The Role of Aspiration Level in Risky Choice - Computer ...

The Role of Aspiration Level in Risky Choice: A Comparisonof Cumulative Prospect Theory and SP/A Theory

Lola L. LopesGregg C. Oden

University of Iowa

Abstract

In recent years, descriptive models of risky choicehave incorporated features that reflect the importanceof particular outcome values in choice. Cumulativeprospect theory (CPT) does this by inserting areference point in the utility function. SP/A(security-potential/aspiration) theory uses aspirationlevel as a second criterion in the choice process.Experiment 1 compares the ability of the CPT and SP/Amodels to account for the same within-subjects dataset and finds in favor of SP/A. Experiment 2replicates the main finding of Experiment 1 in abetween-subjects design. The final discussion bracketsthe SP/A result by showing the impact on fit of bothdecreasing and increasing the number of freeparameters. We also suggest how the SP/A approachmight be useful in modeling investment decision makingin a descriptively more valid way and conclude withcomments on the relation between descriptive andnormative theories of risky choice.

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Formal models of decision making under risk can befound in three disciplinary guises. Until quiterecently, almost all economists believed that decisionmakers both should and do select risks that maximizeexpected utility. In contrast, investmentprofessionals have seen investors as selectingportfolios that achieve an optimal balance betweenrisk and return. Psychologists, too, have exploredexpected utility theory and portfolio theory aspossible descriptive models, and they have alsodeveloped original information processing modelsfocused on how people choose rather than what peoplechoose.

For the most part, these three disciplines havebeen isolated from one another, although psychologistshave been more eclectic in their outlooks than others.Recently, however, both psychologists and economistshave been exploring a nonlinear modification of theexpected utility model that we term the“decumulatively weighted utility” model. At present,this model has not affected practice in finance, butwe believe the model can also be applied at the levelof the individual investor.

In what follows, we first describe decumulativelyweighted utility generically and then present twospecific psychological instantiations of the model,cumulative prospect theory, (Tversky & Kahneman,1992), and SP/A theory (Lopes, 1987; 1990; 1995).Second, we test the ability of these two theories toaccount for the same set of data. Third, we suggesthow these results might apply in the investmentcontext. Finally, we comment on fitting and testingcomplex, nonlinear models.

Decumulatively Weighted UtilityIn a nutshell, the expected utility model asserts

that when people choose between alternativeprobability distributions over outcomes (i.e.,lotteries or gambles), they should (the economicmodel) and do (the psychological model) make theirchoices so as to maximize a probability weighted(i.e., linear) average of the outcome utilities.Although much evidence supports the idea thataveraging rules can model people’s choices and

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judgments reasonably well in a variety of tasks(Anderson, 1981), the linearity assumption of expectedutility theory has not fared so well. Less than adecade after the publication of von Neumann andMorgenstern’s (1947) axiomatization of expectedutility theory, Allais (1952/1979) demonstrated thatlinearity failed in qualitative tests comparingchoices in which “certainty” was an option to choicesin which it was not.

For almost three decades afterwards, economistsignored these failures of linearity whereaspsychologists assumed that they represented subjectfailures rather than model failures. In the 1980s,however, some economists began to take the failuresseriously and to seek out variants of expected utilitythat could better account for behavior. An obviouscandidate at the time was prospect theory (Kahneman &Tversky, 1979), but this model violated stochasticdominance, “an assumption that many theorists [were]reluctant to give up” (Tversky & Kahneman, 1992, p.299). Instead, these economists began to explore thenormatively more acceptable idea of decumulativelyweighted utility. The first economic applications wereproposed independently by Quiggen (1982) Allais(1986), and Yaari (1987), followed quickly by manyothers (Chew, Karni & Safra, 1987; Luce, 1988;Schmeidler, 1989; Segal, 1989).

The best way to understand decumulative weightedutility is to start with the structure of the expectedvalue model:

EV = pi viΣi = 1

n, Eq. 1

in which the v i are the n possible outcomes listed inno particular order and the pi are the outcomes’associated probabilities. Expected utility theorysimply substitutes utility, u(v) , for monetaryoutcomes:

EU = pi u viΣi = 1

n. Eq. 2

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

WU = w pi u viΣi = 1

n, Eq. 3

but this is the move that leads to violations ofstochastic dominance.

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Decumulative weighted utility models recast theissue of transforming raw probabilities to one oftransforming decumulative probabilities:

DWU = h pjΣ

j = i

nu vi – u vi – 1Σ

i = 1

nEq. 4

= h(Di) u(vi) – u(vi – 1)Σi = 1

n.

In such models, the v i are ordered from lowest (worst

outcome) to highest (best outcome). Di = pjΣj = i

n is the

decumulative probability associated with outcome v i ;that is, Di is the probability of obtaining an outcomeat least as high as outcome v i . Thus, D1 (thedecumulative probability of the worst outcome, v 1) is 1(you get at least that for sure) and Dn+1 (thedecumulative probability of exceeding the best outcome,v n) is zero.

The function, h, maps decumulative probabilitiesonto the range (0,1) and so preserves dominance. Itcan also provide an alternative to using curvature inthe utility function to model risk attitudes. Forexample, in expected utility theory, if u(v) is aconcave function of v , decision makers will prefersure things to actuarially equivalent lotteries, apattern termed “risk aversion.” Decumulative weightingcan predict the same pattern even while assuming thatu(v) = v (a variant of decumulatively weightedutility that we call decumulative weighted value) byletting h(D) be a convex function of D. Although thepredicted 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 analoguesin the decumulative weighted value (or utility) model.These are given in Table 1. Both models canaccommodate risk neutrality (expected value maximizingbehavior) and both can accommodate the rejection ofsure things in favor of actuarially equivalentlotteries (termed “risk seeking” in expected utilitytheory and “potential-mindedness” by us). Both modelscan also predict the “cautiously hopeful” (our term)pattern first described by Markowitz (1959) in whichsubjects buy both insurance and lottery tickets,

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thereby paying premiums sometimes to gamble and othertimes to avoid gambling.

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Cumulative Prospect TheoryIn cumulative prospect theory (CPT), Tversky and

Kahneman (1992) reformulated the original prospecttheory in terms of (de)cumulative weighted utility.The utility function, u(v) , was unchanged from theoriginal, being concave (risk averse) for gains andconvex (risk seeking) for losses, with the lossfunction assumed to be steeper than the gain function( λ > 1):

u(v) = {vα if v ≥ 0

– λ( – v)β if v < 0 . Eq. 5

The decumulative weighting function was also takento differ for gains and losses. For gains, thehypothesized function has an inverse S-shape thatreinforces risk aversion for most lottery types buttends toward risk seeking for long shots (lotteriesthat have small probabilities of large prizes):

w+(D) = Dγ

Dγ + (1 – D)γ 1 γ1 γ. Eq. 6

For losses, the weighting function is cumulativerather than decumulative and S-shaped rather thaninverse S-shaped. It reinforces risk seeking for mostlotteries but tends toward risk aversion for longshots (lotteries that have small probabilities of verylarge losses):

w–(P) = Pδ

Pδ + (1 – P)δ1 δ1 δ

. Eq. 7

Thus, the utility functions and (de)cumulativeweighting functions of CPT are largely (but notperfectly) mirror-imaged from gains to losses,producing what Tversky and Kahneman (1992) term “afour-fold pattern” in the predicted pattern of lotterypreferences.

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There are two general features of CPT that shouldalso be noted here. The first is that CPT is based ona psychophysical principle of diminishing sensitivityfrom a reference point. In the case of utility, thereference point is usually assumed to be zero (thestatus quo). In the case of (de)cumulative weights,there are reference points at 0 and at 1. In bothcases, the rate of change in the perceived magnitude(of value or of likelihood) is assumed to be greatestnear the reference point and to diminish as one movesaway.

