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Testing risk aversion and nonexpected utility theories
JOURNALOF
-Behevior tit-
Journal of Economic Behavior & Organization Vol. 33 (1998) 285-302
ELSEVIER
Testing risk aversion and nonexpected utility theories Edna Loehman* Department of Agricultural Economics, Purdue University, W. Lafayette IN 47907-l 133, USA Received
16 December
1992; received in revised form 15 October
1996; accepted
6 November
1996
Abstract This paper
tests
risk
aversion
and compares
several
nonexpected
utility
theories
using
paired
hypotheses associated with risk aversion are found not to hold in preference data. Expected utility with rank dependent probability is found to provide a smooth representation of preference for a wide range of preference types. Published by Elsevier Science B.V.
comparisons.
Economic
JEL classijication: Keywords:
C52; C91; D81
Risk aversion; Nonexpected
utility; Risk reference
1. Testing risk aversion and nonexpected
utility theories
Stemming from the Allais paradox, both psychologists and economists have noted the inadequacy of expected utility theory to describe risk behavior. Kahneman and Tversky (1979) introduced new paradoxes including the reflection effect: the same person can be observed to be both risk-averse, and risk-seeking depending on whether gambles are presented as gains or as losses. Thus, Schoemaker (1989) suggests that risk aversion may not be an intrinsic behavior. At the same time, nonexpected utility theories have become fairly accepted as an alternative to expected utility. There are now a variety of them (see Camerer, 1989a, 1989b for a summary). These theories use a subjective transformation of probability to represent preference. Generally, an S-shaped function probability transformation - first * Corresponding
author.
0167-2681/98/$19.00 Published by 1998 Elsevier Science B.V. All rights reserved PII SO167-268 1(97)00097-8
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concave and then convex with an inflection around p=O.40 - has been obtained in empirical work (Wu and Gonzalez, 1996). However, empirical work (e.g. Currim and Sarin, 1989) has assumed a parametric concave form for the utility function in order to measure the probability transformation. The nature of risk preference is the main subject addressed by this paper. Several alternative nonexpected utility theories of probability transformation are compared here for gambles involving mixed gains and losses. Concomitantly, risk aversion is examined as an appropriate description of behavior. The Pratt-Arrow coefficient of risk aversion (Pratt, 1964, Arrow, 1965) describing the relative concavity of the utility function has been based on asset integration, an assumption that has not held up well in experimental studies (Thaler, 1987). Theories which do not depend on asset integration therefore provide possible superior explanations of behavior. How to combine certainty and risk without asset integration - combining aspects of both the Kahneman and Tversky (1979) model of utility and the Friedman and Savage (1948) model of utility over wealth - is demonstrated in this paper (see Appendix A for theory). Results of this paper suggest that modification of the utility function as related to risk aversion is needed in addition to nonexpected utility transformations of the probability function. For combined gains and losses, results clearly show that traditional economic definitions of risk aversion do not hold behaviorally, since the shape obtained for utility is concave for losses and small gains and convex for larger gains. A test of insurance preference comparing pure loss gambles, however, suggests the inadequacy of a single theory of utility that encompasses certainty, gains, and losses compared separately and in combination. In addition, this paper demonstrates several empirical innovations. First, the method for determining the form of utility and probability functions is nonparametric; therefore no restrictive assumptions about concavity need to be made. Second, the method of paired comparisons is used to measure nonexpected utility theories, thus avoiding the problems of certainty equivalents.‘.’ Third, both utility and probability scale values are estimated simultaneously from paired comparisons data, again avoiding restrictive assumptions.s Finally, measurement is performed for each individual separately, in contrast to empirical studies (Wu and Gonzalez, 1996, Currim and Sarin, 1989) that pooled preference data to estimate preference parameters for a composite individual.
’ Paired comparisons is considered by psychologists to be a reliable method of eliciting modeling preference; see David (1988). It includes a method for utility measurement. Here we use the method of response, measuring utilities in different ways. ’ The certainty equivalent has been the predominant tool for expected utility measurement (see e.g. Becker et al., 1964). The certainty equivalent remains valid in nonexpected utility context (Chew, 1983, Loehman, 1994). So, this concept could also be applied for the measurement of nonexpected utility. However, validity problems have been found with use of the certainty equivalent; see ðer and Plott (1979). a The certainty equivalent was used to compare Prospect Theory and expected utility by Currim and Satin, 1989. They compared lotteries over gains and losses separately; the Prospect Theory utility function shape for separate gains and losses was confirmed by their study. Subjective probabilities and utility were not measured simultaneously, and risk aversion was assumed. They assumed that preference is represented by an exponential utility function. The risk-aversion coefficient was first estimated from certainty equivalents. For the assumed utility, subjective probabilities were then measured.
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Table 1 Lottery comparisons” Lottery
Set I
Set II
1 2 3 4 5
($100, $0;0.5,0.5) ($50, $50;0.5,0.5) ($200,-$100;0.5,0.5) ($300,-$200;0.5,0.5) ($150, -$50;0.5,0.5)
($250,-$200;0.5,0.5) ($50,~$100;0.75,0.25) ($36,~$200;0.9,0.1) ($200-$50;0.25,0.75) ($300-$19.50;0.1,0.9)
aDollar amounts are listed followed by the probabilities
of each outcome.
2. Preference elicitation instrument A preference elicitation instrument was designed both to test risk aversion and to measure alternative nonexpected utility theories. Due to problems with the certainty equivalent, paired comparison was selected as the method of preference elicitation.4 2.1. Design Table 1 shows the compared gambles. Two sets of lotteries (I and II) were presented for paired comparison. In order to test for risk neutrality and consistency with secondorder stochastic dominance, all lotteries in each set have the same expected value. Outcomes in Set I have 50-50 probabilities and an expected value of $50. In Set II, probabilities are varied in order to estimate subjective probability values. One certainty equivalent was also elicited; this certainty equivalent corresponds to having a 50-50 chance at the combination of the best and worst outcomes. Respondents were categorized as to whether they liked or disliked having a chance at this lottery. For respondents who liked the chance, the certainty equivalent is the minimum selling price; for those who disliked, the certainty equivalent is the maximum willingness to pay to avoid the lottery. Two personality trait questions were also included to compare selfreport with measures of risk preference. Questions about insurance preference were included to test risk preferences for a pure loss situation. Insurance gambles are similar in form to those used by Slavic et al. (1977): each gamble has the same expected value of -$lO, but probabilities and loss levels vary. The amount of loss ranged from -$40 to -$4000 and corresponding probabilities ranged from 0.25 to l/400. For each contract, the respondent was asked to state a preference between taking the risk and buying insurance at a cost of $10 to avoid the loss. The instrument was given to 21 faculty members in a variety of academic departments at Purdue University; about a third of the respondents were in the business school. Responses were returned anonymously by mail.
4 No demand revelation incentives were used. Becker et al. (1964) developed an incentive-compatible method for elicitation of certainty equivalents. However, many studies have used hypothetical lotteries. Examples cited here include experiments by Kabneman and Tversky, Lopes, and Slavic et al.
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2.2. Risk aversion results Several theories of risk preference can be tested using the data. Besides concavity of utility (indicated by the relationship between the certainty equivalent and the expected value), other theories are stochastic dominance (Rothschild and Stiglitz, 1970) and the Yaari (1969) acceptance set. Responses are shown in Table 2, ordered by the certainty-equivalent response described above. None of the respondents was risk-neutral according to the certainty equivalent (requiring a certainty equivalent of $50 for this comparison). Risk aversion in the sense of a certainty equivalent less than $50 was obtained for 17 out of 21 respondents. Seven of these respondents had a negative certainty equivalent; that is, they would pay to get rid of this risky lottery.
Table 2 Responses
to questions,
risk preference.
Certainty
Ordering
Ordering
Insurance
equivalent’
(1)
(II)
YIN’
Risk taking’
Luck’
-$50 -$50D -$50 -%SoA -$25 -$20 -$20
15234 12534 12534 21534 15342” 12534 12534 21534 21534 12354 15234 15234 21534 12534 21534 21534 31254 12534” 13524 21354” 12453”
31245 54213 5423 1 54213 45213 54213 53241 54213 5423 I 52143? 45213 32145? 53214 45321? 34521 15423 54213 52431? 41235? 235 14 13542
313” 313 214 412 313 412 412 313 6/O 313 412 O/6 313 412 511 313 610 5/l 313 610 indiff.
