabilities on preferences on the loss side allows one to conjecture that here risk and uncertainty may not be clearly differentiated by decision makers. The basic motivation for our experiment was to examine closely this last issue. This, however, involved acquiring a better understanding of the way people take probabilities into account, and the experiment was designed with this purpose in mind. Moreover, by investigating both the gain and the loss side, we tried to gain some insight into the following general issue: what is the relation between people’s behavior under risk and/or uncertainty in the domain of gains and their behavior in such situations in the domain of losses? METHODS

Subjects We used a sample of n = 134 college students, economics or computer science majors with some background in probability theory but not in decision theory. Questionnaires The experiment required subjects to make the type of binary choices used by certainty equivalence methods for utility assessment in the EU model; related choices were grouped in questionnaires to be filled out by the subjects; there were 10 different questionnaires. In questionnaire denoted by Qu.Gn (resp. Qu.Ln) the subjects had to make a series of binary choices between a risky prospect yielding gain (resp. loss) 1000’ with probability II, and nothing with probability (1 II), and a sure prospect yielding gain G (resp. loss L) for G (resp. L) varying from 0 to 1000 by successive increments of 50. According to the questionnaire, prospects were both positive or both negative, and probability II was given values i/2, ?& i/4, 1/6,or was unknown, which last is henceforth indicated by II = *. 1 Units are French francs; FF 1000 = $150 in 1983.

BEHAVIOR,

RISK, AND UNCERTAINTY

3

Probability II was not announced to the subjects who had to deduce it from the description of a simple event-generating device (dice throwing, coin tossing, etc.); the case of an unknown probability, II = *, corresponded to the drawing of a blue ball from an urn containing unknown proportions of blue and red balls. Figure 1 reproduces one of the questionnaires (translated from the French). Procedure

Each week, for 10 successive weeks, the subjects had to fill out two questionnaires, one of type Qu.G, the other of type Qu.Ln,, with II’ # II; each questionnaire was repeated after several weeks2; upper indexj = 1,2 distinguished the two copies of the same questionnaire. This duplication was intended to provide an estimation of individual variations in choices. With hypothetical payoffs, subjects do not display the same risk attitudes as with real payoffs (Slavic, 1969); moreover, careful choices are not to be expected at all when prospective gains or losses are small. Finding it impossible to offer the subjects payoffs both real and high, we devised the following scheme, making all payoffs but one hypothetical: (a) subjects were informed, at the beginning of the experiment, that after its completion, one of the subjects, one of the questionnaires, and a sure prospect value would be randomly selected; (b) that this subject would not be allowed to reconsider the preference he had previously indicated; (c) that he would receive the corresponding real payoff (conditionally to the realization of the event of probability Tl, if he had chosen the risky prospect); and (d) that should a questionnaire involving losses be selected, a IOOO-unitbonus would ensure him a final net gain. According to the isolation effect, a phenomenon identified by Kahneman and Tversky (1979), it could be expected that subjects would nonetheless consider they were being offered the questionnaire’s actual binary choices-and in particular merely ignore the possible bonus-while reacting as in the case of real payoffs.’ We are able to verify that the isolation effect did indeed work as expected. Preliminary

Data Processing

About two-thirds of the subjects switched their preference marks directly from the risky prospect column to the sure prospect column in all 2 We used the following sequence: GM, L,; G,, L,; G,, L,; G,, L,; G,, L,; Gv,, L,; G,, L,; G,, L,; GH, L,,; G,, L,. 3 See the Appendix for a description of the isolation effect’s presumed action.

QUESTIONNAIRE The experimenter is about to draw a card at random from a standard 52-card pack (13 spades, 13 hearts, 13 clubs, 13 diamonds). He asks you beforehand to choose between the following two propositions: Proposition A: You will receive F.F. 1000 if the card drawn is a diamond, and nothing otherwise. Proposition B: You will receive in all cases an amount G. Depending on the amount G proposed, what is your answer? (Indicate your choice by a mark (x) in the appropriate box).

(+) When the mark is made in the third or fourth column, the experimenter selects in your stead either proposition A or proposition B. FIG. 1. Qu.G, questionnaire with an example of a subject’s answer. G = gain, FF = French franc. A

BEHAVIOR,

RISK, AND UNCERTAINTY

5

the questionnaires, thereby assessing the certainty equivalents with a 50unit precision; on the other hand, in each questionnaire, a certain number (10% at most) of the subjects displayed an “indecision interval,” i.e., made one or several intermediate marks in the equivalence and noncomparability columns, suggestive of either intransitive indifference or incomplete preference.4 To sum up this information, we used two characteristics for each questionnaire form: the length of the indecision interval displayed (if any). equal to 50 (k - 1) units, where k is the number of marks in the last two columns; the so-called certainty equivalent of the risky prospect, defined, depending on the case, as the switchover value or as the midvalue of the indecision interval, and denoted by gin,i for Qu.G$ (resp. I& for Qu.L$ and subject i (examples are given in Fig. 2; in Fig. 1 the CE is 200, the length of the indecision interval is 100). Putting together each subject’s answers to pairs of identical questionnaires, we further calculated