The second feature of CPT is that it, like all itspredecessors in the weighted value family, is a one-criterion model. Although there is much room in themodel for psychological variables to operate, in theend, all these factors are melded into a singleassessment of lottery attractiveness.

SP/A TheorySP/A theory (Lopes, 1987; 1990; 1995) is a dual

criterion model in which the process of choosingbetween lotteries entails integrating two logicallyand psychologically separate criteria:

SP/A = f[SP, A], Eq. 8where SP stands for a security-potential criterion andA for an aspiration criterion.

The SP (security-potential) criterion is modeled bya decumulatively weighted value rule (i.e., the modelis identical to Equation 4 except that the utilityfunction is assumed to be linear 1) :

SP = h(Di) vi – vi – 1Σi = 1

n. Eq. 9

The decumulative weighting function, h(D) , has theform:

h(D) = wDqS + 1 + 1 – w 1 – 1 – D

qP + 1Eq. 10

for both gains and losses. The equation is derived fromthe idea that subjects assess lotteries from the bottomup (a security-minded analysis), or the top down (apotential-minded analysis), or both (a cautiouslyhopeful analysis) 2. The parameters qs and qp representthe rates at which attention to outcomes diminishes asthe evaluation process proceeds up or down. Theparameter, w, determines the relative weight of the S

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and P analyses. If w = 1, the decision maker isstrictly security-minded. If w = 0, the decision makeris strictly potential-minded. If 0< w<1, the decisionmaker is cautiously hopeful, with the degrees of cautionand of hope depending on the relative magnitudes of wand 1– w. Although Equation 10 omits subscripts onparameters for notational simplicity, SP/A theoryfollows CPT in allowing qs , q p, and w to assumedifferent values for gains and for losses, moderatingthe relative importance of security and potential in theoverall SP assessment. Unlike CPT, however, thedecumulative weighting function of SP/A theory does notswitch between inverse S-shaped for gains and S-shapedfor losses.

The A (aspiration) criterion operates on a principleof stochastic control (Dubins & Savage, 1976) in whichsubjects are assumed to assess the attractiveness oflotteries by the probability that a given lottery willyield an outcome at or above the aspiration level, α :

A = p(v > α) Eq. 11For present purposes, we treat the aspiration level ascrisp, which is to say, a discrete value that eitheris or is not satisfied. In principle, however, theaspiration level may be fuzzy: some outcomes maysatisfy the aspiration level completely, others to apartial degree, and still others not at all. To modelthis, Equation 11 would need to incorporate aparticular probability, pi , according to the degreethat its associated outcome, v i , satisfies theaspiration level (Oden & Lopes, 1997).

SP/A theory and CPT share some significantpsychological features: they both model the process bywhich subjects integrate probabilities and values by a(de)cumulative weighting rule, and they both include apoint on the value dimension that has specialsignificance to subjects (the reference point for CPTand the aspiration level for SP/A). Indeed, theaspiration level may be considered to be a kind ofreference point. However, the theories differ three waysin how these features function.

The first difference is in how the reference point(or aspiration level) exerts its impact. In CPT, thereference point is incorporated into the utilityfunction and influences subjects by marking aninflection point about which outcomes are first

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organized into gains and losses, and then scalednonlinearly in accord with a principle of diminishingsensitivity. In SP/A theory, the aspiration levelparticipates in a direct assessment of lotteryattractiveness reflecting a principle of stochasticcontrol and separate from the decumulatively weighted SPassessment. Because SP and A embody different criteria,each may favor a different lottery. When this happens,SP/A theory predicts conflict, a prediction that doesnot follow from single-criterion models such as CPT.

The second difference is that CPT predicts a four-fold pattern across gain preferences and losspreferences. Although some small imperfections in thesymmetry of the pattern might obtain due to smalldifferences in the value and weighting functions forgains and losses, the overall pattern should be one ofreflection between gains and losses. SP/A theory, incontrast, allows considerable asymmetry between gainsand losses. In the most commonly observed case, subjectsappear to avoid risks strongly for gains but to bemore-or-less risk neutral for losses (Cohen, Jaffray, &Said, 1987; Hershey & Schoemaker, 1980; Schneider &Lopes, 1986; Weber & Bottom, 1989). Protocols suggestthat this is because security-minded or cautiouslyhopeful subjects set modest aspiration levels for gains,allowing the SP and A criteria to reinforce one another.For losses, however, the same subjects set highaspiration levels, hoping to lose little or nothing, andthereby setting up a conflict between the A and the SPcriteria (Lopes, 1995).

The third difference is that SP/A theory predictsnonmonotonicities in preference patterns that depend onwhether or not the aspiration level is guaranteed to bemet (by boosting all outcomes above the aspirationlevel) or guaranteed not to be met (by pushing alloutcomes below the aspiration level) no matter whichlottery of a pair is chosen. For example, consider acautiously hopeful decision maker choosing between $50for sure versus a 50/50 chance of $100, else nothing.Suppose also that the decision maker wants to win “atleast something.” Although the SP assessment could favorthe long shot mildly, the A assessment would favor thesure thing strongly, leading to a choice of the surething. If $50 were added to all outcomes, however,

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(e.g., $100 for sure versus a 50/50 chance at $150, else$50) the A assessment would “drop out” (since bothoptions satisfy the aspiration level with certainty)leaving the SP assessment to carry the day. CPT, incontrast, is qualitatively unaffected by cases in whichoutcomes are all pushed upward or downward so long as nooutcomes cross the reference point. The experiments thatfollow use this third difference to distinguish betweenthe two theories and test their abilities to account forsubjects’ choices among a set of multioutcome lotteries.

Experiment 1Method

Stimuli and task. Subjects chose betweenactuarially equivalent pairs of five-outcome lotteriescomprising three positive (gain) sets and threenegative (loss) sets. The standard positive lotteriesare shown in Figure 1. The tally marks representlottery tickets yielding the outcomes shown at theleft. Each of these lotteries has 100 tickets and eachhas an expected value of approximately $100. The namesindicated for the lotteries are for exposition onlyand were not used with subjects.

- - - - - - - - - - - - - -Insert Figure 1 about here- - - - - - - - - - - - - -

Scaled positive and shifted positive lotterieswere created by transforming the outcomes in thestandard positive lotteries linearly. (Examples areshown in Figure 2.) To create the shifted lotteries,standard positive outcomes were increased by $50(bringing the expected value of the shifted positivelotteries to $150). To create the scaled positivelotteries, standard positive outcomes were multipliedby 1.145 (bringing the expected value of the scaledlotteries to $114.50). The multiplicative constant waschosen to equate the maximum outcomes ($398) in thescaled and shifted sets.


Standard negative, scaled negative and shiftednegative lotteries were created by appending a minussign to the outcomes in the respective positive sets.

Design and subjects . Lotteries within stimulus set

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were paired in all possible combinations ( 6C2 = 15pairs per set) and pairs were arrayed vertically onsheets of US letter paper. Two replications werecreated for each set differing in the order of thelotteries on the page.

Pairs from the positive sets were randomizedtogether (within replication) with the constraint thatno particular lottery appeared on adjacent pages.Pairs from the negative sets were randomizedsimilarly. Each replication consisted of 45 pairs (3sets x 15 pairs per set).

The subjects for this experiment were 80undergraduate students at the University of Wisconsinwho served for extra credit in introductory psychologycourses.

Procedure. Subjects were run in groups of two tofour. Each subject was given a notebook containingpractice materials and the randomized stimulus pairs.At the beginning of the experiment, subjects wereshown how to interpret the positive lotteries and weretold that the amount of prize money for each of thelotteries in a pair was the same. Then they were toldthat we were interested in their preferences fordistributions (i.e., how the prizes are distributedover tickets) and were given three positive pairs forpractice. Subjects were asked to indicate whether theywould prefer the top lottery or the bottom lottery ifthey were allowed to draw a ticket from either forfree and keep the prize.