RT RT RS RT RT RT RT RT RA RT RT RT RT RT RT RT RA RT RT RT RT
N L N L L N L L N N N N N N L N N L N L L
SOB $OG $0 $OE $0 $0 $10 $lOC $15 $25F $100 $100 $100 $150
purchases
*‘_ ’ indicates WTP to remove lottery ‘+’ indicates WTP to sell lottery “Yes’ to insurance for larger losses, ‘no’ for smaller losses “a nonmonotonic insurance choice pattern. ‘RT = Risk Trader RA = Risk Avoider RS = Risk Seeker
‘L = Lucky U = Unlucky N = Not lucky or unlucky
? indicates an intransitive order; indicated order is a transitive guess x indicates inconsistency between the CE and the ranking
Risk attitudes
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A risk-averse person should rank a certainty of $50 above all other lotteries in set I; only one person exhibited this order. Instead, 16 of the 17 ‘risk averters’ preferred to take the chance of $100 versus $0 rather than receive a certain $50 (indicating convexity in the utility function for gains of $100 or more). Among the four subjects with certainty equivalents greater than $50, three were riskseeking in the classical sense since they also preferred the $lOO/$O lottery to the $50 certainty. Ranking by variance (here equivalent to second-order stochastic dominance since means are constant), the preference order over Set I would be 21534. Only six subjects (all with certainty equivalents under $50) exhibited this order. Ranking by variance, the rank order for the second set of lotteries is 23541; no one had this ranking. Thus, no subject completely satisfied stochastic dominance criteria. Only three respondents were consistent with risk aversion by choosing fair insurance in all cases. The predominant pattern was to insure only for the higher loss/lower probability cases.5 Considering the acceptance set definition of risk aversion, more risk-averse persons should purchase insurance more often. Conversely, less risk-averse persons should have a larger set of risks that they would accept without insurance. For a given respondent, the larger the number of ‘yeses’ in Table 2, the smaller the risk acceptance set. Respondents with more than four ‘yeses’ for insurance purchase are found entirely among those with certainty equivalents greater than $0, who would be designated as less risk-averse according to the certainty equivalent. For self-report, only two reported themselves to be risk avoiders, and only one reported being a risk seeker. Instead, most were ‘risk traders,’ willing to small risks provided the potential loss is not too great. Four of the 21 respondents had inconsistent responses for the certainty equivalent. That is, consistency requires for Set I: CE > CE <
$50 if and only if lottery 4 preferred to lottery 2; $50 if and only if lottery 2 preferred to lottery 4.
Four other respondents had intransitive responses to the pairwise questions for Set II. Modeling each individual separately, intransitive responses cannot be described with a monotone utility theory. In contrast, in an econometric study inconsistencies would be included in the error term. To describe the nature of observed risk preferences, respondents seem to tradeoff gains and losses. In Set I, 13 respondents (more than half) put lottery 1 as best; this lottery had an intermediate gain, and zero loss. The set of lotteries (1,2) was preferred to (3,4,5) by 14 of the 21 subjects. In set II, the set of lotteries (5,4) was preferred by nearly half the subjects over the set (3,1,2); lotteries 5 and 4 had the highest gain outcomes. The ranking54213 for five subjects indicates -a desire for gain mixed with avoiding loss; a higher chance of loss is taken in order to obtain a higher gain. ’ These results arc opposite to those obtained by Slavic et al. (1977) who found (for college students) that most respondents chose insurance for higher probability/lower loss cases.
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3. Comparison
of alternative
nonexpected
utility theories
Four types of preference models will be compared here using the paired comparisons data described above. The models represent alternative assumptions about the nature of transformation of probability using a common set of assumptions about utility.6 In all cases, both the utility and the probability transformation are assumed to be monotone scales. However, neither is required to have any particular shape; that is, a parametric representation is not used. Following Kahneman and Tversky (1979), utility is normalized in all models so that a zero outcome receives a utility of zero (see Fig. 1). VALUE
LOSSES
GAINS
A hypotheticalvalue funnion. (p. 279, Pmwect Theov)
..-.
:’
_:’ ,_: _./ _:’ /.,. .:’ _’
,.” ...’
: .‘. .:’
:/
t.0
.5 STATED
PROBABILITY:
p
-A hypothctkal rei5hcin5 function. (P. =J, Rorpst
Theorl)
Source: Kahneman & Tversky, 1979
Fig. 1. Kahneman
b A unifying
approach
to several generalizations
and Tversky’s Prospect model
of expected utility is provided by Chew and Epstein (1989).
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Thus, asset integration is not assumed. Appendix A discusses how to combine a normalized utility with a Friedman-Savage utility over wealth without asset integration. For each model, scaling assumptions (Krantz et al., 1971) are imposed in order to determine a unique utility scale between the worst and best outcomes. Probability is scaled between zero and one. A probability of zero is assumed to be not transformed, as is a probability of one. (That is, a certain gain or loss is included on the same utility scale as risk.) The models below differ in whether subjective probabilities are required to sum to one. Summing to one is a desirable mathematical property (see Appendix A). However, Kahneman and Tversky’s model suggests that behavior may not satisfy this requirement.7 A behavioral property such as optimism or pessimism is a different story than computational error! More complex theories such as the Chew and SDM models below assume different probability transformations for gains and losses. That probability transformation is independent of outcome is assumed for EURDP and Prospect Theory; the utility function reflects differences gain and loss values. Measurement is obtained by applying a nonlinear programming algorithm separately for each individual respondent to a paired comparisons design. The result is a piecewise linear representation of utility and probability transformation. 3.1. Alternative
models
Compared lotteries have two outcomes x and y, respectively with probabilities p and 1-p. Lottery comparisons include certainty and gain/loss combinations. Below, x denotes loss while y denotes a gain. F represents a cumulative probability distribution that combines the gain and loss with the given probabilities. For these two outcomes, the cumulative distribution is: F(t) = 0, t < x; F(t)=p,x
’ Kahneman and Tversky (1979) suggested the certainty bias as an explanation of the Allais Paradox: comparing certainty to risk may be different from comparing risk to risk (see also McCord and De Neufville, 1986). In Prospect Theory, this is represented by a discontinuity in the subjective probability transformation at zero and one. No discontinuity in probability is modeled here. Instead, following psychologists such as Bimbaum (1982), we make a distinction between utility functions that represent preference over gambles and functions that represent certain wealth.
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case so that measurement
defines a unique scale: U(300) - U(-200)
= 500
3.1.1. Prospect-type model (PT). The composition is of the form WJ
(1)
+ n(l - PHY)
= GW)
where the subjective probabilities K(P) and 7r(1-p) may sum to less than one. n(O)=O, n(l)=1 are assumed for measurement purposes. r(p) is assumed to be piecewise linear.8 3.1.2. Expected utility with rank dependent probability The composition is of the form
W)
= S(P)+)
(EURDP).
(2)
+ (1 - S(P)MY)
Expected utility is a special case when S is the identity function. utility, symmetry
In contrast to expected
S(p) = 1 - S( 1 - p) is not required.’ The subjective transformation S(F) for ELJRDP is assumed to satisfy S(O)=O, S( l)=l, and piecewise linearity (S(F(t))=S(p) for t between x and y). Subjective probabilities S(p) and 1-S(p) sum to one. 3.1.3.
Chew model.
Following Chew (1983), subjective The assumed form is U(F)
=
a(x)p a(x)p
+ &(Y)(l
Note that the implied subjective 3.1.4.
Subjective
probabilities
-P)
u(x) +
probabilities
could vary with the outcome Q(Y)(l
-PI
a(x)p + a(y)(l
-PI
u(Y)
level.
(3)
sum to one.
Distribution Model (SDM).
A simpler model than the Chew model, following Currim and Satin (1989), allows subjective probabilities for gains and losses to differ. The form of the model is W)
= ~I@)+)
+ rgiTg(t- P)~(Y).
(4)
This model is similar to Prospect Theory except that two probability scales are used. The subjective probabilities may sum to less than one. A rationale for why probabilities
’ Recent work by Tversky and Kahneman (1992) generalizes Prospect Theory to more outcomes and transformation of the cumulative distribution. When gains and losses are mixed, they also find different preference patterns than in the original experiment, depending on the probability of loss. 9 Loehman (1994) interpreted alternative types of symmetry in terms of optimism and pessimism. The implications of alternative combinations of probability and utility shapes were interpreted in terms of risk aversion.