Gn,i = i CSA,i+ .&,;I

or

L,,i = i

called the CE (for subject i) of Qu.Gn’s or Qu.L,‘s risky prospect. The data thus collected constitute a 134-sized multidimensional sample; in the following data analysis, we generally have to consider only one-or two -dimensional components (or combinations of components) of this sample, which are denoted in the following natural way: Gn will be the 134-sized sample formed by the Gn,i’s, L, - L,, will be the 134-sized sample formed by the (L,,i - Lnr,i)‘s, etc. RESULTS Preliminary

Statistical

Tests5

The following statistical analysis of the data rests on several basic assumptions, which we examine immediately. 4 Some subjects made several marks in the equivalence column; others made one mark in the noncomparability column and none in the equivalence column. Thus, subjects did not make a clear distinction between indifference and noncomparability, which fact induced us to merge the corresponding columns for our analysis. Moreover, although the layout of the questionnaires deliberately prompted the subjects to display monotonic preferences, two of the subjects switched several times from the first column to the second and back in some of the questionnaires. These answers were interpreted as resulting from a misunderstanding of the questions. Consequently the two subjects were discarded from the sample, reducing its size from 136 to 134. J In all tests, the significance level required was .05. We abbreviate: t test for Student t test and F test for Fisher-Snedecor variance ratio test. The nonparametric tests used are sign tests (Siegel, 1956).

First subject: Certainty equivalent CE = 450 Length of indecision interval l,, = 0

Second subject: CE = 425 I,, = not defined

FIG. 2. Examples of assessments of certainty equivalents and of indifference interval lengths. G = gain; FF = French franc. 6

BEHAVIOR,

RISK, AND UNCERTAINTY

7

Third subject: CE = 425 I,, = 50

G = FF600

X

Fourth subject: CE.= 425 l,, = 50

FIG. 2-Continued.

(i) Normality of Some Empirical Distributions By x2 tests we ascertained the goodness of fit between the empirical distributions of samples G,,,, G,/,, G*, and L,,* and those of Gaussian distributions, contrary to the other Gn’s and Ln’s whose means were too

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COHEN,

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off-centered with respect to the (0- 1000) range; however, since their distribution are unimodals and the sample is large, the robust tests used below are nonetheless valid. We also ascertained the goodness of fit to Gaussian distributions of the empirical distributions of differences L, - L,, G% - G,, and L, - Lv, in the entire sample, as well as the distributions of L, - LI/, and La - L, in PI--a subsample to be defined below. (ii) Invariability of Subjects’ Preferences The use of averaging in estimating subject i’s CE in Qu.Gn by Gn,i = f (& + g&J rests on the idea that subjects have definite, invariable preferences, which they are merely unable to express precisely; can it, however, be assumed that g& and g& are two realizations of identical random variables? Denoting their theoretical means by mh,i and m$i, we tested the hypothesis mh,i = rnki for all i by t tests for each Qu.Gn (resp. each Qu.Ln) separately. For nine questionnaires, t was definitely smaller than the critical value 1.96; for the last one, Qu.G,, however, we found t = 2.13. Since it is not statistically abnormal to obtain one excessive value out of 10, we concluded that the hypothesis should be accepted globally. (iiz’) Equal Accuracy of All Subjects Admitting thus that CE’s are random variables, their standard deviations can be used to characterize the accuracy of subject answers in the corresponding questionnaires: the smaller the standard deviation, the greater the accuracy. The standard deviation of subject i’s CE in Qu.Gn (resp. Qu.Ln) can be estimated by the absolute values of discrepancies

Al,i = & [l&i - G,J)* Agn,i= $ [gA,i- g&](resp. By t tests and rank tests we showed that discrepancies Agn,i and Aln,i could all be assumed to have mean 0 and to be mutually independent; thus if all subjects have equal accuracy, mean values ai = A

2 Agni + C A1n.i ( n ) n form a 134-sized sample from a population with an approximately Gaussian distribution with mean 0 and estimated standard deviation. By a x2 test we checked that this hypothesis could be accepted (x2 = 4.64 < 16.9, critical value), hence that equal accuracy of all subjects,could be assumed. We therefore estimated the (common) individual standard deviation in

BEHAVIOR,

RISK,

Qu.Gn by sn defined by srf = &

9

AND UNCERTAINTY

5 (Agn,i - Z&)* and used the I-1

similar definition for that in Qu.Ln (their values are given in Table 1); their mean (quadratic) value over the 10 questionnaires was denoted by S; thus,

‘/2 ;=(-lpi; 10 1. (iv) Equal Accuracy in All Experiments The individual standard deviation slightly increases with the average CE value: in a linear regression sn = am + 6, with m equal to G or G, we found a = 0.09, b = 93.7 (r* = .28); a t test, however, showed that a = 0.09 was not significantly different from zero. Moreover, by a series of F tests the equality between the individual standard deviation and its mean value over all the questionnaires were tested; the equality hypothesis was accepted every time. The mean individual standard deviation is S = 125. (v) Significant Diversities among Subject Preferences We now come to the crucial issue: is the accuracy of the subjects sufftcient to reveal true preference dissimilarities among them? For each questionnaire Qu.Gn or Qu.L, separately we tested the assumption that all subjects had the same theoretical certainty equivalent. We used the fact that under this assumption another possible estimator of the common individual

standard deviation is un, defined by crA = 2.