Next, subjects were shown exemplars of negativelotteries and were told that these represented losses.They were then asked to indicate for a set of threemore practice pairs which of each pair they wouldprefer if they were forced to draw a ticket from oneor the other and pay the loss out of their ownpockets.

Stimulus notebooks were divided into five sections,the first containing the practice pairs and theremaining four containing the four sets of stimuluspairs (two positive replications and two negativereplications). Positive and negative replications werealternated with half of the subjects beginning with apositive replication and the other half beginning witha negative replication. Each set was preceded by a

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colored sheet announcing that “The next set oflotteries are all win [or loss] lotteries”. Subjectswent through the notebooks at their own pace,indicating preferences for the top or bottom lotteryby circling “T” or “B” on a separate answer sheet. Thetask took about an hour for most subjects. All choiceswere hypothetical.

Results and DiscussionThe data from Experiment 1 are shown in Figure 3

for gains (left panel) and for losses (right panel).Lotteries are listed along the abscissas in the orderof subject preferences for the standard lotteries. Thedata have been pooled over subjects, replications, andstimulus pair. Each data point represents theproportion of times the average subject chose thelottery out of the total number of times that thelottery was available for choice. Each lottery waspresented 10 times (5 pairs x 2 replications) so thatthe maximum number of choices per subject for a givenlottery was 10 and the minimum was zero.


The data reveal four patterns that are of specialsignificance. (1) For both gains and losses, there areobvious main effects for lotteries [F(5,395) = 93.08and 25.00, respectively, p < .0001 for both] as wellas interactions between lottery and condition[F(10,790) = 48.99 and 8.09, respectively, p < .0001for both]; (2) For both gains and losses, the data forstandard and scaled stimuli are virtually identical[F(5,395) = 1.925, p = .09 and F(5,395) = 0.602, p =.69, respectively]; (3) The slopes of the preferencefunctions for the standard and scaled stimuli aresteeper for gains than for losses; and (4) thepreference functions for the shifted stimuli are non-monotonically related to the preference functions forstandard and scaled stimuli, especially for gains, andthe pattern of non-monotonicity reverses between gainsand losses. Preference for lower risk lotteriesdecreases for gains and increases for losses whereaspreference for higher risk lotteries increases forgains and decreases for losses. As will be seen, thesedifferences between preference functions for shifted

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lotteries and preference functions for standard andscaled lotteries are critical to disentangling theroles of decumulative (or cumulative) weighting andaspiration level in risky choice.

In what follows, we use the Solver function ofMicrosoft Excel) to fit both CPT (Tversky & Kahneman,1992) and SP/A theory (Lopes, 1990; Oden & Lopes,1997) to the data. Solver is an iterative curvefitting procedure that adjusts free parameters tooptimize the fit of a model to a data set according towhatever criterion the user specifies. We used root-mean-squared-deviation (RMSD) between predicted andobtained. In order to lessen the possibility offinding only a local minimum, good practice requiresstarting with one’s best guesses of parameter valuesand then, once a minimum is found, checking the fit bysystematically altering parameter values and rerunningthe program to make sure that a better fit cannot befound. The values we report are the best that we couldfind.

For both CPT and SP/A, we fit the models to theaggregate choice proportions (given in Table 2) forthe two-alternative choice task that subjectsperformed and then pooled the predictions acrosschoice pairs to obtain means (as are shown for theobtained data in Figure 3). Although we had too fewreplications to fit single subject data, visualinspection of single subject means revealed that thepatterns of primary interest were evident at thesingle subject level, being especially clear forstrongly security-minded subjects and somewhatattenuated for subjects whose preferences tendedtoward cautious hopefulness or potential-mindedness.


Fitting CPT. As noted previously, CPT is a one-criterion model in which both values and probabilitiesare transformed psychologically during the lotteryevaluation process. The utility function (Equation 5)has three parameters: α defines the curvature of thefunction for gains (or values above the referencepoint); β defines the curvature of the function forlosses (or values below the reference point); and λdefines the relative slope of the two functions, with

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the loss function specified to be steeper than thegain function ( λ > 1).

Weights in CPT also are defined separately forgains and for losses, as shown in Equations 6 and 7.For gains, the function is decumulative with aparameter, γ, regulating both the curvature andcrossover point of the inverse S-shaped weightingfunction. For losses, the function is cumulative withan analogous parameter, δ, regulating curvature andcrossover points.

CPT was fit simultaneously to the data for thethree scaling conditions (standard, scaled, andshifted) and for both outcome types (gains andlosses), estimating a single set of six parametervalues. The fitting process can best be understood byreference to Table 3. Matrix A shows the pair choicedata for the scaled positive pairs. Each entry is theproportion of times that subjects preferred the columnlottery to the row lottery. Lotteries are orderedacross the columns in descending order of preferenceand down the rows in ascending order of preference.Complementary pairs of entries sum to 1.00, e.g.,entries (1,1) and (6,6) in which riskless (RL) andlong shot (LS) lotteries are opposed. The value .500is entered in the minor diagonal where opposinglotteries are identical. These pairs were not includedin the stimulus set for obvious reasons.


Matrix B gives the best fitting predictions of CPTfor scaled positive pairs obtained iteratively byminimizing the root mean square deviation betweenobtained and predicted choice proportions. The utilityand weighting functions of CPT were fit using the fivevalue and weight parameters described above toestimate CPT attractiveness 3 values for the sixvarious lotteries. These values are shown as thecolumn and row headings in Matrix B.

The second and final step was to use the CPTattractiveness values as input to a pair-choiceprocess that we modeled using the logistic function ofCPT difference scores shown below 4:

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p(CPT 1 > 2) = 11 + e– k (CPT 1 – CPT 2 )

Eq. 12

The function predicts the proportion of times thatlottery 1 is preferred to lottery 2 based on the twoindividual attractiveness values. The function has asingle parameter, k , that is inversely related to thevariance of the distribution of difference scores,CPT1 - CPT 2. Although it might seem reasonable toallow CPT to fit separate k parameters for gains andfor losses, the second k would be redundant with λ andwould, in the present case, allow λ to fall below 1(see Footnote 5.)

Figure 4 shows the best-fitting predictions of CPTto the data for gains (left panel) and for losses(right panel). Parameter values are in Table 2.Although the RMSD of .0810 is respectable for fitting90 data points with six parameters, a comparison ofpredictions and data (Figure 3) reveals a qualitativediscrepancy for the shifted gain lotteries. AlthoughCPT is able to capture the general flattening of thispreference function relative to the standard andscaled functions, all but the prediction for the longshot are monotonically decreasing. In contrast,subjects’ preferences for the shifted rectangular,bimodal, and long shot lotteries were all greater thanfor the shifted short shot. Moreover, thenonmonotonicity that is induced for the shifted longshot comes at the expense of incorrectly predictingnonmonotonicity for the standard and scaled long shotsas well.


A second issue concerns the CPT parameter values(see Table 3). Beginning with the parameters for theutility functions, note that although the function forgains is sharply curved ( α = .426), the function forlosses is close to linear ( β = .942). Second, lookingat the probability weights, note that the function forgains shows considerable nonlinearity ( γ = .685)whereas the function for losses is again essentiallylinear ( δ = .980). Finally, looking at λ , theparameter that determines the relative slopes of theutility functions for gains and losses, note that it

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has reached its floor value of 1.00 5. Although thevalues of these parameters are consistent with theobserved fact that the loss data are essentiallylinear and relatively shallow in slope, there isnothing in CPT that would lead one to expect thislarge asymmetry between gains and losses. Indeed, CPTis well-known for its prediction of reflection inpreferences between gains and losses (i.e., the four-fold pattern).