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can sum to less than one is that the zero outcome, remaining probability weight. 3.2. Nonlinear programming
293
with a utility of zero, can receive the
algorithm
A nonlinear programming method is used to estimate utility and subjective probability values simultaneously for each respondent for each alternative theory.” Two problems are solved simultaneously: (1) minimize the length of the piecewise linear utility function on [-200,300] constrained by Set I preference orders; (2) minimize the length of the piecewise linear subjective probability transformation function on [O,l] constrained by Set II preference orders. Note that if feasible, a linear utility function and the identity transformation on probability would minimize the length criteria. To solve the utility and probability measurement problems simultaneously, an iterative algorithm to search for a fixed point is used. Initially, taking S(OS)=OS, utility scale values are determined by constrained optimization with Set I preferences. Then, these estimated utility values are used with Set II preferences and constrained maximization to determine S(p), for p=O.1,0.25,0.5,0.75,0.9. The resulting value of S(0.5) is then used in the utility problem with Set I, and so on. When the value of S(0.5) obtained from Set II no longer changes from one iteration to the next, the resulting solutions for utility and subjective probability solve both problems simultaneously. For the Chew model, the weights a(x) and utility values U(X) were determined simultaneously from a single optimization problem that minimized the distance between the preference functional and a functional with both linear utility and untransformed probability. Monotonicity was imposed as constraints for the weights and utility. Constraints are defined as follows: Monotqnicity of utility and probability transformation form constraints; for example, u(100) > 450). The specified preference orders define other constraints, depending on the preference functional to be estimated. For example, for EURDP the constraint for lottery 5 preferred to lottery 3 in set I is represented as: S(O.5)4-50)
+ (1 - S(O.5))4
150) > S(O.5)4-100)
+ (1 - S(O.5))~(200).
Table 3 summarizes successes and failures for EURDP and Prospect models. The ‘no solution’ situations for the nonlinear program coincide for the most part with inconsistent or intransitive rankings in Table 2. The EURDP model was more successful in that two more respondents could be modeled than with the Prospect-type model. The Chew and SDM models were less successful in obtaining a solution, probably because more parameters had to be estimated. Convergence of the iterative procedure at the third decimal place required about 3 to I8 iterations depending on the model. The Prospect-type model required the least iteration. The SDM model required the most because of more parameters. The EURDP model was intermediate in terms of iterations. lo The preference instrument is available from the author. The GAMS program Detailed results for Chew and SDM models are also available.
is available
from the author.
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Table 3 EURDP and Prospect models for all respondents,
and subjective probability
Certainty equivalent
EURDP
Prospect
-$50
no solution 0.20 0.5 0.20 inconsistent 0.23 0.5 0.29 0.49 inconsistent 0.27 inconsistent 0.33 inconsistent 0.48 0.22 0.49 inconsistent inconsistent inconsistent inconsistent
no solution 0.52 0.5 0.53 inconsistent 0.53 0.5 0.52 0.5 1 inconsistent 0.57 inconsistent no solution inconsistent 0.5 1 no aolution 0.52 inconsistent inconsistent inconsistent inconsistent
-$50D -$50 -ssoA -$25 -$20 -$20 SOB $OC $0 $OE $0 $0 $10 $lOC $15 $25F $100 $100 $100 $150
3.3. Modeling
transformation
of p=O.5
results
Table 3 shows that the EURDP model generally gave subjective transformations S(0.5) that were less than 0.5, while the Prospect model gave transformations greater than 0.5. Clearly, assumptions about the form of the preference functional affect the estimated subjective probabilities. Tables 4 and 5 show results for the Prospect and EURDP models for selected respondents (A-F in Table 2). Utility slopes rather than scale values are given in these tables in order to compare models easily to the risk-neutral case having a slope of one. Concavity is indicated when the slope decreases, while convexity is indicated when the slope increases. Respondents A and D have a certainty equivalent of -$50, the most risk-averse level exhibited. Respondents B,E, and G have a certainty equivalent of $0. Respondents C and F have positive certainty equivalents. Respondents A and B have the same preference order but different certainty equivalents. Respondents A,B,C, and G have the same ranking over set I. Respondents A,B,D,F have the same ordering over Set II. 3.3.1. Comparison of models for Respondent B. Figs. 2-5 illustrate the shapes of utility and probability transformation functions for all models for this respondent. For both EURDP and Prospect models, the shape of the utility function is concave for losses and small gains but has a convex portion for gains above $100.
E. Loehman/J. Table 4 Prospect-type
model for respondents
of Economic Behavior & Org. 33 (1998) 285-302
A-G.
A
B
C
D
E
F
G
probability 0.25 0.36 0.52 0.76 0.898
0.08 0.27 0.51 0.745 0.899
0.31 0.39 0.52 0.755 0.894
0.24 0.48 0.54 0.73 0.889
0.16 0.30 0.52 0.759 0.899
0.12 0.27 0.51 0.752 0.899
1.20 1.20 1.20 0.92 0.85 0.85 0.85 0.85
1.66 0.77 0.72 0.70 0.60 0.71 0.71
1.27 1.27 1.18 0.75 0.89 0.87 0.8 1 0.81
1.26 0.97 1.06 0.93 0.87 0.99 1.08 0.78
1.25 1.25 1.25 0.87 0.79 0.83 0.83 0.83
C
D
E
F
G
Objective probability 0.1 0.25 0.5 0.75 0.9
Subjective 0.32 0.40 0.53 0.76 0.895
Outcome -100 -50 0 50 100 150 200 300
Utility slope 1.65 I .25 1.65 1.25 0.76 1.25 0.76 0.91 0.65 0.75 0.7 I 0.83 0.7 1 0.83 0.83 0.71
Table 5 EURDP model for respondents
29.5
1.66
A-G. A
B
Objective probability 0.1 0.25 0.5 0.75 0.9
Subjective probability 0.13 0.13 0.16 0.21 0.20 0.29 0.60 0.58 0.84 0.83
0.09 0.249 0.48 0.72 0.901
0.13 0.16 0.20 0.60 0.84
0.11 0.17 0.27 0.57 0.87
0.12 0.22 0.36 0.62 0.84
0.11 0.257 0.49 0.72 0.89
Outcome -100 -50 0 50 100 150 200 300
Utility slope 2.65 1.75 2.65 1.75 0.42 1.75 0.42 0.64 0.08 0.24 0.08 0.52 0.35 0.52 0.35 0.52
1.24 1.24 1.24 0.91 0.82 0.83 0.83 0.83
2.64 2.64 0.42 0.40 0.13 0.06 0.35 0.35
1.82 1.82 1.82 0.45 0.45 0.45 0.45 0.64
1.44 1.44 1.44 1.06 0.59 0.64 0.64 0.64
I .27 I .27 1.27 0.85 0.78 0.82 0.82 0.82
Clearly, different modeling restrictions give different results! These two models have very different shapes for subjective probabilities. For EURDP, the shape is actually similar to that proposed for the original Prospect theory: higher probabilities are deflated, with the biggest change around p=O.5, and lower probabilities are somewhat inflated. For the Prospect-type model, the shape obtained is different from that originally postulated for PT: Prospect results for Respondent B are that low values are inflated while higher values are slightly deflated.
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Fig. 2. Prospect model for respondent
B(CE= $0).
Fig. 3. EURDP Model for respondent
B(CE=$O).
Pessimism should imply that losses with high probabilities would be inflated, and that gains with high probabilities would be deflated. With either model, the indicated behavior is neither optimism nor pessimism. In the paired comparison design, the same probability value occurs with both gains and losses. Since expected value is constant, a high probability is associated with a low loss or gain, whereas a low probability is associated with a high gain or loss. Thus, the EURDP results imply that medium-to-high probabilities associated with low losses or gains are strongly discounted, while low probabilities associated with high gains and losses are slightly inflated. The Prospect-type results imply the opposite: that high gains and losses are strongly inflated with little effect on low losses or gains; the utility function shape with greater slopes for large losses and large gains reinforces the probability transformation. For the SDM model, utility is concave, and loss and gain subjective probabilities differ in shape: deflation of high probability (low value) losses and deflation of low probability (high value) gains. The Chew model has a highly irregular subjective probability shape.
