Lopes 1999 Journal of Mathematical Psychology

8/3/2019 Lopes 1999 Journal of Mathematical Psychology

1/28

Journal of Mathematical Psychology 43, 286313 (1999)

The Role of Aspiration Level in Risky Choice:A Comparison of Cumulative Prospect Theory

and SPA Theory

Lola L. Lopes and Gregg C. Oden

University of Iowa

In recent years, descriptive models of risky choice have incorporatedfeatures that reflect the importance of particular outcome values in choice.Cumulative prospect theory (CPT) does this by inserting a reference point in

the utility function. SPA (security-potentialaspiration) theory uses aspira-tion level as a second criterion in the choice process. Experiment 1 comparesthe ability of the CPT and SPA models to account for the same within-sub-jects data set and finds in favor of SPA. Experiment 2 replicates the mainfinding of Experiment 1 in a between-subjects design. The final discussionbrackets the SPA result by showing the impact on fit of both decreasing andincreasing the number of free parameters. We also suggest how the SPAapproach might be useful in modeling investment decision making in a

descriptively more valid way and conclude with comments on the relationbetween descriptive and normative theories of risky choice. 1999 Academic Press

Formal models of decision making under risk can be found in three disciplinaryguises. Until quite recently, almost all economists believed that decision makersboth should and do select risks that maximize expected utility. In contrast, invest-

ment professionals have seen investors as selecting portfolios that achieve anoptimal balance between risk and return. Psychologists, too, have explored expectedutility theory and portfolio theory as possible descriptive models, and they havealso developed original information processing models focused on how peoplechoose rather than what people choose.

For the most part, these three disciplines have been isolated from one another,although psychologists have been more eclectic in their outlooks than others.Recently, however, both psychologists and economists have been exploring a non-linear modification of the expected utility model that we term the ``decumulatively

Article ID jmps.1999.1259, available online at http:www.idealibrary.com on

2860022-249699

30.00

Copyright 1999 by Academic PressAll rights of reproduction in any form reserved.

Deirdre Huckbody ran the subjects and coded the data as part of an independent study during hersenior year at the University of Wisconsin.

Requests for reprints should be sent to Lola Lopes at the College of Business Administration, Universityof Iowa, Iowa City, IA 52242 (lola-lopesuiowa.edu).


2/28

weighted utility'' model. At present, this model has not affected practice in finance,but we believe the model can also be applied at the level of the individual investor.

In what follows, we first describe decumulatively weighted utility generically and

then present two specific psychological instantiations of the model, cumulativeprospect theory (Tversky 6 Kahneman, 1992), and SPA theory (Lopes, 1987,1990, 1995). Second, we test the ability of these two theories to account for thesame set of data. Third, we suggest how these results might apply in the investmentcontext. Finally, we comment on fitting and testing complex, nonlinear models.

Decumulatively Weighted Utility

In a nutshell, the expected utility model asserts that when people choose betweenalternative probability distributions over outcomes (i.e., lotteries or gambles), theyshould (the economic model) and do (the psychological model) make their choicesso as to maximize a probability weighted (i.e., linear) average of the outcomeutilities. Although much evidence supports the idea that averaging rules can modelpeople's choices and judgments reasonably well in a variety of tasks (Anderson,1981), the linearity assumption of expected utility theory has not fared so well. Less

than a decade after the publication of von Neumann and Morgenstern's (1947)axiomatization of expected utility theory, Allais (19521979) demonstrated thatlinearity failed in qualitative tests comparing choices in which ``certainty'' was anoption to choices in which it was not.

For almost three decades afterward economists ignored these failures of linearity,whereas psychologists assumed that they represented subject failures rather thanmodel failures. In the 1980s, however, some economists began to take the failuresseriously and to seek out variants of expected utility that could better account forbehavior. An obvious candidate at the time was prospect theory (Kahneman 6Tversky, 1979), but this model violated stochastic dominance, ``an assumption thatmany theorists [were] reluctant to give up'' (Tversky 6 Kahneman, 1992, p. 299).Instead, these economists began to explore the normatively more acceptable idea ofdecumulatively weighted utility. The first economic applications were proposedindependently by Quiggin (1982), Allais (1986), and Yaari (1987), followed quicklyby many others (Chew, Karni, 6 Safra, 1987; Luce, 1988; Schmeidler, 1989; Segal,

1989).The best way to understand decumulative weighted utility is to start with the

structure of the expected value model:

EV= :n

i=1

pivi, (1)

in which the vi are the n possible outcomes listed in no particular order and the piare the outcomes' associated probabilities. Expected utility theory simply substitutesutility, u(v), for monetary outcomes:

EU= :n

i=1

piu(vi). (2)

287ASPIRATION LEVEL AND RISKY CHOICE


3/28

Weighted utility models (e.g., prospect theory) substitute decision weights, w(p), forprobabilities,

WU= :n

i=1

w(pi) u(vi), (3)

but this is the move that allows violations of stochastic dominance.Decumulative weighted utility models recast the issue of transforming raw

probabilities to one of transforming decumulative probabilities:

DWU= :

n

i=1h

\:

n

j=ipj

+ (u(v i)&u(v i&1))

= :n

i=1

h(D i)(u(v i)&u(v i&1)). (4)

In such models, the vi are ordered from lowest (worst outcome) to highest (bestoutcome). Di=

nj=i pj is the decumulative probability associated with outcome vi;

that is, Di is the probability of obtaining an outcome at least as high as outcome vi.Thus, D1 (the decumulative probability of the worst outcome, v1) is 1 (you get atleast that for sure) and Dn+1 (the decumulative probability of exceeding the bestoutcome, vn) is zero.

The function, h, maps decumulative probabilities onto the range (0, 1) and sopreserves dominance. It can also provide an alternative to using curvature in theutility function to model risk attitudes. For example, in expected utility theory, ifu(v) is a concave function of v, decision makers will prefer sure things to actuariallyequivalent lotteries, a pattern termed ``risk aversion.'' Decumulative weighting canpredict the same pattern even while assuming that u(v)=v (a variant of decumulativelyweighted utility that we call decumulatively weighted value) by letting h(D) be aconvex function of D. Although the predicted behavior is the same, we call it``security-mindedness'' in order to distinguish between utility-based and probability-based mechanisms.

The other major risk attitudes also have analogues in the decumulative weighted

value (or utility) model. These are given in Table 1. Both models can accommodate

TABLE 1

Risk Attitudes in the Expected Utility and Decumulative Weighted Value Models

Expected utility Decumulative weighted valueAssumes h(D)=D Assumes u(v)=v

Risk attitude u(v) is: Risk attitude h(D) is:Risk neutral linear Risk neutral linearRisk averse concave Security-minded convex

Risk seeking convex Potential-minded concaveMarkowitz type S-shaped Cautiously hopeful inverse S-shaped

288 LOPES AND ODEN


4/28

risk neutrality (expected value maximizing behavior) and both can accommodatethe rejection of sure things in favor of actuarially equivalent lotteries (termed ``riskseeking'' in expected utility theory and ``potential-mindedness'' by us). Both models

can also predict the ``cautiously hopeful'' (our term) pattern first described byMarkowitz (1959), in which subjects buy both insurance and lottery tickets, therebypaying premiums sometimes to gamble and other times to avoid gambling.