In all, then, CPT does reasonably well in fittingthe data if one considers only RMSD. When one looks atqualitative effects, however, CPT fails with theshifted gains. Moreover, in order to get a reasonablefit, CPT must make use of parameter values that areinconsistent with the underlying psychophysicalprinciple (diminishing sensitivity from a referencepoint) on which both utility and weighting functionsare theorized to depend.

Fitting SP/A theory. As explained previously, SP/Atheory proposes that risky choice involves twocriteria, one based on a comparison of decumulativelyweighted averages of probabilities and outcomes (theSP criterion) and the other based on a comparison ofprobabilities of achieving an aspiration level (the Acriterion). The SP assessment process (Equation 4) hasthree parameters: q defines the degree of attentionto different outcomes in assessments of security( qs )and potential ( qp), whereas w defines therelative importance of security and potentialassessments overall. In principle, all threeparameters might differ between gains and losses. TheA assessment process (Equation 5) has a singleparameter, α , the aspiration level, which can alsodiffer for gains ( α+) and losses ( α - ). Although αmight need to be fit as a free parameter in some cases(e.g., with continuous outcome distributions or withmanipulated aspiration levels), our stimuli and choiceconditions justified fixing α+ at 1 and α - at zero 6.


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In fitting SP/A theory to the data, we modeled theSP criterion and the A criterion separately (but notsequentially) and integrated the results into a finalchoice. The procedure is schematized in Table 5.Matrix A gives the scaled positive data and Matrix Bgives the best-fitting predictions based on just theSP criterion. The column and row headings areestimated SP attractiveness values. We let w differbetween gains ( w+) and losses ( w- ) but, in order tohold our parameters to six, set qs = qp = q and used

the same single value for both gains and losses . Theentries in the cells are choice proportions, p(SP 1>2),obtained by applying the logistic function todifference scores just as we did in fitting CPT:

p(SP 1 > 2) = 11 + e–k(SP1 – SP2)

. Eq. 13

The parameter, k , is inversely related to thevariance of the distribution of difference scores, SP 1- SP 2

Matrix C shows best-fitting predictions based onjust the A criterion. The row and column headings showthe probability that the row (or column) lottery willyield a value that meets the aspiration level (e.g.,the riskless gain lottery, column 1, guarantees anonzero payoff whereas the peaked gain lottery, column2, has only a .96 probability of a nonzero payoff).The table entries are choice proportions, p(A 1>2) ,

obtained by submitting the A values to a relativeratio process having a parameter, t , 0 < t thatcontrols contrast. Equation 14 shows the process forgains:

p(A 1 > 2) =

A1t+

A1t+ + A2

t+

. Eq. 14

The equation for losses looks a little different but hasthe same structure:

p(A 1 > 2) = 1 –

1 – A1t–

1 – A1t–

+ 1 – A2t– Eq. 15

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=1 – A2

t–

1 – A1t–

+ 1 – A2t– .

The difference between gains and losses reflects thefact that gains engender an approach/approach processbased on relative lottery goodness, A, whereas lossesengender an avoidance/avoidance process based onrelative lottery badness, 1-A . In other words, theprobability of choosing Lottery 1 over Lottery 2 forgains reflects the degree to which Lottery 1 is betterthan Lottery 2, whereas the probability of choosingLottery 1 over Lottery 2 for losses reflects thedegree to which Lottery 2 is worse than Lottery 1.

Matrix D combines the p(SP 1>2) values and the

p(A 1>2) values according to: p SP/A1 > 2 = Eq. 16

p(SP1 > 2)p(A1 > 2)

1/2

p(SP1 > 2)p(A1 > 2)1/2

+ 1 – p(SP1 > 2) 1 – p(A1 > 2)1/2 .

This rule (which is useful for cases in which both thedomain and the range of the function are 0 to 1)displays both averaging properties [p(SP/A 1>2) lies

between p(SP 1>2) and p(A 1>2) ] and Bayesian properties

(the impact of an input value depends on itsextremity). The exponent, 1/2, sets the weight of thetwo input quantities to be equal. Although one mightimagine that this could be a free parameter in themodel, our experience suggests that it shares variancewith the SP and A parameters, muddying the fittingprocess when it is included.

Figure 5 shows the best fitting predictions of SP/Atheory to the data for gains (left panel) and forlosses (right panel). The RMSD of predicted toobtained is .0681. (relative to .0810 for CPT). As canbe seen by inspection of the figures, SP/A has alsodone a better job of capturing the qualitativefeatures of the data. In particular, SP/A predicts thenonmonotonic increases in preference for the shiftedrectangular, bimodal, and long shot gain lotteries.

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The parameter values for the SP/A model are inTable 6. Although the values are generally reasonable,the values for the SP component are not far from anexpected value fit ( w = .50, qs = qp = 1). We

believe this reflects the simplifying constraints thatwe imposed on the parameters of the SP criterion. Weshall have more to say about the matter in the finaldiscussion.


Experiment 2It is sometimes thought that the opportunities for

comparison offered by repeated measures designs createchoice patterns that might not occur if subjects madeonly a single choice. The purpose of Experiment 2 wasto replicate the main finding of Experiment 1 (i.e.,that shifted lotteries are evaluated differently thanstandard or scaled lotteries) using a between subjectsdesign. We also wanted to collect subjects’ reasonsfor their choices. Because between-subject experimentsare costly in terms of the required number ofsubjects, we used only positive lotteries.Method

The stimuli were the 45 pairs comprising thestandard, scaled, and shifted positive sets. Each pairwas printed separately on a single sheet of US letterpaper. Sheets were randomized and distributed at thebeginning of experimental sessions to subjects whowere participating in other related experiments. In agiven session, different subjects had different pairs.Consequently, it was necessary to describe how tointerpret the lotteries in very general terms, nevermentioning particular outcomes or numbers of tickets.

As in Experiment 1, subjects were asked to markwhich of the two lotteries they would prefer if theywere allowed to draw a ticket from either for free andkeep the prize for themselves. They were also asked towrite a sentence or two explaining the basis for theirpreference.

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A total of 433 subjects from the University ofWisconsin—Madison and the University of Iowaparticipated in the experiment for extra coursecredit. Six of these subjects indicated by theirwritten responses that they had not understood how tointerpret the lotteries, leaving 427 usable responsesranging over the 45 choice pairs.

Results and DiscussionThe data from Experiment 2 are shown in Figure 6.

Clearly, the means are noisier than the means fromExperiment 1, a result one might anticipate not onlyfrom the different subjects contributing to each datapoint, but also from the reduced amount of data goinginto each data point (400 choices per data point inthe within-subject case versus about 24 choices perdata point in the between subject case). Despite thenoise, however, a chi square analysis shows that thekey differences between the three scaling conditionswere replicated. Specifically, (1) the patterns ofpreferences for standard stimuli and scaled stimuli donot differ significantly from one another, χ2(1) =1.65, p > .05; whereas (2) the pattern of preferencesfor shifted stimuli differs significantly from theoverall pattern of preference for standard and scaledstimuli, χ2(1) = 25.13, p < .001.


Table 7 illustrates the reasons that subjects gavefor their choices, taking the long shot (LS) and shortshot (SS) lotteries as examples. (These are thelotteries that are shown in Figure 2.) Protocols forthe scaled condition are on the left and for theshifted condition are on the right. In each set, thefirst four subjects (plain text) chose with themajority whereas the last two subjects (italic) chosewith the minority.


In the scaled condition, the majority of subjects(5 of 8) preferred the short shot to the long shot.Protocols 1 through 4 show clearly that such subjects

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are concerned with achieving a nonzero outcome. Interms of SP/A theory, the A criterion seems to beoutweighing the SP criterion. Neither of the remainingtwo subjects seems particularly concerned withavoiding zero (the A criterion). Consequently, theextra high outcomes in the long shot have more impact(the SP criterion).