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Fig. 4. SDM Model for respondent
B(CE=
$0).
Fig. 5. Chew model for respondent
B(CE=
$0).
297
3.3.2. Results for other respondents. Figs. 6 and 7 give EURDP models for Respondents A and C. Respondent A has a much smaller certainty equivalent than Respondent C; the utility function has a more concave shape for losses, as well as a greater transformation of probability. Respondent B with an intermediate certainty equivalent has a utility shape between that of A and C in concavity.
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Fig. 6. EURDP model for respondent
A (CE= $0).
Fig. 7. EURDP model for respondent
C (CE= $10)
Respondent D has a preference pattern that differs from Respondent A in only one place for set I, and A and D have the same certainty equivalent; the fitted models for these two respondents are quite similar, both for EURDP and Prospect models. Preference orders for B and G (both with a certainty equivalent of $0) differ in one place in Set II. Subjective probabilities for the Prospect model are quite different, and utility values are similar. The EURDP models for these two respondents differ in both utility and subjective probability. Respondent E has a very different preference order from B and G; all three have a certainty equivalent of $0. The EURDP models for E, B, and G are very different in both utility and subjective probability. For the Prospect model, probability shapes are very different but similar utility models are obtained for respondents E, B, and G. 3.3.3. Insurance purchase. Insurance questions compare a loss certain premium of xp. Insurance purchase the estimated nonexpected utilities. For xp=10; insurance purchase is expected
of -x with probability p with paying a was predicted with each model by comparing example, for EURDP, cI(F)=u(-x)S(p), for to occur if U(F) is less than ~(-10). Linear
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299
extrapolation was used for ranges of outcomes larger than $300 and probabilities smaller than 0.1. Insurance predictions were not successful. Both EURDP and Prospect models predicted that purchase of insurance would occur for all cases. The explanation is the concavity of the utility function in the loss range. That is, the shape of the utility function obtained for mixed gain-loss lotteries does not explain choices for pure loss comparisons.
4. Conclusions Classical definitions of risk aversion are not upheld by the preference data for this study. Therefore, nonexpected utility theories with more degrees of freedom in representing behavior are more appropriate than expected utility. In contrast to econometric studies that pool preference responses, it is important to note the wide variety of preference types exhibited. Among the 21 respondents in this study, none had identical preferences. This paper applied four different nonexpected utility theories to model individual preference responses over mixed gain/loss lotteries. All theories used a normalized representation of utility as in Prospect theory. EURDP and Prospect theories differed as to whether or not subjective probabilities were required to sum to one. Other tested theories had more complex representations of subjective probability transformation. In principle, any of the tested models could be estimated using preference information. However, EURDP and Prospect models were more successful in representing preferences than the more complex probability transformation models; their complexity was not needed to describe subject preferences over gains and losses in a smooth way.’ ’ The most striking result is the consistent shape of the EURDP utility function over a wide range of preference types: concave for losses and small gains and convex for larger gains. The EURDP model distinguished among different preference types by the degree of change in concavity for the utility shape. EURDP exhibited more differentiation among preference types than the Prospect model. The Prospect-type model produced less regular shapes. While it cannot be proved that either model is right or wrong, there is clearly a difference in terms of descriptive power.” Quite different subjective probability transformations were obtained with EURDP and Prospect models. With EURDP, the subjective transformation of p=OS was deflated in all exhibited cases. With the Prospect model, the subjective transformation of S(O.5) was inflated in all exhibited cases. Thus, the assumption that subjective probabilities sum to one has a strong effect on subjective probability estimates.
” The observed shape of the utility function for Prospect and EURDP models is consistent with a cubic representation such that the utility is concave for negative and small outcomes and convex for larger outcomes. With a negative Pratt-Arrow coefficient, the third-order (cubic) term can keep the model from being purely concave. ‘* Thus, to select among alternative models, suggested criteria are: the relative number of successes in representing individual preference orders, whether there is a pattern for utility and probability shapes that includes a variety of preference types, and whether the pattern is interpretable in terms of risk preference.
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Considering insurance results here and KT’s (1979) results for pure gains and pure losses, representation of preferences seems to depend very much on the set of risks being compared: different representations may apply for pure loss, pure gain, and mixed gain-loss situations. Shape may also depend on the range of gain or loss. More studies of the type presented here - nonparametric and focusing on individual behavior - can help to elucidate the nature of risk preference.
Acknowledgements Thanks to Monte Vandeveer for his contribution algorithm.
to the development
of the estimation
Appendix A Combining
normalized
and Friedman-Savage
utility
The discussion below shows how to combine risk and certainty (represented by Friedman-Savage-type utility functions over wealth) using normalized KahnemanThis combination provides an alternative to asset Tversky-type utility functions. integration since risk and wealth are not added directly. Suppose preferences over probability distributions F for given wealth W can be represented by a real-valued Frechet-differentiable preference functional U(F;W) (see Chew et al., 1987). Then, there exists a utility function u(x;W) defined over risks x in support of F, and a transformation S of cumulative probability distributions F, such that U(F; W)
J
u(x; W)&(F).
for
J
B(F)
= 1.
Discontinuities in F and discrete outcomes are allowed by this representation (Loehman, 1994). Let F. represent the distribution that gives 0 with a probability of one. Define the value function v(w) as follows:
v(W) = jiim” U(F; W) Define a normalized
preference &yF.
functional
>
w)
=
fi(F; W) as follows for v’(W)
w; WI-
VW) v’(W)
> 0
E. L.oehman/J. of Economic Behavior & Org. 33 (1998) 285-302
Since it is a monotone transformation, the normalized preference same ranking of distributions as U. By definition of v(w),
I!@-;W) =
& By definition
functional
301
produces the
0
of U, U(F; W) = q!&(F; W) + v(W)
where C#J= v’(W) is a constant Furthermore, define
when wealth is held constant.
qx.
Then, by the properties
>
w)
4x; w - VW) v’(W)
=
of the Riemann
integral and definition
ti(F; W) =
of U,
ii@; W)dS(F) J’
Taking limits of both sides as F-+Fc By definition of ii,
implies that ic(x; W) is zero at x=0.
u(x; W) = f#G(x; W) + v(W) That is, in expressing preference over risk, the functional U(fl can be expressed with EURDP as a linear combination over a normalized utility ic over risks x and a FriedmanSavage utility v over wealth W. The normalized utility is indexed by W, that is, a wealthier person may have a different normalization than a less wealthy person. This result is illustrated in the Appendix A Figure.
vi
certain outcomes
References Arrow, K.J., 1965. Aspects of the Theory of Risk Bearing. Helsinki: Yrjo Fahnsoon Foundation. Becker, Gordon M., Morris, H., DeGroot Jacob Marschak, 1964. Measuring utility by a single-response sequential method, Behavioral Science 9, 226-232. Birnbaum, M.H., 1982. Controversies in psychological measurement. In: Wegener, B. (Ed.), Social Attitudes and Psychophysical Measurement, Erlbaum.