Cumulative Prospect Theory

In cumulative prospect theory (CPT), Tversky 6 Kahneman (1992) reformulatedthe original prospect theory in terms of (de)cumulative weighted utility. The utility

function, u(v), was unchanged from the original, being concave (risk averse) forgains and convex (risk seeking) for losses, with the loss function assumed to besteeper than the gain function (*>1):

u(v)={v:

&*(&v);if v0if v


5/28

The second feature of CPT is that it, like all its predecessors in the weightedvalue family, is a one-criterion model. Although there is much room in the modelfor psychological variables to operate, in the end, all these factors are melded into

a single assessment of lottery attractiveness.

SPA Theory

SPA theory (Lopes, 1987, 1990, 1995) is a dual criterion model in which theprocess of choosing between lotteries entails integrating two logically and psycho-logically separate criteria

SPA= f[SP, A], (8)

where SP stands for a security-potential criterion and A for an aspiration criterion.The SP (security-potential) criterion is modeled by a decumulatively weighted

value rule (i.e., the model is identical to Eq. (4) except that the utility function isassumed to be linear1):

SP= :n

i=1

h(Di)(v i&vi&1). (9)

The decumulative weighting function, h(D), has the form

h(D)=wDqs+1+(1&w)[1&(1&D)qp+1] (10)

for both gains and losses. The equation is derived from the idea that subjects assesslotteries from the bottom up (a security-minded analysis), or the top down(a potential-minded analysis), or both (a cautiously hopeful analysis).2 The param-eters qs and qp represent the rates at which attention to outcomes diminishes as theevaluation process proceeds up or down. The parameter, w, determines the relativeweight of the S and P analyses. If w=1, the decision maker is strictly security-

minded. If w=0, the decision maker is strictly potential-minded. If 0


6/28

Unlike CPT, however, the decumulative weighting function of SPA theory doesnot switch between inverse S-shaped for gains and S-shaped for losses.

The A (aspiration) criterion operates on a principle of stochastic control (Dubins6

Savage, 1976) in which subjects are assumed to assess the attractiveness oflotteries by the probability that a given lottery will yield an outcome at or abovethe aspiration level, ::

A=p(v:). (11)

For present purposes, we treat the aspiration level as crisp, which is to say,

a discrete value that either is or is not satisfied. In principle, however, the aspirationlevel may be fuzzy: some outcomes may satisfy the aspiration level completely,others to a partial degree, and still others not at all. To model this, Eq. (11) wouldneed to incorporate a particular probability, pi, according to the degree that itsassociated outcome, vi, satisfies the aspiration level (Oden 6 Lopes, 1997).

SPA theory and CPT share some significant psychological features: they bothmodel the process by which subjects integrate probabilities and values by a

(de)cumulative weighting rule, and they both include a point on the value dimen-sion that has special significance to subjects (the reference point for CPT and theaspiration level for SPA). Indeed, the aspiration level may be considered to be akind of reference point. However, the theories differ three ways in how thesefeatures function.

The first difference is in how the reference point (or aspiration level) exerts itsimpact. In CPT, the reference point is incorporated into the utility function and

influences subjects by marking an inflection point about which outcomes are firstorganized into gains and losses, and then scaled nonlinearly in accord with a principleof diminishing sensitivity. In SPA theory, the aspiration level participates in adirect assessment of lottery attractiveness reflecting a principle of stochastic controland separate from the decumulatively weighted SP assessment. Because SP and Aembody different criteria, each may favor a different lottery. When this happens,SPA theory predicts conflict, a prediction that does not follow from single-criterionmodels such as CPT.

The second difference is that CPT predicts a four-fold pattern across gain preferencesand loss preferences. Although some small imperfections in the symmetry of the patternmight obtain due to small differences in the value and weighting functions for gainsand losses, the overall pattern should be one of reflection between gains and losses.SPA theory, in contrast, allows considerable asymmetry between gains and losses.In the most commonly observed case, subjects appear to avoid risks strongly forgains but to be more-or-less risk neutral for losses (Cohen, Jaffray, 6 Said, 1987;

Hershey 6 Schoemaker, 1980; Schneider 6 Lopes, 1986; Weber 6 Bottom, 1989).Protocols suggest that this is because security-minded or cautiously hopeful sub-

jects set modest aspiration levels for gains, allowing the SP and A criteria toreinforce one another. For losses, however, the same subjects set high aspirationlevels, hoping to lose little or nothing, and thereby setting up a conflict between theA and the SP criteria (Lopes, 1995).



7/28

The third difference is that SPA theory predicts nonmonotonicities in preferencepatterns that depend on whether or not the aspiration level is guaranteed to be met(by boosting all outcomes above the aspiration level) or guaranteed not to be met

(by pushing all outcomes below the aspiration level) no matter which lottery of apair is chosen. For example, consider a cautiously hopeful decision maker choosingbetween 850 for sure versus a 5050 chance of8100, else nothing. Suppose also thatthe decision maker wants to win ``at least something.'' Although the SP assessmentcould favor the long shot mildly, the A assessment would favor the sure thingstrongly, leading to a choice of the sure thing. If850 were added to all outcomes,however, (e.g., 8100 for sure versus a 5050 chance at 8150, else 850) the A assess-ment would ``drop out'' (since both options satisfy the aspiration level with certainty)

leaving the SP assessment to carry the day. CPT, in contrast, is qualitatively unaffectedby cases in which outcomes are all pushed upward or downward so long as no out-comes cross the reference point. The experiments that follow use his third differenceto distinguish between the two theories and test their abilities to account for subjects'choices among a set of multioutcome lotteries.

EXPERIMENT 1

Method

Stimuli and task. Subjects chose between actuarially equivalent pairs of five-out-come lotteries comprising three positive (gain) sets and three negative (loss) sets.The standard positive lotteries are shown in Fig. 1. The tally marks represent lotterytickets yielding the outcomes shown at the left. Each of these lotteries has 100

tickets and each has an expected value of approximately 8100. The names indicatedfor the lotteries are for exposition only and were not used with subjects.

Scaled positive and shifted positive lotteries were created by transforming theoutcomes in the standard positive lotteries linearly. (Examples are shown in Fig. 2.)To create the shifted lotteries, standard positive outcomes were increased by 850(bringing the expected value of the shifted positive lotteries to 8150). To create thescaled positive lotteries, standard positive outcomes were multiplied by 1.145

(bringing the expected value of the scaled lotteries to 8114.50). The multiplicativeconstant was chosen to equate the maximum outcomes (8398) in the scaled andshifted sets.

Standard negative, scaled negative, and shifted negative lotteries were created byappending a minus sign to the outcomes in the respective positive sets.

Design and subjects. Lotteries within stimulus set were paired in all possiblecombinations ( 6C2=15 pairs per set) and pairs were arrayed vertically on sheets of

U.S. letter paper. Two replications were created for each set differing in the orderof the lotteries on the page.

Pairs from the positive sets were randomized together (within replication) withthe constraint that no particular lottery appeared on adjacent pages. Pairs from thenegative sets were randomized similarly. Each replication consisted of 45 pairs(3 sets_15 pairs per set).

292 LOPES AND ODEN


8/28

FIG. 1. Standard positive stimulus set. The tally marks represent lottery tickets yielding the out-comes shown at the left. Each lottery has 100 tickets and an expected value of approximately 8100.