In the shifted condition, the majority of subjects(7 of 10) preferred the long shot to the short shot.In particular, protocols 7, 8, and 10 convey the sensethat the $50 guaranteed outcome is good enough (the Acriterion is fully satisfied for both lotteries)allowing the subjects to choose the slightly riskierlong shot (the SP criterion). In contrast, protocol 11suggests that the subject has adjusted his or heraspiration level upward, so that $50 is now equivalentto “not winning” (the A criterion) whereas protocol 12reveals a subject who focused on the relativemagnitudes of high and low outcomes, but placed moreweight on the low outcomes (the SP criterion).

In sum, then, Experiment 2 confirms that preferencepatterns differ for shifted lotteries and for standardor scaled lotteries even when the data are gathered ina between-subjects design. Moreover, the protocolsshow, as SP/A theory predicts, that this resultreflects differences in the relative impacts of SP andA criteria under the shifted and scaled or standardconditions. In the shifted condition, all lotteriessatisfy the A criterion for gains and none satisfy itfor losses, allowing the SP criterion to manifestitself more strongly in either case. In the scaled andstandard conditions, however, there are largedifferences in the degree to which the A criterion issatisfied, reducing the importance of the SP criterionoverall.

DiscussionThe model comparison in Experiment 1 showed that, on

six parameters, SP/A does a better job than CPT offitting the present set of choice data. Not only isthe RMSD for SP/A 16% smaller, the model also captures(as CPT does not) the nonmonotonic relation betweenpreferences for shifted lotteries and preferences forstandard and scaled lotteries. Experiment 2 reinforcedthis finding by replicating the critical

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nonmonotonicity for gain lotteries in a betweensubjects design. It also provided protocols confirmingthat the nonmonotonicity may arise because adding $50to standard positive lotteries eliminates aspirationlevel as a consideration for most subjects, thusenhancing the impact of the decumulatively weighted SPcriterion. In what follows, we discuss theimplications of this result for modeling investmentrisk. We also provide comments on fitting complexmodels along with two instructive comparisons to thesix-parameter SP/A model. We end with a discussion ofthe relation between descriptive and normativetheories.

Risk Taking and Aspiration Level It has often been pointed out that when people are

in economic difficulty, they tend to take risks thatthey would avoid under better circumstances. Thistendency appears among sophisticated managers introubled firms (Bowman, 1980; 1982) as well as amongunsophisticated subsistence farmers (Kunreuther &Wright, 1979). Experimental studies using managers assubjects have also confirmed the tendency towardrisk-taking for losses, at least when ruin is not atissue (Laughhunn, Payne & Crum, 1980; Payne, Laughhunn& Crum, 1981). Standard thinking in investment theorywould not lead one to expect risk taking inthreatening situations. Instead, hard-pressed decisionmakers should value low risk over high expected returnand choose accordingly.

The S-shaped utility function of prospect theoryseems to provide an explanation for this paradoxicalrisk-taking: people take risks when they face lossesbecause their utility function for losses is “riskseeking” (i.e., convex). Though one can criticize thecircularity of the “explanation,” it at least predictspreferences better than the more standard assumptionof uniform “risk aversion” (i.e., diminishing marginalutility). But predicting preferences is only half thestory, especially when predictions fail, as they oftendo in experimental studies of preferences for losseswith students (Cohen, Jaffray & Said, 1987; Hershey &Schoemaker, 1980; Schneider & Lopes, 1986; Weber &Bottom, 1989) as well as with managers (MacCrimmon &Wehrung, 1986). The other half of the story can be

21

found in protocol data. Studies by Mao (1970) and byPetty and Scott (cited in Payne, Laughhunn & Crum,1980) suggest that managers tend to define investmentrisk as the probability of not achieving a target rateof return (that is to say, an aspiration level).

No one can doubt that expected return (i.e.,expected or mean value) is a central and well-understood concept for managers, but the concept ofrisk is less well understood. In portfolio theory, forexample, risk is usually equated with outcome variance(Markowitz, 1959) but this is not entirelysatisfactory descriptively since it treats wins andlosses alike. Other approaches to defining risk try tobypass this objection by restricting the variancecomputation to losses (i.e., the semivariance) or bycomputing risk as a probability weighted average ofdeviations below a target level (Fishburn, 1977). Few,however, have explored the possibility of modelingrisk as the raw probability of not achieving anaspiration level. One who has is Manski (1988) whodeveloped the idea in what he called a utility massmodel. Another approach that incorporates rawprobabilities comes from Weber (1988) who augmented anexpectation model with weighted probabilities ofwinning, losing, and breaking even.

SP/A theory incorporates both notions, each in aseparate criterion. On the SP side, a security-mindedweighting function (or a cautiously hopeful functiondisplaying more caution than hope) pays more attentionto the worst outcomes than to better outcomes. On theA side, the model operates on the probability ofachieving the aspiration level. Although normativemodels usually focus on a single criterion,descriptive models must go where subjects lead. In thecase at hand, the subjects seem to be saying that theyunderstand and use the term “risk” in bothdistributional and aspirational senses. For example,in Table 6, two subjects refer explicitly to risk. InProtocol 8, one subject uses the concept in the Acriterion sense: risk is the chance of winning less.In Protocol 10, however, another subject does notcount chances, but rather focuses on differences inprize amounts, an SP-focused analysis.

Practitioners work at the boundary between normative

22

and descriptive. Clients expect guidance (thenormative function) in how to achieve their personalgoals (the descriptive function). For the client whois concerned about not meeting a target return, thereseems little point in discussing variance. It wouldseem better for the professional to recognize in aclient’s spoken desires the relevance of thosemathematical rules that seem most applicable and,then, to explain in a simple fashion, properties ofthe rules that may not be self-evident.

Although much has been claimed since von Neumann andMorgenstern (1947) about the dire consequences ofviolating linearity, recent examination of alternativerules based on decumulative weighted utility andaspiration criteria (e.g., Manski, 1988; March, 1996;Yaari, 1987) suggests that these alternatives areneither better nor worse than maximizing expectedutility. They are, however, different and seem to comecloser to doing what people want done.

Pushing the Model TestsIn Experiment 1, we fit the CPT and SP/A models on

the same number of free parameters even though SP/Atheory could reasonably use several more. In order tobetter illuminate the roles of the variouspsychological components of the theory, we now bracketthe six-parameter SP/A fit by comparing it to a zero-parameter fit and a ten-parameter fit.

The top panel of Figure 7 shows what happens whenthe SP criterion of SP/A theory is neutralized bysetting its parameters to yield the expected value forall lotteries ( w = .50, qs = qp = 1.00), thus

allowing the A criterion to dominate. We set theaspiration level here as we did previously: α gains >0; α losses = 0). The choice rule is also zero-parameter, assuming that subjects choose whicheverlottery has the higher value on the A criterion and isindifferent if lotteries tie on aspiration level.


The A criterion alone produces a reasonablequantitative fit, with an RMSD of .1206. Although itmay seem surprising that such a simple mechanism doesso well in fitting complex data given the very

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complicated models that have been favored for riskychoice recently (including both CPT and SP/A), theresult maps well onto the classic finding ofinformation processing studies using duplex bets(Payne & Braunstein, 1971; Slovic & Lichtenstein,1968) that probability of winning dominates the choiceprocess. Still, the best that aspiration can do byitself with the shifted data is to fit a flat line.Aspiration alone also predicts a mirror symmetry(reflection in preferences) between gains and losseswhereas the actual loss preference functions are muchflatter than the gain preference functions for allthree lottery types.