&

for Qu.Gn, and by the similar expression for

2 (Gn,i - &I* 1-l

Qu.Ln. Table 1 gives the values of sn and an. We thus tested their equality by an F test; the equality hypothesis was rejected every time. Thus the scattering of subjects’ certainty equivalents in the various questionnaires cannot be considered to result merely from random variation. Subjects have diverse attitudes, and they reveal them with sufficient accuracy for the analysis of the data to produce reliable information concerning these diversities. Inadequacy

of Conventional

Definitions to Risk

of Attitudes

with Respect

Three attitudes with respect to risk are classically distinguished and

10

COHEN, JAFFBAY, AND SAID TABLE 1 DISTRIBUTIONS OF CERTAINTY EQUIVALENTS AND TESTS OF DIVERSITY AMONG SUBJECT PREFERENCES”

Probability of gain or loss II

CE empirical distributions Gn or L, Mean En or En

Individual standard deviation sn

F

240 200 193 210 229

145 132 122 124 110

2.74* 2.30* 2.50* 2.87* 4.33*

250 228 219 202 246

130 128 115 130 110

3.70* 3.17* 3.63* 2.41* 5.00*

Standard deviation % Gain side

Unknown % = 500 x % = 333 x ‘92= 250 x ‘% = 167 x

10-3 10-3 10-r lO-3

374 488 406 369 327 Loss side

Unknown % = 500 x 10-3

!h = 333 x 10-s ‘A = 250 x 10-r L/6= 167 x lo-’

353 374 325 304 237

a F tests test, for each questionnaire, the theoretical equality of sn and on (which should hold if all subjects had the same theoretical CE). * p < .OOl; df = (133, 133).

defined as follows: with gains measured positively and losses negatively, a subject is risk-averse (resp. risk-neutral, risk-seeking), if, for any prospect, his certainty equivalent is smaller than (resp. equal to, greater than) the mathematical expectation of that prospect. Note that these definitions are intrinsic, i.e., that their meaning is independent of any particular decision model. We basically used these definitions; however, to compensate for the fact that questions were restricted to round numbers, we considered subjects to be risk-neutral up to a 25unit difference between mathematical expectations and certainty equivalents and came up with the proportions given in Table 2. The instability of risk attitudes is striking: on the gain side, subjects move from risk aversion to risk seeking when II decreases; the opposite is true on the loss side. A possible explanation for this phenomenon (suggested by a referee) is the existence of a positive bias in the subjects’ expression of their CE’s, the proximity of the zero level barrier restricting the length of negative deviations; this would, however, imply that individual standard deviation becomes smaller when II decreases; as seen above, data do not show this decrease to be significant, and therefore the barrier effect can only be of minor importance. The reason why subjects’ risk attitudes are not correctly conveyed by

BEHAVIOR,

RISK,

11

AND UNCERTAINTY

TABLE

2

ATTITUDES WITH RESPECTTO RISK Probability II of gain or loss Gain side ‘vi 1% 1% 1% Loss side % ‘$5 1% 1%

Subjects’ risk attitudes (%) Averse

Neutral

40 10 5 2

28 38 39 23

32 52 56 75

11

19 36.5 42 33.5

70 36.5 19 15.5

27 39

51

Seeking

the conventional definitions may simply be that these definitions, despite their intrinsic character, take their origins in the EU model, and therefore share in its deficiencies. Nonexistence of a Reflection Effect It is clear from Tables 1 and 2 that subjects behave quite differently on the gain side and on the loss side; Kahneman and Tversky (1979) hypothesized a reflection effect: subjects who are risk-averse in the domain of gains become risk-seeking in the domain of losses and vice versa; their claim, which was entirely based on an across-subjects analysis of their data, was, however, not confirmed by a within-subjects analysis conducted by Hershey and Schoemaker (1980). We made a within-subjects analysis of risk-attitudes in symmetrical questionnaires Glc,and L,, the results of which are shown in Table 3. As seen in Table 3, only a minority of the subjects (28.5 + 9 + 3.5 = 41%) behaved in conformity with the reflection effect; actually about as many (39%) had a constant attitude. Moreover, Table 3 exhibits a striking feature-its lines, as well as its columns, are roughly proportionalwhich suggests a totally adverse hypothesis: independent attitudes on the gain side and on the loss side6; this hypothesis was indeed accepted in a x2 independence test (x2 = 5.17 < 9.46, critical value) motivating further study of this issue. 6 Similar tables comparing within-subject risk attitudes in Qu.Gn and Qu.L, for II = %, %, and */6also show the absence of any reflection effect. However, due to an extremely unequal distribution of the subjects among the various attitudes (some classes have 0, I, or 2 members) they are not as strikingly suggestive of independence (and x2 tests cannot be performed).