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ofEconomic
Behavior & Org. 33 (1998) 285-302
Camerer, C., 1989a. Experimental markets for insurance, Journal of Risk and Uncertainty 2, 265-300. Camerer, C.F., 1989b. An experimental test of several generalized utility theories. J. of Risk and Uncertainty 2, 61-104. Chew, Soo Hong, 1983. A generalization of the quasi-linear mean with applications to the measurement of income inequality and decision theory resolving the Allais Paradox, Econometrica 5 1, 106-1092. Chew, S.H., Kami, E., Safra, Z., 1987. Risk aversion in the theory of expected utility with rank dependent probabilities, Journal of Economic Theory 42, 370-381. Chew, S.H., Epstein, LG., 1989. A unifying approach to axiomatic non-expected utility theories, J. of Econ. Theory 49(2), 207-240. Currim, Imran S., Rakesh K. Satin, 1989. Prospect Versus Utility, Management Science 35, 1. David, H.A., 1988. The Method of Paired Comparisons. Charles Griffin, London. Friedman, M., Savage, L., 1948. The utility analysis of choices involving risk, Journal of Political Economics 56, 279-304. ðer, D.M., Plott, C.R., 1979. Economic theory of choice and the preference reversal phenomenon, International Economic Review 69, 623-638. Kahneman. D., Tversky, A., 1979. Prospect theory: An analysis of decision under risk, Econometrica 47, 2633 291. Krantz, D.H., Lute, R.D., Suppes, I?, Tversky, A., 1971. Foundations of Measurement, Academic Press. Loehman, E., 1994. Rank dependent expected utility: Stochastic dominance, risk preference, and certainty equivalence, Journal of Mathematical Psychology 38, 159-197. McCord, Mark and De Neufville, Richard, 1986. Lottery equivalents: Reduction of the certainty effect problem in utility assessment, Management Science 32, 1. Pratt, John W., 1994. Risk aversion in the small and in the large, Economettica 2, 1-2. Rothschild, M., Stiglitz, J.E., 1970. Decreasing risk: I. A definition, Journal of Economic Theory 2, 225-243. Schoemaker, P.J.H., 1989. Conceptions of risk-taking: Do intrinsic risk-attitudes matter?, Center for Decision Research, University of Chicago. Slavic, Paul, Fischhoff Baruch, Lichtenstein Sarah, Corrigan Bernard Combs Barbara 1977. Preferences for insuring against probable small losses: Insurance implications, Journal of Risk and Insurance 44, 237-258. Thaler, R., 1987. The psychology of choice and the assumptions of economics. In Roth, A.E. (Ed.), Laboratory Experiments in Economics, Six Points of View, Cambridge University Press. Tversky, A., Kahneman, D., 1992. Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5, 297-323. Yaari, M.E., 1969. Some remarks on measures of risk aversion and on their uses, Journal of Economic Theory 1, 3 15-329. Wu, George, Richard, Gonzalez, 1996. Curvature of the probability weighting function Management Science 429120. 16761690.
-Behevior tit-
Journal of Economic Behavior & Organization Vol. 33 (1998) 285-302
ELSEVIER
Testing risk aversion and nonexpected utility theories Edna Loehman* Department of Agricultural Economics, Purdue University, W. Lafayette IN 47907-l 133, USA Received
16 December
1992; received in revised form 15 October
1996; accepted
6 November
1996
Abstract This paper
tests
risk
aversion
and compares
several
nonexpected
utility
theories
using
paired
hypotheses associated with risk aversion are found not to hold in preference data. Expected utility with rank dependent probability is found to provide a smooth representation of preference for a wide range of preference types. Published by Elsevier Science B.V.
comparisons.
Economic
JEL classijication: Keywords:
C52; C91; D81
Risk aversion; Nonexpected
utility; Risk reference
1. Testing risk aversion and nonexpected
utility theories
Stemming from the Allais paradox, both psychologists and economists have noted the inadequacy of expected utility theory to describe risk behavior. Kahneman and Tversky (1979) introduced new paradoxes including the reflection effect: the same person can be observed to be both risk-averse, and risk-seeking depending on whether gambles are presented as gains or as losses. Thus, Schoemaker (1989) suggests that risk aversion may not be an intrinsic behavior. At the same time, nonexpected utility theories have become fairly accepted as an alternative to expected utility. There are now a variety of them (see Camerer, 1989a, 1989b for a summary). These theories use a subjective transformation of probability to represent preference. Generally, an S-shaped function probability transformation - first * Corresponding
author.
0167-2681/98/$19.00 Published by 1998 Elsevier Science B.V. All rights reserved PII SO167-268 1(97)00097-8
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concave and then convex with an inflection around p=O.40 - has been obtained in empirical work (Wu and Gonzalez, 1996). However, empirical work (e.g. Currim and Sarin, 1989) has assumed a parametric concave form for the utility function in order to measure the probability transformation. The nature of risk preference is the main subject addressed by this paper. Several alternative nonexpected utility theories of probability transformation are compared here for gambles involving mixed gains and losses. Concomitantly, risk aversion is examined as an appropriate description of behavior. The Pratt-Arrow coefficient of risk aversion (Pratt, 1964, Arrow, 1965) describing the relative concavity of the utility function has been based on asset integration, an assumption that has not held up well in experimental studies (Thaler, 1987). Theories which do not depend on asset integration therefore provide possible superior explanations of behavior. How to combine certainty and risk without asset integration - combining aspects of both the Kahneman and Tversky (1979) model of utility and the Friedman and Savage (1948) model of utility over wealth - is demonstrated in this paper (see Appendix A for theory). Results of this paper suggest that modification of the utility function as related to risk aversion is needed in addition to nonexpected utility transformations of the probability function. For combined gains and losses, results clearly show that traditional economic definitions of risk aversion do not hold behaviorally, since the shape obtained for utility is concave for losses and small gains and convex for larger gains. A test of insurance preference comparing pure loss gambles, however, suggests the inadequacy of a single theory of utility that encompasses certainty, gains, and losses compared separately and in combination. In addition, this paper demonstrates several empirical innovations. First, the method for determining the form of utility and probability functions is nonparametric; therefore no restrictive assumptions about concavity need to be made. Second, the method of paired comparisons is used to measure nonexpected utility theories, thus avoiding the problems of certainty equivalents.‘.’ Third, both utility and probability scale values are estimated simultaneously from paired comparisons data, again avoiding restrictive assumptions.s Finally, measurement is performed for each individual separately, in contrast to empirical studies (Wu and Gonzalez, 1996, Currim and Sarin, 1989) that pooled preference data to estimate preference parameters for a composite individual.
’ Paired comparisons is considered by psychologists to be a reliable method of eliciting modeling preference; see David (1988). It includes a method for utility measurement. Here we use the method of response, measuring utilities in different ways. ’ The certainty equivalent has been the predominant tool for expected utility measurement (see e.g. Becker et al., 1964). The certainty equivalent remains valid in nonexpected utility context (Chew, 1983, Loehman, 1994). So, this concept could also be applied for the measurement of nonexpected utility. However, validity problems have been found with use of the certainty equivalent; see ðer and Plott (1979). a The certainty equivalent was used to compare Prospect Theory and expected utility by Currim and Satin, 1989. They compared lotteries over gains and losses separately; the Prospect Theory utility function shape for separate gains and losses was confirmed by their study. Subjective probabilities and utility were not measured simultaneously, and risk aversion was assumed. They assumed that preference is represented by an exponential utility function. The risk-aversion coefficient was first estimated from certainty equivalents. For the assumed utility, subjective probabilities were then measured.
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287
Table 1 Lottery comparisons” Lottery
Set I
Set II
1 2 3 4 5
($100, $0;0.5,0.5) ($50, $50;0.5,0.5) ($200,-$100;0.5,0.5) ($300,-$200;0.5,0.5) ($150, -$50;0.5,0.5)
($250,-$200;0.5,0.5) ($50,~$100;0.75,0.25) ($36,~$200;0.9,0.1) ($200-$50;0.25,0.75) ($300-$19.50;0.1,0.9)
aDollar amounts are listed followed by the probabilities
of each outcome.
2. Preference elicitation instrument A preference elicitation instrument was designed both to test risk aversion and to measure alternative nonexpected utility theories. Due to problems with the certainty equivalent, paired comparison was selected as the method of preference elicitation.4 2.1. Design Table 1 shows the compared gambles. Two sets of lotteries (I and II) were presented for paired comparison. In order to test for risk neutrality and consistency with secondorder stochastic dominance, all lotteries in each set have the same expected value. Outcomes in Set I have 50-50 probabilities and an expected value of $50. In Set II, probabilities are varied in order to estimate subjective probability values. One certainty equivalent was also elicited; this certainty equivalent corresponds to having a 50-50 chance at the combination of the best and worst outcomes. Respondents were categorized as to whether they liked or disliked having a chance at this lottery. For respondents who liked the chance, the certainty equivalent is the minimum selling price; for those who disliked, the certainty equivalent is the maximum willingness to pay to avoid the lottery. Two personality trait questions were also included to compare selfreport with measures of risk preference. Questions about insurance preference were included to test risk preferences for a pure loss situation. Insurance gambles are similar in form to those used by Slavic et al. (1977): each gamble has the same expected value of -$lO, but probabilities and loss levels vary. The amount of loss ranged from -$40 to -$4000 and corresponding probabilities ranged from 0.25 to l/400. For each contract, the respondent was asked to state a preference between taking the risk and buying insurance at a cost of $10 to avoid the loss. The instrument was given to 21 faculty members in a variety of academic departments at Purdue University; about a third of the respondents were in the business school. Responses were returned anonymously by mail.
