Psychological findings and observations ofeveryday decisions point to the existence of risk asa primitive concept and its key role in decision-making. This makes risk perception the focus ofmany investigations. Numerous attempts havebeen made to define the concept of risk. In thisarticle, two different measures of perceived riskare discussed.

The first measure is represented by thebilinear expected/weighted risk models (BER)based on the expectation principle (e.g. Coombsand Lehner, 1984; Jia and Dyer, 1996; Jia, Dyerand Buttler, 1999; Luce and E.U. Weber, 1986;Sarin and M. Weber, 1993). In this approach,different psychological transformations on payoffsand probabilities are introduced to account forinconsistencies between actual risk rates and theexpectation principle (e.g. Brachinger andM. Weber, 1997; E.U. Weber, Anderson andBirnbaum, 1992). In general, such models of

perceived risk reflect extensions introduced intothe models of expected or non-expected utilitypreferences (which range from expected value andexpected/weighted utility to configural weightingand rank dependent utility models).

In contrast, the second approach to perceivedrisk deals with inconsistencies between actual riskrates and the expectation principle by relyingon a different rule for combining payoffs andprobabilities (Mellers and Chang, 1994; Soko-∏owska and Pohorille, 2000). In the models ofperceived risk proposed in this approach, it isassumed that payoffs and probabilities are treatedas separate dimensions of a risky situation. Thesedimensions are combined additively in riskjudgment. Such a model can be called thedimensional model of perceived risk (DIM).

The first purpose of the research presentedhere is to compare how well both kinds of modelsaccount for risk ratings. The second purpose is to

Original Papers

Joanna Soko∏owska*Andrew Pohorille**

How information about probabilities and pay-offsis combined in risk judgment?

In most theories of risk judgment, it is assumed that probabilities and payoffs combine multiplicatively.However, there is empirical evidence that risk judgment is better represented by an additive combination ofprobability and utility. So far, this additive relation has been demonstrated only for risk rates averaged overall participants. Thus, it may be argued that the observed additive combination rule is an artifact ofaveraging individual strategies focused either on probabilities or payoffs. In this experiment, both theaverage and individual risk rates were compared with prediction of the multiplicative and additive models.The additive model provides good description of actual risk rates, both for the averages and for a majority ofparticipants individually, whereas the multiplicative model performs poorly. Moreover, the analysis of therelative importance of payoffs and probabilities in risk judgments points to individual differences inweighting both factors but does not support concerns that the additive relation between probability andamount of loss results from averaging individual strategies, in which participants pay attention to only oneof these two dimension.

Keywords: risk perception, dimensional models of risk perception, the relative importance of utilityand probability of loss in risk judgment

Polish Psychological Bulletin

2004, vol. 35 (3) 141–152

* Warsaw School of Social Psychology, Warsaw, Poland

**NASA – Ames Research Center, Moffett Field, USA

AUTHOR NOTE

Correspondence concerning this article should be addressed to Joanna Soko∏owska, Department of Psychology, Warsaw School of

Social Psychology, Chodakowska 19/31, 03-815 Warsaw, Poland, e-mail: joanna@psychpan.waw.pl

examine the validity of empirical support for theDIM model found in previous research.Specifically, it is tested whether such supportcould result from averaging of individualstrategies, in which participants attend to onlyone dimension, either probability or amount ofloss. Finally, the merits and limitations of the DIMmodel are discussed.

1. The bilinear expected/weighted risk modelsOne of the earliest definitions of risk was

proposed by Markowitz (1959) who related therisk of an investment to the uncertainty orvariability of a return. However, his definition, aswell as other models of risk, in which riskperception is related to the moments of outcomesdistribution, have not been supported by empiricaldata (e.g. Coombs and Lehner, 1981, 1984; Keller,Sarin and M. Weber, 1986; Soko∏owska andPohorille, 2000).

One reason which can explain the lowpsychological accuracy of these models is theconfusion of dangers and opportunities. Coombsand Lehner (1981, 1984) have shown that thesame changes in expectation or dispersionresulting from transformations of good and badoutcomes had asymmetric effects on perceivedrisk. This led to a conclusion that perceived risk isdirectly related to outcomes and probabilities.

In line with the above findings, Coombs andLehner, (1984) have proposed a bilinear weightedrisk model, which differentiates the impact ofpositive and negative aspects of a risky situation.In this model, risk is a linear combination ofcontributions from “bad” and “good” outcomes,weighted by their probabilities, i.e. payoffs andprobabilities combine multiplicatively. The additivecombination of expected gains and expected lossesmeans that gains reduce risk independently of themagnitude or probability of losses.1 It is alsoassumed that people base their judgment of risk onsubjective reception of stimuli rather than onobjective values of payoffs and probabilities. Theprocesses which reflect perception of thesequantities are left undefined in Coombs’ andLehner’s model.