The bottom panel of Figure 7 shows what happens witha full, ten-parameter fit of SP/A theory. The fit(RMSD = .0484) is obviously much better than the six-parameter fit. What interests us more, however, isthat removing constraints on the SP parameters revealstheoretically meaningful values (see Table 8). Whereaspreviously the SP criterion came very close to anexpected value criterion, the new parameter valuessuggest important process differences between gainsand losses. For gains, it appears that the bottom-up(security) evaluation is more important than the top-down (potential) evaluation whereas, for losses, thetop-down (potential) evaluation appears more importantthan the bottom-up (security) evaluation. Similarly,differences in the w parameter also suggest that theimportance of security is greater for gains than forlosses (w + > w –). Thus, SP/A parameters confirm theCPT-based intuition that subjects do, indeed, evaluatehigh-risk options more favorably for losses than forgains.

- - - - - - - - - - - - - Insert Table 8 about here- - - - - - - - - - - - -

There is, however, an important difference betweenthe mechanisms used by CPT and SP/A to account forthese differences between gains and losses. CPT’s zeroreference point provides the rationale for qualitativeinversions of its utility function (from concave forgains to convex for losses) and its decumulativeweighting function (from inverse S-shaped for gains toS-shaped for losses). Thus, CPT specifies that the

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value processing mechanisms and probability weightingmechanisms used by subjects differ qualitatively forgains and losses. SP/A, on the other hand, allows therelative attention paid to worst outcomes and bestoutcomes to shift as a function of domain (gainsversus losses) and also allows the relative importanceof the SP and A components to differ between gains andlosses. But domain-mediated reference effects in SP/Atheory are potentially applicable to a broader rangeof domain differences.

For example, Edwards and von Winterfeldt (1986)propose that a person’s risk attitude may be differentin different “transaction streams.” Choices involvingamounts in what they call the “quick cash” and “playmoney” streams (the former being what people haveavailable in their wallets and the latter being moneyreserved for enjoyment) should be less risk averse(i.e., less security-minded) than choices involving“capital assets” and “income and fixed expenditures”streams. Similarly, MacCrimmon and Wehrung (1986)found that executives are more risk averse in makingdecisions about their own personal investments thanthey are about business investments. Shifts of thesesorts in one’s willingness to accept risk need notinvolve gain/loss shifts. Instead, they may involveonly differences in outcome scale (large versus smalltransaction stream) or differences in real-worldexpectations or consequences (personal decisionsversus business decisions). The parameters of SP/Atheory, while restricting the decumulative weightingfunction to an inverse S-shape for both gains andlosses, nevertheless allow for modeling this broaderclass of reference effects through differences in therelative attention paid to bad versus good outcomes inthe SP assessment and through the relative importanceaccorded to SP and A assessments in the final choice.

The Importance of Normative Theory for Description andVice Versa

In most of economics, expected utility theoryremains the workhorse of academic research--which isto say, of normative research--despite its poor fit todata from psychological experiments. Recently,however, a number of economists have turned theirattention to testing expected utility theory in

25

laboratory settings involving stylized economic games.Up to now, this new enterprise has been doggedlyempirical and intently focused on theoreticallyappropriate task instantiation and on experimentalrigor and control. Despite this attention to detail,however, the predictions of the theory have frequentlynot been borne out, leaving the experimentalists withthe not inconsiderable task of persuading theircolleagues that the model’s failures are meaningfuland should not be overlooked (for reviews, see theessays by Smith, 1982; 1989; 1991; and the variouschapters in Kagel & Roth, 1995). There has not,however, been a commensurate effort from theseresearchers to develop better theory, although Roth(1995, p. 18) has pointed out that at least some ofthose who ran the earliest economic experimentsexpected that experimental data would contribute tothe development of both better descriptive theoriesand better normative theories.

Most economists view their discipline as one thatdeals with ideally rational behavior and, thus, attachlittle significance to discrepancies between what thetheory predicts and what people actually do.Psychologists, on the other hand, view their task asone of predicting behavior and describing itscognitive sources in psychologically meaningful terms,whether or not that behavior is rational. The utilityand probability weighting functions of CPT rest onperceptual concepts. The SP and A components of SP/Atheory rest on attentional and motivational concepts.Thus, both theories provide a psychological groundingthat allows each to appeal directly to intuitions viaeasily understood and compellingly named components.

Intuitiveness is not enough, however. Mathematicalanalysis of the sort pursued here is necessary tospecify the quantitative mechanisms from whichtheoretical predictions flow and to confirm that it isthese specific mechanisms that provide the bestaccount of behavior. History shows that it is easy toconflate phenomena with explanations, especially whenthe explanations appeal to intuition. Thus, thephenomenon of risk aversion became conflated with theidea of diminishing marginal utility (concavity)because the intuition was powerful and, indeed, isaccurate, that constant marginal gains or losses in

26

assets are more noticeable to poor people than to richpeople. Conflation of phenomena with explanation isespecially hazardous to theoretical advancement inthat it suppresses interest in psychologicallyimportant alternative explanations, such as theaspiration and decumulatively weighted utilitymechanisms on which this paper has focused.

The model comparisons we presented pitted competingpsychological mechanisms against one another whileconstraining them to the same number of freeparameters. For the data set at hand, SP/A theoryprovided the better fit, both quantitatively andqualitatively. However, a more important comparisonmay reside in the relative strengths and weaknesses ofthe three different parameterizations of SP/A theory.On the one hand, the zero-parameter model, relyingsolely on the aspiration level mechanism, didsurprisingly well in providing a rough fit to thedata. That a mechanism as simple as this wasoverlooked as an alternative to more complicatedaccounts is testimony to the unhealthy power that the“best existing theory” has to stifle research intoalternatives. On the other hand, the 6 and 10parameter versions of SP/A theory show the necessityof the SP component for modeling preferences among theshifted lotteries and for capturing the relativeflattening of preferences for loss lotteries. Althoughthe possible contributions of aspiration level shouldnot have been overlooked, theorists andexperimentalists since Bernoulli have not been foolishin pursuing weighted utility models. Aspiration aloneis simply too simple.

SP/A theory is a descriptive theory, through andthrough. Its dual choice criteria--the security-potential criterion and the aspiration criterion--areboth included because each seems necessary toadequately capture human choices under risk. It isworth noting, however, that even though these twocriteria are inconsistent with expected utilitymaximization except in special cases, the rationalityof each has been defended recently on normativegrounds, [e.g., see, Manski’s (1988) utility massmodel and Yaari’s (1987) decumulatively weighted valuemodel). Although there is still a great divide between

27

normative and descriptive theories of risky choice,perhaps we are seeing the first evidence thatdescriptive research is finally, as Roth (1995, p. 22)put it, “speaking to theorists.”

28

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Footnotes

Requests for reprints should be sent to Lola Lopesat the College of Business Administration, Universityof Iowa, Iowa City, IA 52242 ([email protected]).Deirdre Huckbody ran the subjects and coded the dataas part of an independent study during her senior yearat the University of Wisconsin.

1Most theorists assume that u(v) is nonlinearwithout asking whether the monetary range underconsideration is wide enough for nonlinearity to bemanifest in the data. We believe that u(v) probablydoes have mild concavity that might be manifest insome cases (as, for example, when someone isconsidering the huge payouts in state lotteries). Butfor narrower ranges, we prefer to ignore concavity andlet the decumulative weighting function carry thetheoretical load.

2We do not provide the derivation of the SP/Adecumulative weighting function at this time becauseit is not relevant to the present focus. Interestedreaders can contact us for details.

3We use the term “attractiveness” to refer toindividual lottery values. Others have used“strength-of-preference” to mean the same thing, butwe prefer to distinquish between individual lotteryassessments (attractiveness) and choices (orpreferences) between lotteries.

4Although CPT is intended to be a theory of riskychoice, the process that maps two or more individuallottery assessments onto choice has not beenspecified. The logistic function that we apply hereand below is commonly used as the cumulativeprobability distribution function in statisticaldecision theory models of the two-alternative choiceprocess (Luce & Galanter, 1963) and is presumed to beneutral with respect to its impact on CPT’s and SP/A’sability to fit the qualitative features of the data.