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COHEN, JAFFRAY, AND SAID TABLE 3

WITHIN-SUBJECT COMPARISON OF RISK ATIITUDES IN QUESTIONNAIRES Qu.G,

AND

Qu.L, (%)” Subjects’ risk attitudes on the gain side Subjects’ risk attitudes on the loss side Averse Neutral

Averse

Neutral

6 (4.5)

5.5 (7.5) 9 (5.5) 4.5

1.5

(3) (E,

Seeking

Seeking

Total

28.5

40

(28) 17.5 (19.5) 24 (22.5) 70

(6) Total

11

a Numbers in parentheses

18

28 32

indicate the products of the margins. x2 = 5.17 (n.s.).

Absence of Correlation

between Attitude in the Domain of Gains and in That of Losses

Lower values of r(Gn, L,,)-type correlation coefftcients stand in contrast, on Table 4, with higher values of r(Gn, Gn,)-type and r(Ln, &)-type ones; F tests showed that only one of the r(Gn, &)-type correlation coeffkients, r(G%, LM), was significant (F = 5.47 > 3.86, critical value); since at the .05 significance level the observation of one excessive value out of 25 is not statistically abnormal, we can conclude that, globally, there is no correlation between each subject’s attitude in the domain of gains and his attitude in the domain of losses. On the contrary, all the TABLE 4 SIGNIFICANCE OF CORRELATION COEFFICIENTS BETWEEN CERTAINTY EQUIVALENTS’

G.

G4 Gl.3

G.

G,

Gn

G,,,

G,

L.

1

.57**

.63** .67**

.60** .60** .63**

.66** .53** .68** .69**

1

1

G, G L. Lb2 Ll4 Lv4 LM

a F tests: df = (1, 132). * p < .05. **p c .OOl.

1

1

L,

h.3

- .07

-.12

-.ll

-.08

-.03 -.06

-.12 -.16 - .20* .67**

- .07 -.14 - .08 -.09

-.13 1

1

-.lO .64**

.75** 1

Lv.

Ll.5

- .09 -.15 - .08 -.07 - .05 .62** .52** .62** 1

- .05 -.15 .003 - .OOl

.002 .57** .56** .67** .55** 1

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r(Gn, Gnu)-type and r(&, &)-type correlation coefficients were found to be highly significant; their values are actually in full agreement with the hypothesis of a consistent attitude on each side separately, since, given subjects’ poor accuracy, they cannot be expected to get much closer to 1 (they should be compared with correlation coefficients of CE’s in identical questionnaires, r(gA, g&> and r(#, I$, which do not exceed 59). Remark. The fact that we found choices in the domain of gains to be mutually correlated but not correlated to choices in the domain of losses suffices to prove that subjects did not take into account the lOOO-unit bonus which transformed losses into gains; it is also clear from the data that subjects did not consider Qu.L, and Qu.G, to be identical. Thus the isolation effect did occur as expected. Attitude with Respect to Uncertainty (i) Definitions The notion of attitude with respect to uncertainty, first introduced by Ellsberg (1961), does not claim to reflect subjects’ absolute behavior under uncertainty but the differences between their behavior with respect to risk and with respect to uncertainty -more precisely, to the extreme situation of uncertainty known as complete ignorance. Ellsberg observed subjects’ choice between a prospect yielding a given outcome with probability ‘/2 and a prospect yielding the same outcome with an unknown probability; in our experiment, however, each subject’s preference was determined indirectly by the comparison of G,,i and G*,i, on the gain side, and of L,,i and L+,i, on the loss side. Types of attitudes with respect to uncertainty are defined as follows: a subject is a pessimist’ when he prefers the risky prospect to the uncertain prospect, an optimist if he has the opposite preference, and a moderate if he has no definite preference. However, to account for the fact that questions were restricted to round numbers, we considered subject i to be a pessimist on the gain side (resp. loss side) when G,,i - G*,i (resp. L*,i - L,,i) was greater than 50, to be an optimist when it was smaller than -50, and to be a moderate otherwise, and came up with the proportions of the three attitudes among subjects given by Table 5. The quasi-proportionality of the lines and the columns of Table 5 suggests independent attitudes on the gain side and on the loss side, confirmed by a x2-independence test (x2 = 5.48 < 9.46, critical value), and

’ Use of the terms “pessimism” and “optimism” in this sense goes back to Hurwicz (see Lute & Raiffa (1957, chap. 13)) and we considered them to be more evocative than Ellsberg’s terms, aversion and preference, for “ambiguity.”