4 No demand revelation incentives were used. Becker et al. (1964) developed an incentive-compatible method for elicitation of certainty equivalents. However, many studies have used hypothetical lotteries. Examples cited here include experiments by Kabneman and Tversky, Lopes, and Slavic et al.
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2.2. Risk aversion results Several theories of risk preference can be tested using the data. Besides concavity of utility (indicated by the relationship between the certainty equivalent and the expected value), other theories are stochastic dominance (Rothschild and Stiglitz, 1970) and the Yaari (1969) acceptance set. Responses are shown in Table 2, ordered by the certainty-equivalent response described above. None of the respondents was risk-neutral according to the certainty equivalent (requiring a certainty equivalent of $50 for this comparison). Risk aversion in the sense of a certainty equivalent less than $50 was obtained for 17 out of 21 respondents. Seven of these respondents had a negative certainty equivalent; that is, they would pay to get rid of this risky lottery.
Table 2 Responses
to questions,
risk preference.
Certainty
Ordering
Ordering
Insurance
equivalent’
(1)
(II)
YIN’
Risk taking’
Luck’
-$50 -$50D -$50 -%SoA -$25 -$20 -$20
15234 12534 12534 21534 15342” 12534 12534 21534 21534 12354 15234 15234 21534 12534 21534 21534 31254 12534” 13524 21354” 12453”
31245 54213 5423 1 54213 45213 54213 53241 54213 5423 I 52143? 45213 32145? 53214 45321? 34521 15423 54213 52431? 41235? 235 14 13542
313” 313 214 412 313 412 412 313 6/O 313 412 O/6 313 412 511 313 610 5/l 313 610 indiff.
RT RT RS RT RT RT RT RT RA RT RT RT RT RT RT RT RA RT RT RT RT
N L N L L N L L N N N N N N L N N L N L L
SOB $OG $0 $OE $0 $0 $10 $lOC $15 $25F $100 $100 $100 $150
purchases
*‘_ ’ indicates WTP to remove lottery ‘+’ indicates WTP to sell lottery “Yes’ to insurance for larger losses, ‘no’ for smaller losses “a nonmonotonic insurance choice pattern. ‘RT = Risk Trader RA = Risk Avoider RS = Risk Seeker
‘L = Lucky U = Unlucky N = Not lucky or unlucky
? indicates an intransitive order; indicated order is a transitive guess x indicates inconsistency between the CE and the ranking
Risk attitudes
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289
A risk-averse person should rank a certainty of $50 above all other lotteries in set I; only one person exhibited this order. Instead, 16 of the 17 ‘risk averters’ preferred to take the chance of $100 versus $0 rather than receive a certain $50 (indicating convexity in the utility function for gains of $100 or more). Among the four subjects with certainty equivalents greater than $50, three were riskseeking in the classical sense since they also preferred the $lOO/$O lottery to the $50 certainty. Ranking by variance (here equivalent to second-order stochastic dominance since means are constant), the preference order over Set I would be 21534. Only six subjects (all with certainty equivalents under $50) exhibited this order. Ranking by variance, the rank order for the second set of lotteries is 23541; no one had this ranking. Thus, no subject completely satisfied stochastic dominance criteria. Only three respondents were consistent with risk aversion by choosing fair insurance in all cases. The predominant pattern was to insure only for the higher loss/lower probability cases.5 Considering the acceptance set definition of risk aversion, more risk-averse persons should purchase insurance more often. Conversely, less risk-averse persons should have a larger set of risks that they would accept without insurance. For a given respondent, the larger the number of ‘yeses’ in Table 2, the smaller the risk acceptance set. Respondents with more than four ‘yeses’ for insurance purchase are found entirely among those with certainty equivalents greater than $0, who would be designated as less risk-averse according to the certainty equivalent. For self-report, only two reported themselves to be risk avoiders, and only one reported being a risk seeker. Instead, most were ‘risk traders,’ willing to small risks provided the potential loss is not too great. Four of the 21 respondents had inconsistent responses for the certainty equivalent. That is, consistency requires for Set I: CE > CE <
$50 if and only if lottery 4 preferred to lottery 2; $50 if and only if lottery 2 preferred to lottery 4.
Four other respondents had intransitive responses to the pairwise questions for Set II. Modeling each individual separately, intransitive responses cannot be described with a monotone utility theory. In contrast, in an econometric study inconsistencies would be included in the error term. To describe the nature of observed risk preferences, respondents seem to tradeoff gains and losses. In Set I, 13 respondents (more than half) put lottery 1 as best; this lottery had an intermediate gain, and zero loss. The set of lotteries (1,2) was preferred to (3,4,5) by 14 of the 21 subjects. In set II, the set of lotteries (5,4) was preferred by nearly half the subjects over the set (3,1,2); lotteries 5 and 4 had the highest gain outcomes. The ranking54213 for five subjects indicates -a desire for gain mixed with avoiding loss; a higher chance of loss is taken in order to obtain a higher gain. ’ These results arc opposite to those obtained by Slavic et al. (1977) who found (for college students) that most respondents chose insurance for higher probability/lower loss cases.
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290
3. Comparison
of alternative
nonexpected
utility theories
Four types of preference models will be compared here using the paired comparisons data described above. The models represent alternative assumptions about the nature of transformation of probability using a common set of assumptions about utility.6 In all cases, both the utility and the probability transformation are assumed to be monotone scales. However, neither is required to have any particular shape; that is, a parametric representation is not used. Following Kahneman and Tversky (1979), utility is normalized in all models so that a zero outcome receives a utility of zero (see Fig. 1). VALUE
LOSSES
GAINS
A hypotheticalvalue funnion. (p. 279, Pmwect Theov)
..-.
:’
_:’ ,_: _./ _:’ /.,. .:’ _’
,.” ...’
: .‘. .:’
:/
t.0
.5 STATED
PROBABILITY:
p
-A hypothctkal rei5hcin5 function. (P. =J, Rorpst
Theorl)
Source: Kahneman & Tversky, 1979
Fig. 1. Kahneman
b A unifying
approach
to several generalizations
and Tversky’s Prospect model
of expected utility is provided by Chew and Epstein (1989).
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291
Thus, asset integration is not assumed. Appendix A discusses how to combine a normalized utility with a Friedman-Savage utility over wealth without asset integration. For each model, scaling assumptions (Krantz et al., 1971) are imposed in order to determine a unique utility scale between the worst and best outcomes. Probability is scaled between zero and one. A probability of zero is assumed to be not transformed, as is a probability of one. (That is, a certain gain or loss is included on the same utility scale as risk.) The models below differ in whether subjective probabilities are required to sum to one. Summing to one is a desirable mathematical property (see Appendix A). However, Kahneman and Tversky’s model suggests that behavior may not satisfy this requirement.7 A behavioral property such as optimism or pessimism is a different story than computational error! More complex theories such as the Chew and SDM models below assume different probability transformations for gains and losses. That probability transformation is independent of outcome is assumed for EURDP and Prospect Theory; the utility function reflects differences gain and loss values. Measurement is obtained by applying a nonlinear programming algorithm separately for each individual respondent to a paired comparisons design. The result is a piecewise linear representation of utility and probability transformation. 3.1. Alternative
models
Compared lotteries have two outcomes x and y, respectively with probabilities p and 1-p. Lottery comparisons include certainty and gain/loss combinations. Below, x denotes loss while y denotes a gain. F represents a cumulative probability distribution that combines the gain and loss with the given probabilities. For these two outcomes, the cumulative distribution is: F(t) = 0, t < x; F(t)=p,x
’ Kahneman and Tversky (1979) suggested the certainty bias as an explanation of the Allais Paradox: comparing certainty to risk may be different from comparing risk to risk (see also McCord and De Neufville, 1986). In Prospect Theory, this is represented by a discontinuity in the subjective probability transformation at zero and one. No discontinuity in probability is modeled here. Instead, following psychologists such as Bimbaum (1982), we make a distinction between utility functions that represent preference over gambles and functions that represent certain wealth.
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292
case so that measurement
defines a unique scale: U(300) - U(-200)
= 500
3.1.1. Prospect-type model (PT). The composition is of the form WJ
(1)
+ n(l - PHY)
= GW)
where the subjective probabilities K(P) and 7r(1-p) may sum to less than one. n(O)=O, n(l)=1 are assumed for measurement purposes. r(p) is assumed to be piecewise linear.8 3.1.2. Expected utility with rank dependent probability The composition is of the form
W)
= S(P)+)
(EURDP).