The subjects for this experiment were 80 undergraduate students at the University

of Wisconsin who served for extra credit in introductory psychology courses.

Procedure. Subjects were run in groups of two to four. Each subject was givena notebook containing practice materials and the randomized stimulus pairs. At thebeginning of the experiment, subjects were shown how to interpret the positivelotteries and were told that the amount of prize money for each of the lotteries ina pair was the same. Then they were told that we were interested in their preferencesfor distributions (i.e., how the prizes are distributed over tickets) and were giventhree positive pairs for practice. Subjects were asked to indicate whether they wouldprefer the top lottery or the bottom lottery if they were allowed to draw a ticketfrom either for free and keep the prize.

Next, subjects were shown exemplars of negative lotteries and were told thatthese represented losses. They were then asked to indicate for a set of three morepractice pairs which of each pair they would prefer if they were forced to draw aticket from one or the other and pay the loss out of their own pockets.

Stimulus notebooks were divided into five sections, the first containing thepractice pairs and the remaining four containing the four sets of stimulus pairs (twopositive replications and two negative replications). Positive and negative replica-tions were alternated with half of the subjects beginning with a positive replicationand the other half beginning with a negative replication. Each set was preceded bya colored sheet announcing that ``The next set of lotteries are all win [or loss]



9/28

FIG. 2. Examples of stimuli from the scaled positive stimulus set and the shifted positive stimulusset. Scaled stimuli are produced from standard stimuli by multiplying outcomes by 1.145. Shifted stimuliare produced from standard stimuli by adding 850 to each outcome.

lotteries.'' Subjects went through the notebooks at their own pace, indicatingpreferences for the top or bottom lottery by circling ``T'' or ``B'' on a separateanswer sheet. The task took about an hour for most subjects. All choices werehypothetical.

Results and Discussion

The data from Experiment 1 are shown in Fig. 3 for gains (left panel) and forlosses (right panel). Lotteries are listed along the abscissas in the order of subjectpreferences for the standard lotteries. The data have been pooled over subjects,replications, and stimulus pair. Each data point represents the proportion of timesthe average subject chose the lottery out of the total number of times that thelottery was available for choice. Each lottery was presented 10 times (5 pairs_2 replications) so that the maximum number of choices per subject for a given

lottery was 10 and the minimum was zero.The data reveal four patterns that are of special significance: (1) For both gains

and losses, there are obvious main effects for lotteries [F(5, 395)=93.08 and 25.00,respectively, p


10/28

FIG. 3. Data from Experiment 1 pooled over subjects, replications, and stimulus pair. Lotteries arelisted along the abscissa in order of average subject preference for standard lotteries. Data are theproportion of occasions on which subjects chose a given lottery out of the 10 occasions on which thatlottery was available for choice.

[F(5, 395)=1.925, p=0.09 and F(5, 395)=0.602, p=0.69, respectively]; (3) The

slopes of the preference functions for the standard and scaled stimuli are steeper forgains than for losses; and (4) the preference functions for the shifted stimuli arenonmonotonically related to the preference functions for standard and scaledstimuli, especially for gains, and the pattern of nonmonotonicity reverses betweengains and losses. Preference for lower risk lotteries decreases for gains and increasesfor losses whereas preference for higher risk lotteries increases for gains anddecreases for losses. As will be seen, these differences between preference functions

for shifted lotteries and preference functions for standard and scaled lotteries arecritical to disentangling the roles of decumulative (or cumulative) weighting andaspiration level in risky choice.

In what follows, we use the Solver function of Microsoft Excel to fit both CPT(Tversky 6 Kahneman, 1992) and SPA theory (Lopes, 1990; Oden 6 Lopes, 1997)to the data. Solver is an iterative curve fitting procedure that adjusts freeparameters to optimize the fit of a model to a data set according to whatevercriterion the user specifies. We used root-mean-squared-deviation (RMSD) betweenpredicted and obtained. In order to lessen the possibility of finding only a localminimum, good practice requires starting with one's best guesses of parametervalues and then, once a minimum is found, checking the fit by systematically alter-ing parameter values and rerunning the program to make sure that a better fitcannot be found. The values we report are the best that we could find.

For both CPT and SPA, we fit the models to the aggregate choice proportions(given in Table 2) for the two-alternative choice task that subjects performed and

then pooled the prediction across choice pair to obtain means (as are shown for theobtained data in Fig. 3). Although we had too few replications to fit single subjectdata, visual inspection of single subject means revealed that the patterns of primaryinterest were evident at the single subject level, being especially clear for stronglysecurity-minded subjects and somewhat attenuated for subjects whose preferencestended toward cautious hopefulness or potential-mindedness.



11/28

TABLE 2

Choice Proportions

Gains Losses

Standard lotteries

RL PK SS RC BM LS LS BM RC SS PK RL

LS 0.931 0.844 0.800 0.869 0.594 0.500 RL 0.781 0.756 0.731 0.606 0.556 0.500BM 0.875 0.819 0.850 0.806 0.500 0.406 PK 0.713 0.700 0.656 0.519 0.500 0.444RC 0.856 0.719 0.631 0.500 0.194 0.131 SS 0.575 0.706 0.613 0.500 0.481 0.394SS 0.844 0.588 0.500 0.369 0.150 0.200 RC 0.606 0.588 0.500 0.388 0.344 0.269

PK 0.713 0.500 0.413 0.281 0.181 0.156 BM 0.569 0.500 0.413 0.294 0.300 0.244RL 0.500 0.288 0.156 0.144 0.125 0.069 LS 0.500 0.431 0.394 0.425 0.288 0.219

Scaled lotteries

RL PK SS RC BM LS LS BM RC SS PK RL

LS 0.856 0.844 0.813 0.806 0.563 0.500 RL 0.781 0.763 0.725 0.644 0.563 0.500BM 0.856 0.788 0.813 0.750 0.500 0.438 PK 0.663 0.675 0.613 0.500 0.500 0.438RC 0.863 0.744 0.650 0.500 0.250 0.194 SS 0.675 0.631 0.575 0.500 0.500 0.356

SS 0.844 0.531 0.500 0.350 0.188 0.188 RC 0.644 0.563 0.500 0.425 0.388 0.275PK 0.769 0.500 0.469 0.256 0.213 0.156 BM 0.488 0.500 0.438 0.369 0.325 0.238RL 0.500 0.231 0.156 0.138 0.144 0.144 LS 0.500 0.513 0.356 0.325 0.338 0.219

Shifted lotteries

RL RC LS BM PK SS RC LS BM SS PK RL

SS 0.788 0.625 0.600 0.625 0.519 0.500 RL 0.725 0.688 0.650 0.675 0.613 0.500PK 0.713 0.550 0.513 0.588 0.500 0.481 PK 0.606 0.588 0.525 0.463 0.500 0.388BM 0.731 0.575 0.575 0.500 0.413 0.375 SS 0.544 0.550 0.538 0.500 0.538 0.325LS 0.788 0.519 0.500 0.425 0.488 0.400 BM 0.594 0.488 0.500 0.463 0.475 0.350RC 0.769 0.500 0.481 0.425 0.450 0.375 LS 0.488 0.500 0.513 0.450 0.413 0.313RL 0.500 0.231 0.213 0.269 0.288 0.213 RC 0.500 0.513 0.406 0.456 0.394 0.275

Note. Lotteries within each matrix are listed in descending order of preference across the columnsand in ascending order of preference down the rows.