Attempts to define such processes have beenmade by researchers who studied the sensitivity ofrisk judgments to certain transformations of riskyoptions (e.g. Keller et al. 1986; Luce, 1980, 1981;Luce and E.U. Weber, 1986; Sarin and M. Weber,1993). For example, it has been observed thatperceived risk increases when all payoffs ofa lottery are multiplied by a constant greater

than 1. One model which is in agreement withthe above empirically proven property of perceivedrisk is the conjoint expected risk model (CER),proposed by Luce and E.U. Weber (1986). In theCER model perception of payoffs is defined aspower functions with different exponents for gainsand losses. Sarin (1984 cf. Sarin and M. Weber,1993, Brachinger and M. Weber, 1997) andM. Weber (1990, cf. Sarin and M. Weber, 1993,Brachinger and M. Weber, 1997) have proposeddifferent models, in which perception of payoffs isdefined as an exponential function. This definitionis in agreement with the empirical finding thatrisk decreases when a positive constant is added toall payoffs.

One of the axioms on which the above modelsare based is the expectation principle2, proposedfor risk by Huang (1971). The validity of suchmodels could be questioned because in severalstudies it has been found that the expectationprinciple is violated in risk judgments (Coombsand Lehner, 1984; Keller et al., 1986; E. U. Weberand Bottom, 1989, 1990). For example, Keller,Sarin and M. Weber (1986) found that onlyapproximately one-third of risk comparisons madeby U.S. and German business students wereconsistent with the expectation principle. E.U.Weber and Bottom (1989) reported violations ofthe expectation principle by 48% of undergra-duates from the University of Illinois. E.U. Weberand Bottom (1990) have shown that suchviolations arise because people do not combineprobabilities according to the probability calculus.To account for this result, they proposed tointroduce a subjective transformation on probabi-lities (risk weights) into the CER model.3

2. The dimensional models of perceived riskAnother reason that casts doubts on the

validity of bilinear expected/weighted risk modelsis their inconsistency with some findings inprocess tracing studies. In line with the cognitivealgebra behind such models, each payoff isweighted by its probability. This requires simul-taneous consideration of both dimensions. Incontrast, in process tracing studies it has beenfound that information about outcomes and theirprobabilities is considered separately. For example,Payne (1976) found that participants first collectedinformation about probability of loosing in every

Joanna Soko∏owska, Andrew Pohorille142

1 In contrast, in Fishburn’s (1982) risk model, the multiplicative

combination of contributions from gains and losses isassumed, i.e. gains affect risk only by modifying the effect

of losses. This was falsified in an experiment by E.U.

Weber and Bottom (1989).

2 When applied to choice, this assumption leads to the well-

known expected utility representation. One earlydemonstration that the expectation principle is violated

for choice was the common ratio test described by Allais

(1979).3 In further research, Weber et al. (1992) proposed a risk model

based on the idea of configural weighting (Quiggin, 1982).In this model, the weight of a given outcome (its relative

impact on risk) depends on its probability and its position

(rank) within the distribution of all outcomes.

gamble and, having this information, they nextlooked for information about the amounts ofpayoff. Such a pattern of information searchsupports dimensional representation of a gamble,in which probabilities and payoffs are treated asseparate dimensions and suggests thatprobabilities and outcomes enter into riskevaluation independently rather than as theirproducts (Slovic and Lichtenstein, 1968a; Slovic,Fischhoff and Lichtenstein, 1977). This idea wasfurther developed by Mellers and her colleagues(Mellers, Chang, Birnbaum and Ordonez, 1992;Mellers and Chang, 1994) who proposed to dealwith inconsistencies between the expectationprinciple and actual risk ratings by adopting theadditive rule for combining these two dimensions.Indeed, in their experiments, actual risk rates wereconsistent with an additive combination of utilityand probability of loss.

The additive pattern of information integra-tion for risk assessment has been investigateddirectly and supported by Joag, Mowen andGentry (1990) on the basis of experimentsinvolving both gambling tasks and simulating anindustrial purchasing task. This issue was alsoinvestigated in two experiments by Soko∏owskaand Pohorille (2000), in which risk judgments fora set of risky investments were collected frommanagers and fitted to both DIM and BERmodels. The best fit was found for a dimensionalmodel.

The additive effect of information aboutprobabilities and payoffs has also been observed inresearch on the joint effect of vagueness on boththe probability and the outcome dimensions inrisk judgment (Casey and Scholz, 1991;Kunreuther Meszaros, Hogarth and Spranca,1995; Kuhn and Budescu, 1996). In these studies,it was found that risk evaluation dependedadditively on vagueness of both dimensions, i.e.no interaction between vagueness of theprobability and payoff dimensions was observed.