(2)
+ (1 - S(P)MY)
Expected utility is a special case when S is the identity function. utility, symmetry
In contrast to expected
S(p) = 1 - S( 1 - p) is not required.’ The subjective transformation S(F) for ELJRDP is assumed to satisfy S(O)=O, S( l)=l, and piecewise linearity (S(F(t))=S(p) for t between x and y). Subjective probabilities S(p) and 1-S(p) sum to one. 3.1.3.
Chew model.
Following Chew (1983), subjective The assumed form is U(F)
=
a(x)p a(x)p
+ &(Y)(l
Note that the implied subjective 3.1.4.
Subjective
probabilities
-P)
u(x) +
probabilities
could vary with the outcome Q(Y)(l
-PI
a(x)p + a(y)(l
-PI
u(Y)
level.
(3)
sum to one.
Distribution Model (SDM).
A simpler model than the Chew model, following Currim and Satin (1989), allows subjective probabilities for gains and losses to differ. The form of the model is W)
= ~I@)+)
+ rgiTg(t- P)~(Y).
(4)
This model is similar to Prospect Theory except that two probability scales are used. The subjective probabilities may sum to less than one. A rationale for why probabilities
’ Recent work by Tversky and Kahneman (1992) generalizes Prospect Theory to more outcomes and transformation of the cumulative distribution. When gains and losses are mixed, they also find different preference patterns than in the original experiment, depending on the probability of loss. 9 Loehman (1994) interpreted alternative types of symmetry in terms of optimism and pessimism. The implications of alternative combinations of probability and utility shapes were interpreted in terms of risk aversion.
E. L.oehman/J. of Economic Behavior & Org. 33 (1998) 285-302
can sum to less than one is that the zero outcome, remaining probability weight. 3.2. Nonlinear programming
293
with a utility of zero, can receive the
algorithm
A nonlinear programming method is used to estimate utility and subjective probability values simultaneously for each respondent for each alternative theory.” Two problems are solved simultaneously: (1) minimize the length of the piecewise linear utility function on [-200,300] constrained by Set I preference orders; (2) minimize the length of the piecewise linear subjective probability transformation function on [O,l] constrained by Set II preference orders. Note that if feasible, a linear utility function and the identity transformation on probability would minimize the length criteria. To solve the utility and probability measurement problems simultaneously, an iterative algorithm to search for a fixed point is used. Initially, taking S(OS)=OS, utility scale values are determined by constrained optimization with Set I preferences. Then, these estimated utility values are used with Set II preferences and constrained maximization to determine S(p), for p=O.1,0.25,0.5,0.75,0.9. The resulting value of S(0.5) is then used in the utility problem with Set I, and so on. When the value of S(0.5) obtained from Set II no longer changes from one iteration to the next, the resulting solutions for utility and subjective probability solve both problems simultaneously. For the Chew model, the weights a(x) and utility values U(X) were determined simultaneously from a single optimization problem that minimized the distance between the preference functional and a functional with both linear utility and untransformed probability. Monotonicity was imposed as constraints for the weights and utility. Constraints are defined as follows: Monotqnicity of utility and probability transformation form constraints; for example, u(100) > 450). The specified preference orders define other constraints, depending on the preference functional to be estimated. For example, for EURDP the constraint for lottery 5 preferred to lottery 3 in set I is represented as: S(O.5)4-50)
+ (1 - S(O.5))4
150) > S(O.5)4-100)
+ (1 - S(O.5))~(200).
Table 3 summarizes successes and failures for EURDP and Prospect models. The ‘no solution’ situations for the nonlinear program coincide for the most part with inconsistent or intransitive rankings in Table 2. The EURDP model was more successful in that two more respondents could be modeled than with the Prospect-type model. The Chew and SDM models were less successful in obtaining a solution, probably because more parameters had to be estimated. Convergence of the iterative procedure at the third decimal place required about 3 to I8 iterations depending on the model. The Prospect-type model required the least iteration. The SDM model required the most because of more parameters. The EURDP model was intermediate in terms of iterations. lo The preference instrument is available from the author. The GAMS program Detailed results for Chew and SDM models are also available.
is available
from the author.
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Table 3 EURDP and Prospect models for all respondents,
and subjective probability
Certainty equivalent
EURDP
Prospect
-$50
no solution 0.20 0.5 0.20 inconsistent 0.23 0.5 0.29 0.49 inconsistent 0.27 inconsistent 0.33 inconsistent 0.48 0.22 0.49 inconsistent inconsistent inconsistent inconsistent
no solution 0.52 0.5 0.53 inconsistent 0.53 0.5 0.52 0.5 1 inconsistent 0.57 inconsistent no solution inconsistent 0.5 1 no aolution 0.52 inconsistent inconsistent inconsistent inconsistent
-$50D -$50 -ssoA -$25 -$20 -$20 SOB $OC $0 $OE $0 $0 $10 $lOC $15 $25F $100 $100 $100 $150
3.3. Modeling
transformation
of p=O.5
results
Table 3 shows that the EURDP model generally gave subjective transformations S(0.5) that were less than 0.5, while the Prospect model gave transformations greater than 0.5. Clearly, assumptions about the form of the preference functional affect the estimated subjective probabilities. Tables 4 and 5 show results for the Prospect and EURDP models for selected respondents (A-F in Table 2). Utility slopes rather than scale values are given in these tables in order to compare models easily to the risk-neutral case having a slope of one. Concavity is indicated when the slope decreases, while convexity is indicated when the slope increases. Respondents A and D have a certainty equivalent of -$50, the most risk-averse level exhibited. Respondents B,E, and G have a certainty equivalent of $0. Respondents C and F have positive certainty equivalents. Respondents A and B have the same preference order but different certainty equivalents. Respondents A,B,C, and G have the same ranking over set I. Respondents A,B,D,F have the same ordering over Set II. 3.3.1. Comparison of models for Respondent B. Figs. 2-5 illustrate the shapes of utility and probability transformation functions for all models for this respondent. For both EURDP and Prospect models, the shape of the utility function is concave for losses and small gains but has a convex portion for gains above $100.
E. Loehman/J. Table 4 Prospect-type
model for respondents
of Economic Behavior & Org. 33 (1998) 285-302
A-G.
A
B
C
D
E
F
G
probability 0.25 0.36 0.52 0.76 0.898
0.08 0.27 0.51 0.745 0.899
0.31 0.39 0.52 0.755 0.894
0.24 0.48 0.54 0.73 0.889
0.16 0.30 0.52 0.759 0.899
0.12 0.27 0.51 0.752 0.899
1.20 1.20 1.20 0.92 0.85 0.85 0.85 0.85
1.66 0.77 0.72 0.70 0.60 0.71 0.71
1.27 1.27 1.18 0.75 0.89 0.87 0.8 1 0.81
1.26 0.97 1.06 0.93 0.87 0.99 1.08 0.78
1.25 1.25 1.25 0.87 0.79 0.83 0.83 0.83
C
D
E
F
G
Objective probability 0.1 0.25 0.5 0.75 0.9
Subjective 0.32 0.40 0.53 0.76 0.895
Outcome -100 -50 0 50 100 150 200 300
Utility slope 1.65 I .25 1.65 1.25 0.76 1.25 0.76 0.91 0.65 0.75 0.7 I 0.83 0.7 1 0.83 0.83 0.71
Table 5 EURDP model for respondents
29.5
1.66
A-G. A
B
Objective probability 0.1 0.25 0.5 0.75 0.9
Subjective probability 0.13 0.13 0.16 0.21 0.20 0.29 0.60 0.58 0.84 0.83
0.09 0.249 0.48 0.72 0.901
0.13 0.16 0.20 0.60 0.84
0.11 0.17 0.27 0.57 0.87
0.12 0.22 0.36 0.62 0.84
0.11 0.257 0.49 0.72 0.89
Outcome -100 -50 0 50 100 150 200 300
Utility slope 2.65 1.75 2.65 1.75 0.42 1.75 0.42 0.64 0.08 0.24 0.08 0.52 0.35 0.52 0.35 0.52
1.24 1.24 1.24 0.91 0.82 0.83 0.83 0.83
2.64 2.64 0.42 0.40 0.13 0.06 0.35 0.35
1.82 1.82 1.82 0.45 0.45 0.45 0.45 0.64
1.44 1.44 1.44 1.06 0.59 0.64 0.64 0.64
I .27 I .27 1.27 0.85 0.78 0.82 0.82 0.82
Clearly, different modeling restrictions give different results! These two models have very different shapes for subjective probabilities. For EURDP, the shape is actually similar to that proposed for the original Prospect theory: higher probabilities are deflated, with the biggest change around p=O.5, and lower probabilities are somewhat inflated. For the Prospect-type model, the shape obtained is different from that originally postulated for PT: Prospect results for Respondent B are that low values are inflated while higher values are slightly deflated.