Fitting CPT. As noted previously, CPT is a one-criterion model in which bothvalues and probabilities are transformed psychologically during the lottery evalua-tion process. The utility function (Eq. (5)) has three parameters: : defines thecurvature of the function for gains (or values above the reference point); ; definesthe curvature of the function for losses (or values below the reference point); and* defines the relative slope of the two functions, with the loss function specified to

be steeper than the gain function (*>1).Weights in CPT also are defined separately for gains and for losses, as shown in

Eqs. (6) and (7). For gains, the function is decumulative with a parameter, #,regulating both the curvature and crossover point of the inverse S-shaped weightingfunction. For losses, the function is cumulative with an analogous parameter, $,regulating curvature and crossover points.

296 LOPES AND ODEN


12/28


13/28

FIG. 4. Predictions of CPT pooled over stimulus pair using the six parameter values shown inTable 4. h, standard; m, scaled; M, shifted.

The second and final step was to use the CPT attractiveness values as input toa pair-choice process that we modeled using the logistic function of CPT differencescores shown below4:

p(CPT1>2)= 11+e&k(CPT1&CPT2 ). (12)

The function predicts the proportion of times that lottery 1 is preferred to lottery2 based on the two individual attractiveness values. The function has a singleparameter, k, that is inversely related to the variance of the distribution of dif-ference scores, CPT1&CPT2 . Although it might seem reasonable to allow CPT to

fit separate k parameters for gains and for losses, the second k would be redundantwith * and would, in the present case, allow * to fall below 1 (see Footnote 5).Figure 4 shows the best-fitting predictions of CPT to the data for gains (left

panel) and for losses (right panel). Parameter value are in Table 2. Although theRMSD of 0.0810 is respectable for fitting 90 data points with six parameters,a comparison of predictions and data (Fig. 3) reveals a qualitative discrepancy forthe shifted gain lotteries. Although CPT is able to capture the general flattening ofthis preference function relative to the standard and scaled functions, all but the

prediction for the long shot are monotonically decreasing. In contrast, subjects'preferences for the shifted rectangular, bimodal, and long shot lotteries were allgreater than for the shifted short shot. Moreover, the nonmonotonicity that isinduced for the shifted long shot comes at the expense of incorrectly predicting non-monotonicity for the standard and scaled long shots as well.

A second issue concerns the CPT parameter values (see Table 4). Beginning withthe parameters for the utility functions, note that although the function for gains is

298 LOPES AND ODEN

4 Although CPT is intended to be a theory of risky choice, the process that maps two or moreindividual lottery assessments onto choice has not been specified. The logistic function that we applyhere and below is commonly used as the cumulative probability distribution function in statisticaldecision theory models of the two-alternative choice process (Luce 6 Galanter, 1963) and is presumedto be neutral with respect to its impact on CPT's and SPA's ability to fit the qualitative features of thedata.


14/28

TABLE 4

Parameter Values for CPT

Parameter Value

: 0.551; 0.970* 1.000# 0.699$ 0.993k 0.739

sharply curved (:=0.426), the function for losses is close to linear (;=0.942).Second, looking at the probability weights, note that the function for gains showsconsiderable nonlinearity (#=0.685), whereas the function for losses is again essen-tially linear ($=0.980). Finally, looking at *, the parameter that determines therelative slopes of the utility functions for gains and losses, note that it has reached

its floor value of 1.00.5

Although the values of these parameters are consistent withthe observed fact that the loss data are essentially linear and relatively shallow inslope, there is nothing in CPT that would lead one to expect this large asymmetrybetween gains and losses. Indeed, CPT is well-known for its prediction of reflectionin preferences between gains and losses (i.e., the four-fold pattern).

In all, then, CPT does reasonably well in fitting the data if one considers onlyRMSD. When one looks at qualitative effects, however, CPT fails with the shifted

gains. Moreover, in order to get a reasonable fit, CPT must make use of parametervalues that are inconsistent with the underlying psychophysical principle (diminishingsensitivity from a reference point) on which both utility and weighting functions aretheorized to depend.

Fitting SPA theory. As explained previously, SPA theory proposes that riskychoice involves two criteria, one based on a comparison of decumulatively weightedaverages of probabilities and outcomes (the SP criterion) and the other based

on a comparison of probabilities of achieving an aspiration level (the A criterion).The SP assessment process (Eq. (4)) has three parameters: q defines the degree ofattention to different outcomes in assessments of security ( qs) and potential (qp),whereas w defines the relative importance of security and potential assessmentsoverall. In principle, all three parameters might differ between gains and losses. TheA assessment process (Eq. (5)) has a single parameter, :, the aspiration level, whichcan also differ for gains (:+) and losses (:&). Although : might need to be fit asa free parameter in some cases (e.g., with continuous outcome distributions or with


5 A somewhat improved RMSD (0.0770) resulted when * was allowed to drop to 0.400 (i.e., for thegain function to be considerably steeper than the loss function). Although this value is consistent withthe observed fact that the standard and scaled gain data are steeper than the standard and scaled lossdata, the value is not consistent with CPT's oft-repeated claim that ``losses loom larger than gains.''


15/28

manipulated aspiration levels), our stimuli and choice conditions justified fixing :+

at 1 and :& at zero.6

In fitting SPA theory to the data, we modeled the SP criterion and the A

criterion separately (but not sequentially) and integrated the results into a finalchoice. The procedure is schematized in Table 5. Matrix A gives the scaled positivedata and Matrix B gives the best-fitting predictions based on just the SP criterion.The column and row headings are estimated SP attractiveness values. We let wdiffer between gains (w+) and losses (w&) but, in order to hold our parameters tosix, set qs=qp=q and used the same single value for both gains and losses. Theentries in the cells are choice proportions, p(SP1>2), obtained by applying thelogistic function to difference scores just as we did in fitting CPT:

p(SP1>2)=1

1+e&k(SP1&SP2 ). (13)

The parameter, k, is inversely related to the variance of the distribution of differencescores, SP1&SP2 .

Matrix C shows best-fitting predictions based on just the A criterion. The row

and column headings show the probability that the row (or column) lottery willyield a value that meets the aspiration level (e.g., the riskless gain lottery, column1, guarantees a nonzero payoff whereas the peaked gain lottery, column 2, has onlya 0.96 probability of a nonzero payoff). The table entries are choice proportions,p(A1>2), obtained by submitting the A values to a relative ratio process having aparameter, t, 0t that controls contrast. Equation (14) shows the process forgains:

p(A1>2)=A t

+

1

A t+

1 +At+

2

. (14)

The equation for losses looks a little different but has the same structure

p(A1>2)=1&

(1&A1)t&

(1&A1)t& +(1&A2)t&

=(1 &A2)

t&

(1&A1)t& +(1&A2)

t&. (15)

The difference between gains and losses reflects the fact that gains engender anapproachapproach process based on relative lottery goodness, A, whereas losses

300 LOPES AND ODEN

6 Our assumption concerning the values of the aspiration level is analogous to Kahneman andTversky's assumption (1979; Tversky 6 Kahneman, 1992) that the reference point of the utility functionis at zero. Although we, as they, might sometimes want to modify this simplifying assumption, in thepresent case the lottery outcomes are spaced widely enough that minimum (or maximum) outcomes aregood approximations for what might be ``at least a small gain'' or ``no more than a small loss'' in choicesbetween more continuous lotteries.