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COHEN, JAFFRAY, AND SAID TABLE 5 WITHIN-SUBJECTCOMPARISONOF AITITUDES WITH RESPECTTO UNCERTAINTY (%)O

Subjects’ attitudes on the gain side Pessimists Moderates Optimists

Subjects’ attitudes on the loss side Pessimists 13.5 (16.5) 13.5 (10.5) (K)

Moderates

21.5 (17)

28.5

Total 58 36.5

(ii) 1.5

(2) Total

Optimists

41.5

&I 2 (1.5) 29.5

5

a Numbers in parentheses indicate the products of the margins. x2 = 5.48 (n.s.).

by the fact that r(Gti - G*,, L, - L,) = - .08. Thus, under uncertainty as under risk, behavior on the gain side is uncorrelated with behavior on the loss side. Predominant attitudes shown by Table 5 are pessimism on the gain side and moderation on the loss side; however, to determine the exact conclusions that can be derived from Table 5 we have yet to take into account subjects’ lack of accuracy, which we now turn to. (ii) Moderation in the Domain of Losses The empirical distribution of L, - L, has mean 21 and standard deviation 136; we have already mentioned the goodness of tit between this distribution and a Gaussian distribution; by using a t test and an F test, respectively, we moreover showed that neither is its mean significantly different from zero nor is its standard deviation significantly different from S (= 125); last, a sign test also concludes to the identity of L, and L,, since L,,i - L*,i > 0 (resp. ~0) for 66 (resp. 53) subjects. Thus although the observed values of differences L,,i - L,,i identify only 42% of the subjects as moderates, it should be concluded that these differences are entirely due to the subjects’ poor accuracy and that subjects, as a whole, do not make any distinction between sustaining a loss with probability !1!2and sustaining the same loss with unknown probability. (iii) Pessimism in the Domain of Gains On the gain side, where 58% of the subjects are identified as pessimists, the predominance of pessimism is clearly significant: strengthening the denomination of pessimists to these subjects for whom GEi.i - G,,i is greater than 125 (mean individual standard deviation) still leaves 48% of

BEHAVIOR,

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pessimists. Furthermore, a sign test on differences G~,i - G*,i is conclusive with p s .OOOl,as is even the sign test on differences (Gti,j - G* i) - 125; thus, subjects as a whole are clearly pessimistic. Strong predominance of pessimism on the gain side has already been observed in several studies (Ellsberg, 1961; MacCrimmon & Larsson, 1979); in none of them, however, was it possible to test the hypothesis of universal pessimism. (iv) Incompatibility of Subject Behavior with the SEU Model In Qu.G, and Qu.L, subjects who assess subjective probabilities to the relevant events are bound to ascribe to them equal probabilities by Laplace’s rule; Laplacian subjects are therefore moderates (but moderates are not necessarily Laplacian subjects). Thus at most a small minority (17%) of subjects behaves under uncertainty in conformity with the SEU model. Again, this conclusion is in accordance with the results of earlier experimental studies (e.g., Ellsberg, 1961; MacCrimmon & Larsson, 1979) which found high proportions of subjects violating Savage’s sure-thing principle (91% in Cohen & Jaffray, 1981). Limited Sensitiveness of Subjects to Probabilities on the Loss Side Table 1 shows clearly that average CE’s of risky prospects vary much less than their mathematical expectations, especially on the loss side. A possible explanation for this fact could be that a good many of the subjects either do not take the probabilities into account or at least do not take them precisely into account. To investigate the validity of this hypothesis we first applied ANOVA tests to groups of empirical distributions (G,, G++ G,) and (G*, G&, G,, Gti) on the gain side as well as to the corresponding group (LM, L,, L,) and (LM, L,, L,, L& on the loss side. As shown in Table 6, the theoretical equality of the distributions was TABLE 6 ANOVAa

CERTAINT~EQUIVALENTDEPENDENCEONPROBABILITY

Groups of probabilities of gain or loss Gain side ‘h, ‘A, % %( K , ‘A, 1% Loss side ‘Yi, 1%)1% Y2,L/, %, Y6

Mean CE

df

Critical value

F

421 397

(2, 399) (3, 532)

3.02 2.6

12.19** 14.3**

334 310

(2, 399) (3, 532)

3.02 2.6

3.4* 8.4**

a The computation of F values requires data appearing in Table 1. * p < .05. ** p<.Ool.

16

COHEN, JAFFFiAY, AND SAID

rejected for each of the groups; note, however, that for the (LM, L,, L,) group, F = 3.4, quite close to the critical value (3.02). We moreover studied the significance of discrepancies between pairs (Gn,, Grin)or (Lnl, Ln”) by counting, for each of these pairs, the number of violations of the CE’s natural order, and performing sign tests. As seen in Table 7, discrepancies were found to be significant, with the exception of the (LM, L%) pair. Concerning the identity of L, and Lti, we furthermore checked the goodness of fit between L, - L, and a Gaussian distribution and showed that the mean of LB - L, was not significantly different from zero, nor was its standard deviation different from S. It can therefore be concluded that subjects on the whole take the precise probabilities of the relevant events into account on the gain side but not on the loss side, where probabilities % and VI (at the least) belong to the same category of belief. The possibility that many of the subjects in fact have even coarser categories of belief is discussed in the next section. Diversity of Subject Sensitiveness to Probabilities (i) Evidence of Subjects’ Diverse Sensitiveness Subject by subject data analysis showed that 14 subjects were expected-gain maximizers, i.e., had CE’s which were equal to (or, allowing TABLE 7 CERTAINTY EQUIVALENT DEPENDENCEON PROBABILITIES:SIGN TESTSOF CE DIFFERJZNCEP Pairs of probabilities of gain or loss (fI’,ff”)