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Fig. 2. Prospect model for respondent
B(CE= $0).
Fig. 3. EURDP Model for respondent
B(CE=$O).
Pessimism should imply that losses with high probabilities would be inflated, and that gains with high probabilities would be deflated. With either model, the indicated behavior is neither optimism nor pessimism. In the paired comparison design, the same probability value occurs with both gains and losses. Since expected value is constant, a high probability is associated with a low loss or gain, whereas a low probability is associated with a high gain or loss. Thus, the EURDP results imply that medium-to-high probabilities associated with low losses or gains are strongly discounted, while low probabilities associated with high gains and losses are slightly inflated. The Prospect-type results imply the opposite: that high gains and losses are strongly inflated with little effect on low losses or gains; the utility function shape with greater slopes for large losses and large gains reinforces the probability transformation. For the SDM model, utility is concave, and loss and gain subjective probabilities differ in shape: deflation of high probability (low value) losses and deflation of low probability (high value) gains. The Chew model has a highly irregular subjective probability shape.
E. L..oehman/J. of Economic Behavior & Org. 33 (1998) 285-302
Fig. 4. SDM Model for respondent
B(CE=
$0).
Fig. 5. Chew model for respondent
B(CE=
$0).
297
3.3.2. Results for other respondents. Figs. 6 and 7 give EURDP models for Respondents A and C. Respondent A has a much smaller certainty equivalent than Respondent C; the utility function has a more concave shape for losses, as well as a greater transformation of probability. Respondent B with an intermediate certainty equivalent has a utility shape between that of A and C in concavity.
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Fig. 6. EURDP model for respondent
A (CE= $0).
Fig. 7. EURDP model for respondent
C (CE= $10)
Respondent D has a preference pattern that differs from Respondent A in only one place for set I, and A and D have the same certainty equivalent; the fitted models for these two respondents are quite similar, both for EURDP and Prospect models. Preference orders for B and G (both with a certainty equivalent of $0) differ in one place in Set II. Subjective probabilities for the Prospect model are quite different, and utility values are similar. The EURDP models for these two respondents differ in both utility and subjective probability. Respondent E has a very different preference order from B and G; all three have a certainty equivalent of $0. The EURDP models for E, B, and G are very different in both utility and subjective probability. For the Prospect model, probability shapes are very different but similar utility models are obtained for respondents E, B, and G. 3.3.3. Insurance purchase. Insurance questions compare a loss certain premium of xp. Insurance purchase the estimated nonexpected utilities. For xp=10; insurance purchase is expected
of -x with probability p with paying a was predicted with each model by comparing example, for EURDP, cI(F)=u(-x)S(p), for to occur if U(F) is less than ~(-10). Linear
E. L.oehman/J. of Economic Behavior & Org. 33 (1998) 28.5-302
299
extrapolation was used for ranges of outcomes larger than $300 and probabilities smaller than 0.1. Insurance predictions were not successful. Both EURDP and Prospect models predicted that purchase of insurance would occur for all cases. The explanation is the concavity of the utility function in the loss range. That is, the shape of the utility function obtained for mixed gain-loss lotteries does not explain choices for pure loss comparisons.
4. Conclusions Classical definitions of risk aversion are not upheld by the preference data for this study. Therefore, nonexpected utility theories with more degrees of freedom in representing behavior are more appropriate than expected utility. In contrast to econometric studies that pool preference responses, it is important to note the wide variety of preference types exhibited. Among the 21 respondents in this study, none had identical preferences. This paper applied four different nonexpected utility theories to model individual preference responses over mixed gain/loss lotteries. All theories used a normalized representation of utility as in Prospect theory. EURDP and Prospect theories differed as to whether or not subjective probabilities were required to sum to one. Other tested theories had more complex representations of subjective probability transformation. In principle, any of the tested models could be estimated using preference information. However, EURDP and Prospect models were more successful in representing preferences than the more complex probability transformation models; their complexity was not needed to describe subject preferences over gains and losses in a smooth way.’ ’ The most striking result is the consistent shape of the EURDP utility function over a wide range of preference types: concave for losses and small gains and convex for larger gains. The EURDP model distinguished among different preference types by the degree of change in concavity for the utility shape. EURDP exhibited more differentiation among preference types than the Prospect model. The Prospect-type model produced less regular shapes. While it cannot be proved that either model is right or wrong, there is clearly a difference in terms of descriptive power.” Quite different subjective probability transformations were obtained with EURDP and Prospect models. With EURDP, the subjective transformation of p=OS was deflated in all exhibited cases. With the Prospect model, the subjective transformation of S(O.5) was inflated in all exhibited cases. Thus, the assumption that subjective probabilities sum to one has a strong effect on subjective probability estimates.
” The observed shape of the utility function for Prospect and EURDP models is consistent with a cubic representation such that the utility is concave for negative and small outcomes and convex for larger outcomes. With a negative Pratt-Arrow coefficient, the third-order (cubic) term can keep the model from being purely concave. ‘* Thus, to select among alternative models, suggested criteria are: the relative number of successes in representing individual preference orders, whether there is a pattern for utility and probability shapes that includes a variety of preference types, and whether the pattern is interpretable in terms of risk preference.
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Considering insurance results here and KT’s (1979) results for pure gains and pure losses, representation of preferences seems to depend very much on the set of risks being compared: different representations may apply for pure loss, pure gain, and mixed gain-loss situations. Shape may also depend on the range of gain or loss. More studies of the type presented here - nonparametric and focusing on individual behavior - can help to elucidate the nature of risk preference.
Acknowledgements Thanks to Monte Vandeveer for his contribution algorithm.
to the development
of the estimation
Appendix A Combining
normalized
and Friedman-Savage
utility
The discussion below shows how to combine risk and certainty (represented by Friedman-Savage-type utility functions over wealth) using normalized KahnemanThis combination provides an alternative to asset Tversky-type utility functions. integration since risk and wealth are not added directly. Suppose preferences over probability distributions F for given wealth W can be represented by a real-valued Frechet-differentiable preference functional U(F;W) (see Chew et al., 1987). Then, there exists a utility function u(x;W) defined over risks x in support of F, and a transformation S of cumulative probability distributions F, such that U(F; W)
J
u(x; W)&(F).
for
J
B(F)
= 1.
Discontinuities in F and discrete outcomes are allowed by this representation (Loehman, 1994). Let F. represent the distribution that gives 0 with a probability of one. Define the value function v(w) as follows:
v(W) = jiim” U(F; W) Define a normalized
preference &yF.
functional
>
w)
=
fi(F; W) as follows for v’(W)
w; WI-
VW) v’(W)
> 0
E. L.oehman/J. of Economic Behavior & Org. 33 (1998) 285-302
Since it is a monotone transformation, the normalized preference same ranking of distributions as U. By definition of v(w),
I!@-;W) =
& By definition
functional
301
produces the
0
of U, U(F; W) = q!&(F; W) + v(W)
where C#J= v’(W) is a constant Furthermore, define
when wealth is held constant.
qx.
Then, by the properties
>
w)
4x; w - VW) v’(W)
=
of the Riemann
integral and definition
ti(F; W) =
of U,
ii@; W)dS(F) J’
Taking limits of both sides as F-+Fc By definition of ii,
implies that ic(x; W) is zero at x=0.
u(x; W) = f#G(x; W) + v(W) That is, in expressing preference over risk, the functional U(fl can be expressed with EURDP as a linear combination over a normalized utility ic over risks x and a FriedmanSavage utility v over wealth W. The normalized utility is indexed by W, that is, a wealthier person may have a different normalization than a less wealthy person. This result is illustrated in the Appendix A Figure.
vi
certain outcomes
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