16/28

TABLE 5

Fitting SPA Theory to the Data for Scaled Gain Pairs

Matrix A: Raw choice proportions

RL PK SS RC BM LS

LS 0.856 0.844 0.813 0.806 0.563 0.500BM 0.856 0.788 0.813 0.750 0.500 0.438RC 0.863 0.744 0.650 0.500 0.250 0.194SS 0.844 0.531 0.500 0.350 0.188 0.188PK 0.769 0.500 0.469 0.256 0.213 0.156

RL 0.500 0.231 0.156 0.138 0.144 0.144Means 0.838 0.628 0.580 0.460 0.271 0.224

Matrix B: Choice predictions based on SP criterion

114.86 113.98 113.60 113.91 113.91 113.88

113.88 0.839 0.537 0.382 0.510 0.509 0.500113.91 0.834 0.528 0.374 0.501 0.500 0.491113.91 0.833 0.527 0.372 0.500 0.499 0.490

113.60 0.894 0.652 0.500 0.628 0.626 0.618113.98 0.818 0.500 0.348 0.473 0.472 0.463114.86 0.500 0.182 0.106 0.167 0.166 0.161

Matrix C: Choice proportions based on A criterion

1.000 0.960 0.960 0.800 0.680 0.620

0.620 0.989 0.984 0.984 0.917 0.705 0.500

0.680 0.975 0.963 0.963 0.823 0.500 0.2950.800 0.892 0.848 0.848 0.500 0.177 0.0830.960 0.595 0.500 0.500 0.152 0.037 0.0160.960 0.595 0.500 0.500 0.152 0.037 0.0161.000 0.500 0.405 0.405 0.108 0.025 0.011

Matrix D: Predictions combining SP and A criteria

RL PK SS RC BM LS

LS 0.956 0.895 0.861 0.773 0.612 0.500BM 0.933 0.844 0.797 0.684 0.500 0.388RC 0.865 0.714 0.646 0.500 0.316 0.227SS 0.779 0.578 0.500 0.354 0.203 0.139PK 0.720 0.500 0.422 0.286 0.156 0.105RL 0.500 0.280 0.221 0.135 0.067 0.044

Means 0.850 0.662 0.590 0.446 0.271 0.181

engender an avoidanceavoidance process based on relative lottery badness, 1&A.In other words, the probability of choosing Lottery 1 over Lottery 2 for gainsreflects the degree to which Lottery 1 is better than Lottery 2, whereas the probabil-ity of choosing Lottery 1 over Lottery 2 for losses reflects the degree to whichLottery 2 is worse than Lottery 1.



17/28

FIG. 5. Predictions of SPA theory pooled over stimulus pair using the six parameter values shownin Table 6. h, standard; m, scaled; M, shifted.

Matrix D combines the p(SP1>2) values and the p(A1>2) values according to

p(SPA1>2)=

[p(SP1>2) p(A1>2)]12

[p(SP1>2) p(A1>2)]12+[(1&p(SP1>2))(1&p(A1>2))]

12 . (16)

This rule (which is useful for cases in which both the domain and the range of thefunction are 0 to 1) displays both averaging properties [p(SA1>2) lies betweenp(SP1>2) and p(A1>2)] and Bayesian properties (the impact of an input valuedepends on its extremity). The exponent, 12, sets the weight of the two input quan-tities to be equal. Although one might imagine that this could be a free parameter

in the model, our experience suggests that it shares variance with the SP and Aparameters, muddying the fitting process when it is included.

Figure 5 shows the best fitting predictions of SPA theory to the data for gains(left panel) and for losses (right panel). The RMSD of predicted to obtained is0.0681 (relative to 0.0810 for CPT). As can be seen by inspection of the figures,SPA has also done a better job of capturing the qualitative features of the data. Inparticular, SPA predicts the nonmonotonic increases in preference for the shifted

rectangular, bimodal, and long shot gain lotteries.

TABLE 6

Parameter Values for SPA Theory (Six-Parameter Fit)

Parameter Value

q 1.053w+ 0.505w& 0.488k 1.694t+ 9.447t& 2.035

302 LOPES AND ODEN


18/28

The parameter values for the SPA model are in Table 6. Although the values aregenerally reasonable, the values for the SP component are not far from an expectedvalue fit (w=0.50, q

s=q

p=1). We believe this reflects the simplifying constraints

that we imposed on the parameters of the SP criterion. We shall have more to sayabout the matter in the final discussion.

EXPERIMENT 2

It is sometimes thought that the opportunities for comparison offered by repeatedmeasures designs create choice patterns that might not occur if subjects made onlya single choice. The purpose of Experiment 2 was to replicate the main finding of

Experiment 1 (i.e., that shifted lotteries are evaluated differently than standard orscaled lotteries) using a between-subjects design. We also wanted to collect subjects'reasons for their choices. Because between-subject experiments are costly in termsof the required number of subjects, we used only positive lotteries.

Method

The stimuli were the 45 pairs comprising the standard, scaled, and shifted

positive sets. Each pair was printed separately on a single sheet of U.S. letter paper.Sheets were randomized and distributed at the beginning of experimental sessionsto subjects who were participating in other related experiments. In a given session,different subjects had different pairs. Consequently, it was necessary to describe howto interpret the lotteries in very general terms, never mentioning particular out-comes or numbers of tickets.

As in Experiment 1, subjects were asked to mark which of the two lotteries they

would prefer if they were allowed to draw a ticket from either for free and keep theprize for themselves. They were also asked to write a sentence or two explaining thebasis for their preference.

FIG. 6. Data from Experiment 2 pooled over subjects and stimulus pair. Data are the proportionof subjects choosing a given lottery out of the total number of subjects who had that lottery availablefor choice. h, standard; m, scaled; M, shifted.



19/28

A total of 433 subjects from the University of WisconsinMadison and theUniversity of Iowa participated in the experiment for extra course credit. Six ofthese subjects indicated by their written responses that they had not understood

how to interpret the lotteries, leaving 427 usable responses ranging over the 45choice pairs.

Results and Discussion

The data from Experiment 2 are shown in Fig. 6. Clearly, the means are noisierthan the means from Experiment 1, a result one might anticipate not only from thedifferent subjects contributing to each data point, but also from the reduced

amount of data going into each data point (400 choices per data point in thewithin-subject case versus about 24 choices per data point in the between-subject

TABLE 7

Illustrative Protocols from Experiment 2

Scaled lotteries (LS vs SS) Shifted lotteries (LS vs SS)

1. Have a greater chance of winning at leastsome money [in the SS]. The greatest chance is8160. [In the LS] lottery, your greatest chanceis zero. (Picks SS)

7. I picked [LS] because winning any amountof money would be exciting for me. If I picked850 that would be great but if I picked any othernumber it automatically gives me more moneyin comparison with the other lottery choices.