Gain side Number of subjects for whom GuPsGu~,iis >o = 0 o = 0
(‘Y&‘/6

(%,%)

(%,%)

106 11 17 8.02*

103 10 21 7.3*

98 7 29 6.1*

92 15 27 5.9*

82 I 45 3.3*

84 15 35 4.4*

105 14 15 8.2*

92 12 30 5.6*

79 13 42 3.4*

95 11 28 6.0*

68 9 57 0.98(n.s.)

90 10 34 5.0*

LIz.is the sign-test statistic (critical value 1.96). *p<.OO1.

(vi, 1%)

(‘vi,‘vi)

( 1%) 1%)

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AND UNCERTAINTY

17

for round numbers, differed at most by 25 units from) the mathematical expectations of the corresponding Qu.Gn’s risky prospects; 8 of these subjects were also found to be expected-loss minimizers (similar detinition) . By the wide range of their CE’s, these subjects differ greatly from the rest of the sample; they also differ from other subjects in individual standard deviation under risk, which averages 65, as opposed to 125 for the whole sample. A likely explanation for this greater-but not perfectaccuracy is that these subjects calculated the mathematical expectations of the risky prospects but were not consistent in their use of round numbers. If this proves to be the case, it implies that the six expectedgain maximizers who did not minimize their expected losses actually renounced on the loss side a normative criterion they had complied with on the gain side-a drastic change in behavior. Despite their high sensitiveness to probabilities, the eight expectedloss minimizers are not numerous enough to account alone for the variation of the mean CE’s with the probabilities; one may, however, wonder if the entire variation is not due to that subsample which includes, aside from them, those subjects who have roughly the same criterion of choice but use it more loosely, and who appear to be risk-neutral or nearly riskneutral on the loss side. Thus, to check the truth of this hypothesis, we selected a subsample of subjects-those subjects satisfying the condition that L,,i c 375 (i.e., 500 minus 125, the mean individual standard deviation)-thus making sure that most actual risk-neutral subjects were excluded from the subsample; we called this subsample Pr. (ii) Insensitiveness to Probabilities on the Loss Side in Subsample $I Subsample P, consists of 62 subjects; on the gain side, its characteristics are not statistically different from whole sample P’s characteristics; on the loss side, however, IA’s (mean CE’s in Pi) can be expected to take lower values than En’s (the corresponding mean CE’s in P) since L,,i s 375 for every subject i in Pi and correlation coefficients r(L, LnV) were found to be high; Table 8 shows that this is indeed the case. The striking fact about z/, is the mutual proximity of &EL, and Lb, and the transposition of z& and & (?& is also smaller than Lh but this is not necessarily significant, since, due to Pi’s mode of definition, it must be biased downward). The reason for these facts cannot be that the CE’s of risk-seeking subjects, being all close to zero, have to be close to each other, since (a) this would also be true for & which, on the contrary, was found to be significantly different from the other mean CE’s, (b) in the subsample selected by the similar criterion on the gain side (G,,i c 375), mean CE’s increase significantly with II, (c) as shown by F tests,

18

COHEN, JAFFBAY, AND SAID TABLE 8 COM~ARISONOFSAMPLEG'S

ANDSUBSAMPLEP'S

LOSSSIDECHARACTERISTICS~

CE distribution in p1 SD

SD in P(sn)

Individual SD in P,

F

216 156

130 128

131 125

.98(KS.) 1.05(n.s.)

304 325

269 240 (biased) 253 233

187 168

130 115

136 112

1.05 .91(tbs.) (n.s.)

237

186

170

110

89

1.52(n.s.)

Probability rl of loss

Mean CE in P (TJ

Unknown !4

353 374

; %

Individual

Mean (&I)

’ F is the ratio of the individual variancesinland P,; df = (133,61);critical values(0.65, 1.58).