3. The relative importance of information aboutamount and probability of loss in risk judgment

One may question empirical support for DIMmodels by arguing that the additive relationbetween probability and payoffs found for averagerisk rates may reflect individual differences infocus of attention on different dimensions ofa risky situation. Specifically, if some participantsconcentrated primarily on probabilities whereasothers on payoffs, analyses of the averages couldlead to a misleading conclusion that participantsevaluate risk by combining additively these twodimensions. Such a concern is well justified inlight of ambiguous findings about the relativeimportance of probabilities and payoffs in riskjudgment.

In some experiments, it has been shown thatperceived risk is determined primarily by theprobability of loss (Slovic, 1967; Slovic andLichtenstein, 1968b; Payne, 1975; Coombs, Donnelland Kirk, 1978). In contrast, other studies indicatethat the most important factor in risk perception isthe magnitude of loss. For example, Shapira(1994) found that managers, who comparedriskiness of two lotteries with the same expectedvalue but different probabilities and magnitudes ofloss, based their judgments on the magnitude ofloss. Furthermore, most executives asked forestimates of the “worst possible outcome” or the“maximum loss” and referred to it in defining risk.Similar finding was reported by Huber, Wider andHuber (1997), who studied hypothetical riskymanagerial decisions. In this experiment, onegroup of participants was given a minimaldescription of a situation and could search foradditional information. Only 22% of them askedfor information about probability. Not one of therespondents asked for precise probabilities.Another group of respondents was given preciseprobability information. Less than 20% of themmentioned the word “probability” or “likelihood”in their verbal protocols.

To address the concerns, which were raisedabove: (1) not only averages but also individualrisk rates were analyzed in this experiment, and(2) the relative importance of the amount and theprobability of loss in risk judgment wereinvestigated for every participant.

Method

1. OverviewHow do people combine information aboutoutcomes and probabilities in risk judgment?

In order to test whether actual risk ratesconform better to the multiplicative or to theadditive rule of information integration two kindsof analyses were performed. First, we graphicallyanalyzed average risk rates collected fromparticipants by plotting them as functions ofamount of loss, separately for different levels ofprobability of loss. We further performed ANOVA,as proposed by Anderson (1974, 1982) andpreviously applied to experiments on riskjudgment (e.g. Shantau, 1974; Joag et al. 1990;Mellers et al. 1992). We extended Anderson’sparadigm by analyzing the differences in risk ratesbetween two different probability levels. Accordingto the BER model, but not the DIM model, thesedifferences should depend on payoffs, thusproviding a sensitive test for distinguishingbetween these two types of models. Second,individual risk rates were directly compared withpredictions of both models.

Risk judgment 143

Factors mediating the relative importance of payoffsand probabilities.

Previous findings suggest that one factorthat affects the relative importance of differentdimensions in risk judgment is how available andeasy to process is the provided information. Forexample, one possible reason for focusing on theamount of loss might be that people havedifficulties in interpreting probabilities, especiallyif they are low (Camerer and Kunreuther, 1989;Magat, Viscusi and Huber, 1987). In a recentexperiment by Kunreuther, Noemsky and Kahne-man (2001), respondents were asked to use a smallprobability value while judging risk. Kunreutheret al. (2001) observed that respondents did notreadily evaluate the small probabilities in theabsence of context information. However, whenindividuals were provided with useful contextinformation, the probabilities became easier toevaluate.4 Another reason for focusing on theamount of loss might be that estimatingprobabilities is difficult in the real world.

A clear example of the impact of easiness ofinformation processing on its use in an evaluationtask has been given by Russo (1977). He foundthat the use of information about price increasedwhen the information was brought together in theform of organized lists by ordering the availablebrands by the increasing unit price. Russo arguedthat such an organization makes price informationeasy to process. Other authors also reported thataccepted pattern of information organizationaffected information processing. (e.g. Bettmanand Kakkar, 1977; Jarvenpaa, 1989, 1990) andthat consumers’ choice depended on the way, inwhich the set of available options was categorized(Fox, Levav, 2003).

Thus, in this experiment we controlled thepossible effect of information organization oninformation processing – different forms of stimulipresentations were included into experimentaldesign.

2. Stimuli presentationBased on the findings that organization of

stimuli affects information processing anddecisions by making particular dimensions moresalient and easier to process, the risky situa-tions judged by participants were presented ina systematic order on a single page of a question-naire. Thus, participants could easily detect andprocess the differences in stimuli. In this experi-

ment, there are two dimensions along whichstimuli can be organized, i.e. amount andprobability of loss. To eliminate the possibility ofasymmetric attention focus on these dimensionsdue to the order of presentation, two systematicorders were used. The presented situations wereorganized either by the amount of loss (order 1) or,alternatively, by the probability of loss (order 2).