5A somewhat improved RMSD (.0770) resulted when λwas allowed to drop to 0.400 (i.e., for the gainfunction to be considerably steeper than the lossfunction). Although this value is consistent with theobserved fact that the standard and scaled gain data

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are steeper than the standard and scaled loss data,the value is not consistent with CPT’s oft-repeatedclaim that “losses loom larger than gains.”

6Our assumption concerning the values of theaspiration level is analogous to Kahneman andTversky’s assumption (1979; Tversky & Kahneman, 1992)that the reference point of the utility function is atzero. Although we, as they, might sometimes want tomodify this simplifying assumption, in the presentcase the lottery outcomes are spaced widely enoughthat minimum (or maximum) outcomes are goodapproximations for what might be “at least a smallgain” or “no more than a small loss” in choicesbetween more continuous lotteries.

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Figure Captions

Figure 1. Standard positive stimulus set. The tallymarks represent lottery tickets yielding the outcomesshown at the left. Each lottery has 100 tickets and anexpected value of approximately $100.

Figure 2. Examples of stimuli from the scaledpositive stimulus set and the shifted positivestimulus set. Scaled stimuli are produced fromstandard stimuli by multiplying outcomes by 1.145.Shifted stimuli are produced from standard stimuli byadding $50 to each outcome.

Figure 3. Data from Experiment 1 pooled oversubjects, replications, and stimulus pair. Lotteriesare listed along the abscissa in order of averagesubject preference for standard lotteries. Data arethe proportion of occasions on which subjects chose agiven lottery out of the 10 occasions on which thatlottery was available for choice.

Figure 4. Predictions of CPT pooled over stimuluspair using the six parameter values shown in Table 4.

Figure 5. Predictions of SP/A theory pooled overstimulus pair using the six parameter values shown inTable 6.

Figure 6. Data from Experiment 2 pooled oversubjects and stimulus pair. Data are the proportion ofsubjects choosing a given lottery out of the totalnumber of subjects who had that lottery available forchoice.

Figure 7. Top panel: SP/A predictions based on theA criterion alone with the SP criterion neutralized.Bottom panel: SP/A predictions based on the ten-parameter values shown in Table 8.

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IIIIII

IIIIII

IIIIII

IIIIII

IIIIII

IIIII

SH

IFT

ED

(S

TA

ND

AR

D +

$50

)

SH

OR

T S

HO

TLO

NG

SH

OT

Ga

in D

ata

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion

Sh Sc St

Loss

Da

ta

.0.2.4.6.81.0

LS

BM

RC

SS

PK

RL

Lott

erie

s

Choice Proportion

Sh St Sc

CPT

Los

s Pr

edic

tion

sC

PT G

ain

Pre

dic

tion

s

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion

Sh St Sc

.0.2.4.6.81.0

LS

BM

RC

SS

PK

RL

Lott

erie

s

Choice Proportion

SP/A

Ga

in P

red

ictio

ns

SP/A

Los

s Pr

edic

tion

s

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion

.0.2.4.6.81.0

LS

BM

RC

SS

PK

RL

Lott

erie

s

Choice Proportion

Betw

een

Su

bje

ct R

eplic

atio

n: G

ain

s

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion

SP/A

Los

s Pr

edic

tion

s (1

0 p

ara

met

e

.0.2.4.6.81.0

LS

BM

RC

SS

PK

RL

Lott

erie

s

Choice Proportion

SP/A

Ga

in P

red

ictio

ns

(10

pa

ram

eter

s

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion

Ga

ins:

Asp

ira

tion

On

ly

.0.2.4.6.81.0

RL

PK

SS

RC

BM

LS

Lott

erie

s

Choice Proportion.0.2.4.6.81.0

LS

BM

RC

SS

PK

RL

Lott

erie

s

Choice Proportion

Loss

es: A

spir

atio

n O

nly

Table 1Risk Attitudes in the

Expected Utility and Decumulative Weighted Value Models

Expected Utility Decumulativ e Weighte d Value Assumes 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-minde d convexRisk seeking convex Potential-minded concaveMarkowitz typ e S-shape d Cautiousl y hopeful invers e S-shaped

43

T

ab

le 2

Ch

oic

e P

rop

ort

ion

sG

ain

s

Lo

sse

sS

tan

da

rd

lott

eri

es

RL

P

K

SS

R

C

BM

L

S

LS

B

M

RC

SS

P

K

RL

LS

.93

1.8

44

.80

0.8

69

.59

4.5

00

R

L.7

81

.75

6.7

31

.60

6.5

56

.50

0B

M.8

75

.81

9.8

50

.80

6.5

00

.40

6

PK

.71

3.7

00

.65

6.5

19

.50

0.4

44

RC

.85

6.7

19

.63

1.5

00

.19

4.1

31

S

S.5

75

.70

6.6

13

.50

0.4

81

.39

4S

S.8

44

.58

8.5

00

.36

9.1

50

.20

0

RC

.60

6.5

88

.50

0.3

88

.34

4.2

69

PK

.71

3.5

00

.41

3.2

81

.18

1.1

56

B

M.5

69

.50

0.4

13

.29

4.3

00

.24

4R

L.5

00

.28

8.1

56

.14

4.1

25

.06

9

LS

.50

0.4

31

.39

4.4

25

.28

8.2

19

Sca

led

lo

tte

rie

s R

L

PK

S

S

RC

B

M

LS

L

S

BM

R

C S

S

PK

R

LL

S.8

56

.84

4.8

13

.80

6.5

63

.50

0

RL

.78

1.7

63

.72

5.6

44

.56

3.5

00

BM

.85

6.7

88

.81

3.7

50

.50

0.4

38

P

K.6

63

.67

5.6

13

.50

0.5

00

.43

8R

C.8

63

.74

4.6

50

.50

0.2

50

.19

4

SS

.67

5.6

31

.57

5.5

00

.50

0.3

56

SS

.84

4.5

31

.50

0.3

50

.18

8.1

88

R

C.6

44

.56

3.5

00

.42

5.3

88

.27

5P

K.7

69

.50

0.4

69

.25

6.2

13

.15

6

BM

.48

8.5

00

.43

8.3

69

.32

5.2

38

RL

.50

0.2

31

.15

6.1

38

.14

4.1

44

L

S.5

00

.51

3.3

56

.32

5.3

38

.21

9S

hif

ted

lo

tte

rie

s R

L

RC

L

S

BM

P

K

SS

RC

L

S

BM

SS

P

K

RL

SS

.78

8.6

25

.60

0.6

25

.51

9.5

00

R

L.7

25

.68

8.6

50

.67

5.6

13

.50

0P

K.7

13

.55

0.5

13

.58

8.5

00

.48

1

PK

.60

6.5

88

.52

5.4

63

.50

0.3

88

BM

.73

1.5

75

.57

5.5

00

.41

3.3

75

S

S.5

44

.55

0.5

38

.50

0.5

38

.32

5L

S.7

88

.51

9.5

00

.42

5.4

88

.40

0

BM

.59

4.4

88

.50

0.4

63

.47

5.3

50

RC

.76

9.5

00

.48

1.4

25

.45

0.3

75

L

S.4

88

.50

0.5

13

.45

0.4

13

.31

3R

L.5

00

.23

1.2

13

.26

9.2

88

.21

3

RC

.50

0.5

13

.40

6.4

56

.39

4.2

75

No

te:

Lo

tte

rie

s w

ith

in e

ach

ma

trix

are

lis

ted

in

de

sce

nd

ing

ord

er

of

pre

fere

nce

acr

oss

th

e c

olu

mn

s a

nd

in

asc

en

din

g o

rde

r o

f p

refe

ren

ce d

ow

nth

e r

ow

s.