2. There is a better chance to win money [in theSS] because there are more tally marks for thehigher value of money than there is for 80.(Picks SS)

8. The amount of money to be won [in LS] isgreater. Even though the chances of winningmay be less, it's worth the risk. (Picks LS)

3. The odds of winning any amount of moneyhere [in the SS] are a lot higher. Only 4 peopleout of 100 didn't get anything. (Picks SS)

9. The [SS] seems to be the better choicebecause the most tickets are for the larger prizeamounts. But the [LS] has larger prizes. Yourchances of winning more money seem to begreater here. (Picks LS)

4. There are more chances for winning money inthe [SS]. The [LS] has a lot of tickets thathave no monetary value. Even though the

amounts are greater in the [LS], your chancesare better in the [SS] to get a ticket worthmoney. (Picks SS)

10. Excluding the extremes (the 8190 top lotteryprize in [SS] and the 850 lottery prize in [LS])it appears that if you win, the prize will be for

more money in [LS]. A better payoff for notthat much more risk. (Picks LS)

5. I was not putting forth any money for thislottery, so even though the [LS] has a greaterchance of getting 80even if I was unfortunateI lose nothing. But if I win, I win substantiallymore money. (Picks LS)

11. I prefer the [SS] because there is more likeli-hood of winning. The best chance in the [LS] is850 while the [SS] is 8190. Although [with theLS] you can win more there is a greater chanceof not winning.

6. There is a better chance of winning more

money [in the LS] and there is more moneyinvolved (higher prizes). (Picks LS)

12. There are less tickets for more money in the

[LS]. I'd rather have a better chance for a littleless money. My odds are better to get 8190 [withSS] than 8398 [with LS]. My odds are equal to

get 8190 [with SS] or 850 [with LS]. I opted forthe 8190 bracket. (Picks SS)

Note. Protocols in italic are from subjects whose preferences went against majority preferences.

304 LOPES AND ODEN


20/28

case). Despite the noise, however, a chi-square analysis shows that the key differen-ces between the three scaling conditions were replicated. Specifically, (1) thepatterns of preferences for standard stimuli and scaled stimuli do not differ signifi-

cantly from one another, /2

(1)=1.65, p>0.05; whereas (2) the pattern of preferencefor shifted stimuli differs significantly from the overall pattern of preference for standardand scaled stimuli, /2(1)=25.13, p


21/28

provided protocols confirming that the nonmonotonicity may arise because adding850 to standard positive lotteries eliminates aspiration level as a consideration formost subjects, thus enhancing the impact of the decumulatively weighted SP

criterion. In what follows, we discuss the implications of this result for modelinginvestment risk. We also provide comments on fitting complex models along withtwo instructive comparisons to the six-parameter SPA model. We end with adiscussion of the relation between descriptive and normative theories.

Risk Taking and Aspiration Level

It has often been pointed out that when people are in economic difficulty, theytend to take risks that they would avoid under better circumstances. This tendency

appears among sophisticated managers in troubled firms (Bowman,1980, 1982) aswell as among unsophisticated subsistence farmers (Kunreuther 6 Wright, 1979).Experimental studies using managers as subjects have also confirmed the tendencytoward risk-taking for losses, at least when ruin is not at issue (Laughhunn, Payne,6 Crum, 1980; Payne, Laughhunn, 6 Crum, 1981). Standard thinking in invest-ment theory would not lead one to expect risk taking in threatening situations.Instead, hard-pressed decision makers should value low risk over high expected

return and choose accordingly.The S-shaped utility function of prospect theory seems to provide an explanation

for this paradoxical risk-taking: people take risks when they face losses becausetheir utility function for losses is ``risk seeking'' (i.e., convex). Though one cancriticize the circularity of the ``explanation,'' it at least predicts preferences betterthan the more standard assumption of uniform ``risk aversion'' (i.e., diminishingmarginal utility). But predicting preferences is only half the story, especially when

predictions fail, as they often do in experimental studies of preferences for losseswith students (Cohen, Jaffray, 6 Said, 1987; Hershey 6 Schoemaker, 1980;Schneider 6 Lopes, 1986; Weber 6 Bottom, 1989) as well as with managers(MacCrimon 6 Wehrung, 1986). The other half of the story can be found inprotocol data. Studies by Mao (1970) and by Petty and Scott (cited in Payne,Laughhunn, 6 Crum, 1980) suggest that managers tend to define investment riskas the probability of not achieving a target rate of return (that is to say, an aspira-tion level).

No one can doubt that expected return (i.e., expected or mean value) is a centraland well-understood concept for managers, but the concept of risk is less wellunderstood. In portfolio theory, for example, risk is usually equated with outcomevariance (Markowitz, 1959) but this is not entirely satisfactory descriptively sinceit treats wins and losses alike. Other approaches to defining risk try to bypass thisobjection by restricting the variance computation to losses (i.e., the semivariance)or by computing risk as a probability weighted average of deviations below a target

level (Fishburn, 1977). Few, however, have explored the possibility of modeling riskas the raw probability of not achieving an aspiration level. One who has is Manski(1988) who developed the idea in what he called a utility mass model. Anotherapproach that incorporates raw probabilities comes from Weber (1988) whoaugmented an expectation model with weighted probabilities of winning, losing,and breaking even.

306 LOPES AND ODEN


22/28

SPA theory incorporates both notions, each in a separate criterion. On the SPside, a security-minded weighting function (or a cautiously hopeful function dis-playing more caution than hope) pays more attention to the worst outcomes than

to better outcomes. On the A side, the model operates on the probability of achiev-ing the aspiration level. Although normative models usually focus on a singlecriterion, descriptive models must go where subjects lead. In the case at hand, thesubjects seem to be saying that they understand and use the term ``risk'' in bothdistributional and aspirational senses. For example, in Table 7, two subjects referexplicitly to risk. In Protocol 8, one subject uses the concept in the A criterionsense: risk is the chance of winning less. In Protocol 10, however, another subjectdoes not count chances, but rather focuses on differences in prize amounts, an

SP-focused analysis.Practitioners work at the boundary between normative and descriptive. Clients

expect guidance (the normative function) in how to achieve their personal goals(the descriptive function). For the client who is concerned about not meeting atarget return, there seems little point in discussing variance. It would seem betterfor the professional to recognize in a client's spoken desires the relevance of thosemathematical rules that seem most applicable and, then, to explain in simple

fashion, properties of the rules that may not be self-evident.Although much has been claimed since von Neumann and Morgenstern (1947)

about the dire consequences of violating linearity, recent examination of alternativerules based on decumulative weighted utility and aspiration criteria (e.g., Manski,1988; March, 1996; Yaari, 1987) suggests that these alternatives are neither betternor worse than maximizing expected utility. They are, however, different and seemto come closer to doing what people want done.

Pushing the Model Tests

In Experiment 1, we fit the CPT and SPA models on the same number of freeparameters even though SPA theory could reasonably use several more. In orderto better illuminate the roles of the various psychological components of the theory,we now bracket the 6-parameter SPA fit by comparing it to a 0-parameter fit anda 10-parameter fit.

The top panel of Fig. 7 shows what happens when the SP criterion of SPAtheory is neutralized by setting its parameters to yield the expected value for alllotteries (w=0.50, qs=qp=1.00), thus allowing the A criterion to dominate. We setthe aspiration level here as we did previously: : gains>0; : losses=0. The choicerule is also zero-parameter, assuming that subjects choose whichever lottery has thehigher value on the A criterion and are indifferent if lotteries tie on aspiration level.