individual standard deviations are not smaller in P, than in P (see their values in Table 8), which excludes a barrier effect at zero. We shall propose another explanation, justified by the following results concerning P,: as shown by Table 9, ANOVA tests conclude to the identity of the empirical distributions in group (L,, L,, L,,), as well as in groups (L,, L,, Lw, LJ and (L+, L,, Lti, L,s8; moreover, sign tests (see Table 10) confirm that the identity of the members of pairs (L,, LH), (L,, L,J, and (L%, L,,J can be accepted (but reject the identity of those of other pairs, including (L,, LJ); finally, the goodness of fit between L, L, (resp. L, - L,) and a Gaussian distribution can be checked, and it is found that its mean, - 13 (resp. -2O), is not significantly different from zero, nor is its standard deviation significantly different from S. Thus members of P, do not make any significant. distinction between sustaining a loss with probability ?I& or with probability Y3, or with an unknown probability. Moreover, since P, contains 62 of the 134 subjects, it can be safely inferred that the equal treatment of a loss with probability Y2 and a loss with an unknown probability, which is known to be true in P, is also true in PI (the bias in L, values precludes the checking of this property directly). When we consider these facts together, we can conclude that members of PI, as a whole, do not make any significant distinction between sustaining a loss with probabilities 1/4,‘15, 1/2,or with an unknown probability.9 * Similar tests applied, on the gain side, show significant discrepancies among the distributions in groups (G,, G,, G,,) and (LM, L,, L,, LA, which fact reinforces our conclusion that members of 9, behave on the gain side like members of the whole sample, P: in particular, they take precise probabilities into account. p Besides P,, one can also consider the symmetric subsample characterized by L,,i > 625, which consists of only seven subjects; although conclusions drawn from such a small sample cannot be decisive, it may be worth noting that its members do not make any statistically significant distinction between sustaining losses with probability %, !4.,unknown, or even ‘A.

BEHAVIOR,

19

RISK, AND UNCERTAINTY TABLE 9

CERTAINTY EQUIVALENT DEPENDENCE ON PROBABILITY ON THE Loss SIDE IN SUBSAMPLE 9,: ANOVA” Groups of probabilities

Mean

of loss %, I%, 1% FL, L/, 'A, '/6

*, vi,%, %

CE

df

242 228 249

(2, 183) (3, 244) (3, 244)

Critical value

F

3.04 2.64 2.64

.22 (n.s.) 1.80 (n.s.) .45 (as.)

L1The computation of F values requires data appearing in Table 8.

Learning and Complete Preferences Besides the 14 expected-gain maximizers, 24 other subjects used this same criterion during the first 2 weeks of experimentation but ceased to apply it during the last 8 weeks. This fact suggests that a learning phenomenon took place during the course of the experiment. Another aspect of learning can be found by examining the number of subjects exhibiting indecision intervals, which dropped rapidly from 13 subjects in each first-week questionnaire to only 3 subjects by the fourth week (with the exception of Qu.LA and Qu.Li where 6 subjects were found; average length of the indifference intervals was also greater; thus subjects were a little more indecisive in choices under uncertainty than under risk; the difference is, however, not conclusive). Thus, after a learning period, nearly all subjects felt able to express a complete preference relation on choices under uncertainty as well as under risk. DISCUSSION AND CONCLUSIONS This study limited its investigation to probability interval (0-S) and TABLE 10 CERTAINTY EQUIVALENT DEPENDENCE ON PROBABILITY ON THE Loss SIDE IN SUBSAMPLE 9, : SIGN TE!RS OF CE DIFFERENCESO

Pairs of probabilities of loss (WJI“) (*,%I Number of subjects for whom L,,,,-L,.,, is >o =o co Z

28 19

15 1.92(n.s.)

c*,w

31 8 23 1.08(n.s.)

Dz is the sign-test statistic (critical value l.%). *p< .ool.

(*,w

41 13 8 4.F

(%,‘h)

('%,%)

(1.4,'%)

24 6 32

40 9 13

- l.M(n.s.)

3.7*

40 9 13 3.7*

20

COHEN,

JAFFRAY,

AND SAID

gain or loss intervals (O-1000); nonetheless, its conclusion that subject choices in the domain of gains are not correlated with those in the domain of losses is so strongly supported by the data that it must be robust; an unforeseen result, this conclusion may have already been detectable, but overlooked, in other studies: for example, in twenty-one out of twentyeight 2 x 2 contingency tables given by Hershey and Schoemaker (1980, Table 4) x2 tests-not performed by them-conclude to the independence of risk attitudes on the gain and on the loss side. Consequences of this independence are far-reaching: The observation of a subject’s choices on, say, the gain side, is of absolutely no usefulness in predicting his choices on the loss side; the reflection effect is only an illusion created by the predominance of the risk-averse attitude on the gain side and that of the risk-seeking attitude on the loss side; moreover, the predominance of these attitudes implies the scarcity of the conjunction of the opposite attitudes, which facts confirm the doubts of Hershey, Kunreuther, and Schoemaker (1982) about the validity of the opposed conclusions obtained by Fishburn and Kochenberger (1980) by pooling heterogeneous data. Limited sensitiveness of subjects to probabilities, especially on the loss side, is the second main conclusion of this study; this is not as clear-cut a conclusion as the first one; the reason for this is that subjects do not behave homogeneously in this respect (for example, expected-gain maximizers are clearly differentiated from other subjects) and that subjects’ lack of accuracy makes it difficult to divide them into appropriate categories with perfect reliability. It is nonetheless ascertained that an important minority of decision makers, who take the precise probabilities into account on the gain side, seem to distinguish only coarse categories of belief on the loss side in comparable situations. Further experiments (involving other outcome ranges, direct comparison of risky prospects, etc.) are clearly needed to establish the generality of this observation. In any case the recourse to separate models to explain choices under risk in the domain of gains and in the domain of losses seems unavoidable. Even in the domain of gains, these models have to be distinctly different from the EU theory, to whose list of known deficiences-inadequacy to explain many results of laboratory experiments (e.g., Allais, 1979) and field studies (Kunreuther, 1976); dependence of utility on the assessment method (de Neufville & McCord, 1982; Hershey et al., 1982)-must now be added yet another defect: incapacity to identify subjects’ attitudes with respect to risk. However, if Kahneman and Tversky (1979) are right in assuming that subjects ponder utilities by weights II@), rather than by the probabilities, p, themselves, then correct definitions of risk attitudes could be based on the comparison of the risky prospect’s CE with [II($)lII(l)] x g, where g is the sure outcome. It can