44

Table 3Fitting CPT to the Data for Scaled Gain Pairs

Matrix A: Raw choice proportionsRL PK SS RC BM LS

LS .856 .844 .813 .806 .563 .500BM .856 .788 .813 .750 .500 .438RC .863 .744 .650 .500 .250 .194SS .844 .531 .500 .350 .188 .188PK .769 .500 .469 .256 .213 .156RL .500 .231 .156 .138 .144 .144Means 838 .628 .580 .460 .271 .224

Matrix B: Choice predictions based on CPTattractiveness values

13.42 12.11 11.72 10.61 9.91 10.1910.19 .916 .805 .756 .577 .449 .500 9.91 .931 .835 .792 .626 .500 .55110.61 .889 .752 .694 .500 .374 .42311.72 .779 .572 .500 .306 .208 .24412.11 .725 .500 .428 .248 .165 .19513.42 .500 .275 .221 .111 .069 .084Means .848 .648 .578 .374 .253 .299

45

Table 4Parameter Values for CPT

Parameter Valueα 0.551β 0.970λ 1.000γ 0.699δ 0.993k 0.739

46

Table 5Fitting SP/A Theory to the Data for Scaled Gain Pairs

Matrix A: Raw choice proportionsRL PK SS RC BM LS

LS .856 .844 .813 .806 .563 .500BM .856 .788 .813 .750 .500 .438RC .863 .744 .650 .500 .250 .194SS .844 .531 .500 .350 .188 .188PK .769 .500 .469 .256 .213 .156RL .500 .231 .156 .138 .144 .144Means .838 .628 .580 .460 .271 .224

Matrix B: Choice predictions based on SP criterion114.86 114.00 113.66 113.9 6 113.97 113.88

113.88 .839 .537 .382 .510 .509 .500113.91 .834 .528 .374 .501 .500 .491113.91 .833 .527 .372 .500 .499 .490113.60 .894 .652 .500 .628 .626 .618113.98 .818 .500 .348 .473 .472 .463114.86 .500 .182 .106 .167 .166 .161

Matrix C: Choice proportions based on A criterion1.000 0.960 0.960 0.800 0.680 0.620

0.620 .989 .984 .984 .917 .705 .5000.680 .975 .963 .963 .823 .500 .2950.800 .892 .848 .848 .500 .177 .0830.960 .595 .500 .500 .152 .037 .0160.960 .595 .500 .500 .152 .037 .0161.000 .500 .405 .405 .108 .025 .011

Matrix D: Predictions combining SP and A criteriaRL PK SS RC BM LS

LS .956 .895 .861 .773 .612 .500BM .933 .844 .797 .684 .500 .388RC .865 .714 .646 .500 .316 .227SS .779 .578 .500 .354 .203 .139PK .720 .500 .422 .286 .156 .105RL .500 .280 .221 .135 .067 .044Means .850 .662 .590 .446 .271 .181

47

Table 6Parameter Values for SP/A Theory

(six parameter fit)

Parameter Valueq 1.053w+ 0.505w- 0.488k 1.694t + 9.447t - 2.035

48

Ta

ble

7Ill

ust

rativ

e P

roto

cols

fro

m E

xpe

rim

en

t 2

Sca

led L

otte

ries

(LS

vs.

SS

)

S

hift

ed L

otte

ries

(LS

vs.

SS

)

1. H

ave

a g

reate

r ch

ance

of w

innin

g a

t least

som

e m

oney

[in th

e S

S].

The g

reate

st c

hance

is$160. [

In th

e L

S] l

otte

ry, y

our

gre

ate

st c

hance

is z

ero

. (P

icks

SS

.)

7. I

pic

ked [L

S] b

eca

use

win

nin

g a

ny

am

ount o

fm

oney

would

be e

xciti

ng fo

r m

e. I

f I p

icke

d $

50

that w

ould

be g

reat b

ut i

f I p

icke

d a

ny

oth

er

num

ber

it auto

matic

ally

giv

es

me m

ore

money

inco

mparis

on w

ith th

e o

ther

lotte

ry c

hoic

es.

2. T

here

is a

bette

r ch

ance

to w

in m

oney

[in th

eS

S] b

eca

use

there

are

more

tally

mark

s fo

r th

ehig

her

valu

e o

f money

than th

ere

is fo

r $0.

(Pic

ks S

S)

8. T

he a

mount o

f money

to b

e w

on [i

n L

S] i

sgre

ate

r. E

ven th

ough th

e c

hance

s of w

innin

g m

ay

be le

ss, i

t’s w

ort

h th

e r

isk.

(P

icks

LS

)

3. T

he o

dds

of w

innin

g a

ny

am

ount o

f money

here

[in th

e S

S] a

re a

lot h

igher.

Only

4 p

eople

out

of 1

00 d

idn’t

get a

nyt

hin

g. (

Pic

ks S

S)

9. T

he [S

S] s

eem

s to

be th

e b

ette

r ch

oic

ebeca

use

the m

ost

tick

ets

are

for

the la

rger

priz

e a

mounts

. But t

he [L

S] h

as

larg

er

priz

es.

Your

chance

s of w

innin

g m

ore

money

seem

to b

egre

ate

r here

. (P

icks

LS

)

4. T

here

are

more

chance

s fo

r w

innin

g m

oney

inth

e [S

S].

The [L

S] h

as

a lo

t of t

icke

ts th

at

have

no m

oneta

ry v

alu

e. E

ven th

ough th

e a

mounts

are

gre

ate

r in

the [L

S],

your

chance

s are

bette

rin

the [S

S] t

o g

et a

tick

et w

ort

h m

oney.

(P

icks

SS

.)

10. E

xclu

din

g th

e e

xtre

mes

(the $

190 to

p lo

ttery

priz

e in

[SS

] and th

e $

50 lo

ttery

priz

e in

[LS

])it

appears

that i

f you w

in, t

he p

rize w

ill b

efo

r m

ore

money

in [L

S].

A b

ette

r payo

ff fo

r not

that m

uch

more

ris

k. (

Pic

ks L

S)

5. I

was

not

put

ting

fort

h an

y m

oney

for

this

lotte

ry, s

o ev

en th

ough

the

[LS

] has

a g

reat

erch

ance

of g

ettin

g $0

—ev

en if

I w

as u

nfor

tuna

te—

I los

e no

thin

g. B

ut if

I w

in, I

win

subs

tant

ially

mor

e m

oney

. (P

icks

LS

)

11. I

pre

fer

the

[SS

] bec

ause

ther

e is

mor

elik

elih

ood

of w

inni

ng. T

he b

est c

hanc

e in

the

[LS

] is

$50

whi

le th

e [S

S] i

s $1

90. A

lthou

gh[w

ith th

e LS

] you

can

win

mor

e th

ere

is a

grea

ter

chan

ce o

f not

win

ning

.

6. T

here

is a

bet

ter

chan

ce o

f win

ning

mor

em

oney

[in

the

LS] a

nd th

ere

is m

ore

mon

eyin

volv

ed (

high

er p

rizes

). (

Pic

ks L

S.)

12. T

here

are

less

tick

ets

for

mor

e m

oney

in th

e[L

S].

I’d r

athe

r ha

ve a

bet

ter

chan

ce fo

r a

little

less

mon

ey. M

y od

ds a

re b

ette

r to

get

$190

[with

SS

] tha

n $3

98 [w

ith L

S].

My

odds

are

equa

l to

get $

190

[with

SS

] or

$50

[with

LS

]. I

opte

d fo

r th

e $1

90 b

rack

et. (

Pic

ks S

S)

Note

: Pro

toco

ls in

italic

are

from

subje

cts

whose

pre

fere

nce

s w

ent a

gain

st m

ajo

rity

pre

fere

nce

s.

49

Table 8Parameter Values for SP/A Theory

(ten-parameter fit)

Parameter Valueqs+ 372.07

qp+ 64.37

qs– 4.86

qp– 16.58

w+ 0.837w– 0.003k+ .043k– .023t + 10.000t - 2.070

50

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