The A criterion alone produces a reasonable quantitative fit, with an RMSD of0.1206. Although it may seem surprising that such a simple mechanism does so well

in fitting complex data given the very complicated models that have been favoredfor risky choice recently (including both CPT and SPA), the result maps well ontothe classic finding of information processing studies using duplex bets (Payne 6Braunstein, 1971; Slovic 6 Lichtenstein, 1968) that probability of winning dominatesthe choice process. Still, the best that aspiration can do by itself with the shifteddata is to fit a flat line. Aspiration alone also predicts a mirror symmetry (reflection



23/28

FIG. 7. Top panel: SPA predictions based on the A criterion alone with the SP criterionneutralized. Bottom panel: SPA predictions based on the ten parameter values shown in Table 8.h, standard; m, scaled; M, shifted.

in preferences) between gains and losses whereas the actual loss preference functionsare much flatter than the gain preference functions for all three lottery types.

The bottom panel of Fig. 7 shows what happens with a full, 10-parameter fit ofSPA theory. The fit (RMSD=0.0484) is obviously much better than the 6-parameterfit. What interests us more, however, is that removing constraints on the SP param-eters reveals theoretically meaningful values (see Table 8). Whereas previously the SPcriterion came very close to an expected value criterion, the new parameter valuessuggest important process differences between gains and losses. For gains, it appearsthat the bottom-up (security) evaluation is more important than the top-down(potential) evaluation whereas, for losses, the top-down (potential) evaluation appears

more important than the bottom-up (security) evaluation. Similarly, differences in thew parameter also suggest that the importance of security is greater for gains than forlosses (w+>w&). Thus, SPA parameters confirm the CPT-based intuition thatsubjects do, indeed, evaluate high-risk options more favorably for losses than for gains.

There is, however, an important difference between the mechanisms used byCPT and SPA to account for these differences between gains and losses. CPT's

308 LOPES AND ODEN


24/28

TABLE 8

Parameter Values for SPA Theory (10-Parameter Fit)

Parameter Value

q+s

372.07q+

p64.37

q&s

4.86q&

p16.58

w+ 0.837w& 0.003k+ 0.043

k& 0.023t+ 10.000t& 2.070

zero reference point provides the rationale for qualitative inversions of its utilityfunction (from concave for gains to convex for losses) and its decumulativeweighting function (from inverse S-shaped for gains to S-shaped for losses). Thus,

CPT specifies that the value processing mechanisms and probability weightingmechanisms used by subjects differ qualitatively for gains and losses. SPA, on theother hand, allows the relative attention paid to worst outcomes and best outcomesto shift as a function of domain (gains versus losses) and also allows the relativeimportance of the SP and A components to differ between gains and losses. Butdomain-mediated reference effects in SPA theory are potentially applicable to abroader range of domain differences.

For example, Edwards 6 von Winterfeldt (1986) propose that a person's risk

attitude may be different in different ``transaction streams.'' Choices involvingamounts in what they call the ``quick cash'' and ``play money'' streams (the formerbeing what people have available in their wallets and the latter being money reservedfor enjoyment) should be less risk averse (i.e., less security-minded) than choicesinvolving ``capital assets'' and ``income and fixed expenditures'' streams. Similarly,MacCrimon 6 Wehrung (1986) found that executives are more risk averse inmaking decisions about their own personal investments than they are about

business investments. Shifts of these sorts in one's willingness to accept risk neednot involve gainloss shifts. Instead, they may involve only differences in outcomescale (large versus small transaction stream) or differences in real-world expecta-tions or consequences (personal decisions versus business decisions). The param-eters of SPA theory, while restricting the decumulative weighting function to aninverse S shape for both gains and losses, nevertheless allow for modeling thisbroader class of reference effects through differences in the relative attention paid to

bad versus good outcomes in the SP assessment and through the relative impor-tance accorded to SP and A assessments in the final choice.

The Importance of Normative Theory for Description and vice versa

In most of economics, expected utility theory remains the workhorse of academicresearchwhich is to say, of normative researchdespite its poor fit to data from



25/28

psychological experiments. Recently, however, a number of economists have turnedtheir attention to testing expected utility theory in laboratory settings involvingstylized economic games. Up to now, this new enterprise has been doggedly empirical

and intently focused on theoretically appropriate task instantiation and on experimen-tal rigor and control. Despite this attention to detail, however, the predictions ofthe theory have frequently not been borne out, leaving the experimentalists with thenot inconsiderable task of persuading their colleagues that the model's failures aremeaningful and should not be overlooked (for reviews, see the essays by Smith,1982, 1989, 1991; and the various chapters in Kagel 6 Roth, 1995). There has not,however, been a commensurate effort from these researchers to develop bettertheory, although Roth (1995, p. 18) has pointed out that at least some of those who

ran the earliest economic experiments expected that experimental data would con-tribute to the development of both better descriptive theories and better normativetheories.

Most economists view their discipline as one that deals with ideally rationalbehavior and, thus, attach little significance to discrepancies between what thetheory predicts and what people actually do. Psychologists, on the other hand, viewtheir task as one of predicting behavior and describing its cognitive sources in

psychologically meaningful terms, whether or not that behavior is rational. Theutility and probability weighting functions of CPT rest on perceptual concepts. TheSP and A components of SPA theory rest on attentional and motivational concepts.Thus, both theories provide a psychological grounding that allows each to appealdirectly to intuitions via easily understood and compellingly named components.

Intuitiveness is not enough, however. Mathematical analysis of the sort pursuedhere is necessary to specify the quantitative mechanisms from which theoretical

predictions flow and to confirm that it is these specific mechanisms that provide thebest account of behavior. History shows that it is easy to conflate phenomena withexplanations, especially when the explanations appeal to intuition. Thus, thephenomenon of risk aversion became conflated with the idea of diminishingmarginal utility (concavity) because the intuition was powerful and, indeed, isaccurate, that constant marginal gains or losses in assets are more noticeable topoor people than to rich people. Conflation of phenomena with explanation isespecially hazardous to theoretical advancement in that it suppresses interest inpsychologically important alternative explanations, such as the aspiration anddecumulatively weighted utility mechanisms on which this paper has focused.

The model comparisons we presented pitted competing psychological mechanismsagainst one another while constraining them to the same number of free parameters.For the data set at hand, SPA theory provided the better fit, both quantitativelyand qualitatively. However, a more important comparison may reside in therelative strengths and weaknesses of the three different parameterizations of SPA

theory. On the one hand, the zero-parameter model, relying solely on the aspirationlevel mechanism, did surprisingly well in providing a rough fit to the data. That amechanism as simple as this was overlooked as an alternative to more complicatedaccounts is testimony to the unhealthy power that the ``best existing theory'' hasto stifle research into alternatives. On the other hand, the 6- and 10-parameterversions of SPA theory show the necessity of the SP component for modeling

310 LOPES AND ODEN


26/28

preferences among the shifted lotteries and for capturing the relative flattening ofpreferences for loss lotteries. Although the possible contributions of aspiration levelshould not have been overlooked, theorists and experimentalists since Bernoulli

have not been foolish in pursuing weighted utility models. Aspiration alone issimply too simple.SPA theory is a descriptive theory, through and through. Its dual choice criteria

the security-potential criterion and the aspiration criterionare both includedbecause each seems necessary to adequately capture human choices under risk. Itis worth noting, however, that even though these two criteria are inconsistent withexpected utility maximization except in special cases, the rationality of each hasbeen defended recently on normative grounds, (e.g., see Manski's (1988) utility

mass model and Yaari's (1987) decumulatively weighted value model). Althoughthere is still a great divide between normative and descriptive theories of riskychoice, perhaps we are seeing the first evidence that descriptive research is finally,as Roth (1995, p. 22) put it, ``speaking to theorists.''