BEHAVIOR,

RISK, AND UNCERTAINTY

21

FIG. 3. Isolation effect scheme on the gain side.

be conjectured that with this new definition subjects’ attitudes become constant; since our data did not allow us to assess the weight functions, this will be the object of another experiment. Our observations concerning the relation between behavior under risk and that under uncertainty have been the following: on the gain side, subjects take probabilities into account under risk but do not use subjective probabilities under uncertainty; on the contrary, on the loss side, choices under uncertainty and under risk do not differ and many subjects exhibit a limited sensitiveness to probabilities. It is therefore our conclusion that subjects never equate uncertainty with risk, but sometimes, on the loss side, equate risk with uncertainty. APPENDIX

The isolation effect, a phenomenon identified by Kahneman and Tversky (1979), is the result of the subjects’ spontaneous simplified representation of binary choices; subjects disregard components common to both alternatives, such as a common bonus or a shared conditional outcome. Thus in our experiment subjects were expected to choose as if payoffs

FIG. 4. Isolation effect scheme on the loss side.

22

COHEN, JAFFRAY, AND SAID

were real, according to the scheme in Fig. 3, in which E is the event specifying that “payoffs of choice (A, B) will be real for that subject.” In choices involving losses they were moreover expected to merely ignore the lOOO-FFbonus according to the scheme in Fig. 4. REFERENCES Allais, M. (1979). The foundations of a positive theory of choice involving risk and a criticism of the postulate and axioms of the American School (from 1952, French version). In M. Allais & D. Hagen (Eds.), Expected ufility hypotheses and the Allais paradox (pp. 27-245). Dordrecht: Reidel. Cohen, M., & JaBray, J. Y. (1981). Experimental results on decision making under uncertainty. Methods of Operations Research Proceedings, 44, 275-289. De Neufville, R., & MC Cord, M. (1982). Utility dependence on probability: An empirical demonstration. Journal of Large Scale Systems. Ellsberg, D. (1961). Risk, ambiguity and the Savage axioms. Quarterly Journal of Economics, 75, 643-669.

Fishbum, P C., & Kochenberger, G. A. (1980). Two-piece Von Neumann-Morgenstem utility function. Decision Science, 10, 503-518. Hershey, J., & Schoemaker, P. (1980). Prospect theory’s reflection hypothesis: A critical examination. Organizational behavior and human performance, 25, 395-418. Hershey, J., Kunreuther, H., & Schoemaker, P. (1982). Sources of bias in assessment procedures for utility functions. Management Science, 25(8), 936-953. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica,

47(2), 263-292.

Kunreuther, H. (1976). Limited knowledge and insurance protection. Public Policy, 24(2). Lute, R. D., & Raitfa, H. (1957). Games and decisions. New York: Wiley. MacCrimmon, K. R., & Larsson, S. (1979). Utility theory: Axioms versus “paradoxes.” In M. Allais & D. Hagen (Eds.), Expected utility hypotheses and the Aliais paradox, (pp. 333-409). Dordrecht: Reidel. Savage, L. J. (1954). The foundations of statistics. New York: Wiley. Schoemaker, P J. (1980). Experiments on decisions under risk: The expected utility hypothesis. Boston: Nijhoff. Siegel, S. (1956). Nonparametric statistics. New York: McGraw-Hill. Slavic, P. (1969). Differential effects of real versus hypothetical payoffs on choices among gambles. Journal of Experimental Psychology, 79, 434-437. Slavic, P., & Lichtenstein, S. (1968). The relative importance of probabilities and payoffs in risk taking. Journal of Experimental Psychology, 78, 1- 18. Slavic, P., C Tversky, A. (1974). Who accepts Savage’s axiom? Behavioral Science, 19, 368-373. von Neumann, J., & Morgenstem, 0. (1947). Theory of games and economic behavior (2nd ed.). Princeton, NJ: Princeton Univ. Press. Yaari, M. E. (1965). Convexity in the theory of choice under risk. Quarterly Journal of Economics,

78, 278-290.

RECEIVED: August 21, 1984