REFERENCES

Allais, M. (19521979). The foundations of a positive theory of choice involving risk and a criticism ofthe postulates and axioms of the American School. In M. Allais, 6 O. Hagen (Eds.), Expected utilityhypotheses and the Allais Paradox (pp. 27145). Dordrecht, Holland: Reidel.

Allais, M. (1986). The general theory of random choices in relation to the invariant cardinal utility functionand the specific probability function (Working Paper C4475): Centre d'Analyse E conomique, E coledes Mines, Paris, France.

Anderson, N. H. (1981). Foundations of information integration theory. New York: Academic Press.

Bowman, E. H. (1980). A risk-return paradox for strategic management. Sloan Management Review,21(3), 1731.

Bowman, E. H. (1982). Risk seeking by troubled firms. Sloan Management Review, 23(4), 3342.

Chew, S.-H., Karni, E., 6 Safra, S. (1987). Risk aversion in the theory of expected utility with rankdependent probabilities. Journal of Economic Theory, 42, 370381.

Cohen, M., Jaffray, J.-Y., 6 Said, T. (1987). Experimental comparison of individual behavior under riskand under uncertainty for gains and for losses. Organizational Behavior and Human DecisionProcesses, 39, 122.

Coombs, C. H., 6 Huang, L. (1970). Tests of a portfolio theory of risk preference. Journal of ExperimentalPsychology, 85, 2329.

Dubins, L. E., 6 Savage, L. J. (1976). Inequalities for stochastic processes: How to gamble if you must(2nd ed.). New York: Dover.

Fishburn, P. C. (1977). Mean-risk analysis with risk associated with below-target returns. AmericanEconomic Review, 67(2), 116126.

Hershey, J. C., 6 Schoemaker, P. J. H. (1980). Prospect theory's reflection hypothesis: A criticalexamination. Organizational Behavior and Human Performance, 25, 395418.

Kagel, J. H., 6 Roth, A. E. (1995). The handbook of experimental economics. Princeton, NJ: PrincetonUniversity Press.

Kahneman, D., 6 Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica,47, 263291.

Kunreuther, H., 6 Wright, G. (1979). Safety-first, gambling, and the subsistence farmer. In J. A.Roumasset, J.-M. Boussard, 6 I. Singh (Eds.), Risk, uncertainty, and agricultural development(pp. 213230). New York, NY: Agricultural Development Council.



27/28

Laughhunn, D. J., Payne, J. W., 6 Crum, R. (1980). Managerial risk preferences for below-target

returns. Management Science, 26(12), 12381249.

Lopes, L. L. (1987). Between hope and fear: The psychology of risk. Advances in Experimental Social

Psychology, 20, 255295.

Lopes, L. L. (1990). Re-modeling risk aversion. In G. M. von Furstenberg (Ed.), Acting under

uncertainty: Multidisciplinary conceptions (pp. 267299). Boston: Kluwer.

Lopes, L. L. (1995). Algebra and process in the modeling of risky choice. In J. Busemeyer, R. Hastie,

6 D. L. Medin (Eds.), Decision making from a cognitive perspective (Vol. 32, pp. 177220). San Diego:

Academic Press.

Lopes, L. L. (1996). When time is of the essence: Averaging, aspiration, and the short run. Organiza-

tional Behavior and Human Decision Processes, 65, 179189.

Luce, R. D. (1988). Rank-dependent, subjective expected-utility representations. Journal of Risk andUncertainty, 1, 305332.

Luce, R. D., 6 Galanter, E. (1963). Discrimination. In R. D. Luce, R. R. Bush, 6 E. Galanter (Eds.),

Handbook of mathematical psychology (Vol. 1). New York: Wiley.

MacCrimon, K. R., 6 Wehrung, D. A. (1986). Taking risks: The management of uncertainty. New York:

Free Press.

Manski, C. F. (1988). Ordinal utility models of decision making under uncertainty. Theory and Decision,

25, 79104.

Mao, J. C. T. (1970). Survey of capital budgeting: Theory and practice. Journal of Finance, 25, 349360.March, J. G. (1996). Learning to be risk averse. Psychological Review, 103(2), 309319.

Markowitz, H. M. (1959). Portfolio selection: Efficient diversification of investments. New York: Wiley.

Oden, G. C., 6 Lopes, L. L. (1997). Risky choice with fuzzy criteria. In R. W. Scholz 6 A. C. Zimmer

(Eds.), Qualitative aspects of decision making (pp. 5682). Lengerich, Germany: Pabst Science

Publishers.

Payne, J. W., 6 Braunstein, M. L. (1971). Preferences among gambles with equal underlying distribu-

tions. Journal of Experimental Psychology,87

, 1318.Payne, J. W., Laughhunn, D. J., 6 Crum, R. (1980). Translation of gambles and aspiration level effects

in risky choice behavior. Management Science, 26(10), 10391060.

Payne, J. W., Laughhunn, D. J., 6 Crum, R. (1981). Further tests of aspiration level effects in risky

choice behavior. Management Science, 27(8), 953958.

Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior and Organization, 3,

323343.

Roth, A. E. (1995). Introduction to experimental economics. In J. H. Kagel 6 A. E. Roth (Eds.), The

handbook of experimental economics (pp. 3109). Princeton, NJ: Princeton University Press.Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica , 57,

571587.

Schneider, S. L., 6 Lopes, L. L. (1986). Reflection in preferences under risk: Who and when may suggest

why. Journal of Experimental Psychology: Human Perception and Performance, 12(4), 535548.

Segal, U. (1989). Axiomatic representation of expected utility with rank-dependent probabilities. Annals

of Operations Research, 19, 359373.

Slovic, P., 6 Lichtenstein, S. (1968). Relative importance of probabilities and payoffs in risk taking.

Journal of Experimental Psychology Monograph, 78(3 (Part 2)), 118.

Smith, V. L. (1982). Microeconomic systems as an experimental science. American Economic Review, 72,

923955.

Smith, V. L. (1989). Theory, experiment, and economics. Journal of Economic Perspectives, 3, 151169.

Smith, V. L. (1991). Rational choice: The contrast between economics and psychology. Journal of Political

Economy, 99, 877897.

312 LOPES AND ODEN


28/28

Tversky, A., 6 Kahneman, D. (1992). Advances in prospect theory: Cumulative representation ofuncertainty. Journal of Risk and Uncertainty, 5, 297323.

von Neumann, J., 6 Morgenstern, O. (1947). Theory of games and economic behavior (2nd ed.).Princeton, NJ: Princeton University Press.

von Winterfeldt, D., 6 Edwards, W. (1986). Decision analysis and behavioral research. New York:Cambridge University.

Weber, E. U. (1988). A descriptive measure of risk. Acta Psychologica, 69(2), 185203.

Weber, E. U., 6 Bottom, W. P. (1989). Axiomatic measures of perceived risk: Some tests and extensions.Journal of Behavioral Decision Making, 2, 113132.

Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55, 95115.

Received: November 19, 1998


Lopes 1999 Journal of Mathematical Psychology

Documents

Transcript of Lopes 1999 Journal of Mathematical Psychology