3. Participants and stimuli188 students of psychology at the Warsaw

School of Social Psychology participated in theexperiment. First, participants were presentedwith a general scenario of a risky investmentsituation. Each participant was asked to assumethat he/she runs a Polish company, which ownsa chain of grocery stores. A foreign investor madean offer to create a joint venture. He would like tobuild a supermarket next to the gas station ownedby his company. Next, participants were presentedwith detailed descriptions of 25 situationscontaining information about the probabilities ofsuccess and failure of such an investment and therelated payoffs.

The situations were described by five levelsof probability (.25; .40; .50; .60; .75) and five levelsof payoffs (–150/350; –250/250; –350/150;–450/250; –550/450 thousands of PLN5).

4. Measure of perceived riskParticipants rated riskiness separately for each

situation on an 11-point scale (from 0 “not riskyat all” to 10 “extremely risky”). Even though thesystematic presentation of lotteries allowed foreasy detection of the payoff and probability ranges,participants were given this information in theinstruction describing their task. The average riskrates for the 25 risky situations given in bothorders of stimuli presentation are listed in Table 1.

Results

1. Graphical analysis and conformity of averagerisk rates to multiplicative and additive modelsanalyzed via ANOVA (Anderson’s paradigm)

In brief the idea of such analysis, mentionedin the Overview, can be explained as follows. Fortwo-outcome lotteries, risk judgment, R, isdetermined by the amounts of loss and gain(al and ag, respectively) and the correspondingprobabilities (pl and pg). In bilinear expected riskmodels (BER) R is described as:

R = wl (pl ) �jl (al ) – wg(pg) �jg(ag) + c, (Eq. 1)

where wl (pl ), wg(pg) and jl(al), jg(ag) representfunctions describing, respectively, the perception

Joanna Soko∏owska, Andrew Pohorille144

4 Authors interpret their findings in line with evaluability

concept introduced by Hsee and his colleagues (Hsee,

1996; Hsee et al. 1999). According to the evaluability

hypothesis, if a dimension is hard to evaluateindependently then a person does not know how good

a given value of the dimension is without making

comparisons which are meaningful to him or her. 5 1 PLN is approximately equal to $0.25.

of probabilities and payoffs. They are combinedin a multiplicative fashion. The additive constant,c, depends on the choice of scale used to measureR. All functions in Eq.(1) are non-negative and theminus sign in front of the term describing positiveaspects of the lottery means that this term reducesperceived risk.

For dimensional models (DIM), risk rates arerepresented using the additive combination rule,i.e. as a sum of contributions from al, pl, ag and pg:

R = wl(pl) + jl(al) – wg(pg) –jg(ag) + c (Eq. 2)

Again positive aspects of a lottery, wg(pg) andjg(ag), reduce risk.

Different specific multiplicative and dimen-sional models are possible, depending on theconcrete functions chosen to describe perceptionsof probabilities and payoffs. In this study we donot address this issue. Instead we focus on dataanalysis that helps us to determine whethermultiplicative or additive rule for combiningpayoffs and probabilities yields better descriptionof empirical risk ratings collected here. Thus, wewant to distinguish between two classes of models(BER and DIM), and we need appropriate toolsfor this task. Statistical analysis based onAnderson’s paradigm is one of them.

In the previous experiments, in whichAnderson’s paradigm was applied, the amount ofnon-negative outcome was kept constant and wasequal either to 0 or to a positive constant (e.g.Joag et al. 1990; Mellers and Chang, 1994). Thisallows for analyzing perceived risk exclusively asa function of negative aspects of a situation. Thusin line with the bilinear expected risk models(BER), perceived risk is described as:

wl(pl) �jl(al), (Eq. 1’)

where jl, and wl, represent psychologicaltransformations on al, and pl.

For dimensional models (DIM), risk rates arerepresented as a sum of contributions from al,and pl:

wl(pl) + jl(al) (Eq. 2’)

Although both models predict that risk ratingsshould change with changes in probability andamount of loss, the nature of these changes isquite different in each model. The BER modelpredicts that the differences between ratings atthe two most extreme levels of probability increaseas the amount of loss increases, i.e. the differencein perceived risk of loosing 550,000 PLN with the

Risk judgment 145Table 1

Average risk rates collected from participants

Lottery descriptionLotteries organized due to

amount of loss – order 1 probability of loss – order 2

Payoffs in thousandsprobability of loss N average risk rates N average risk rates

of PLN

450/–550 0.75 112 7.88 76 7.780.60 112 7.26 76 6.50

0.50 112 6.34 76 6.01

0.40 112 5.69 76 5.34

0.25 112 4.75 74 4.43

250/–450 0.75 112 7.78 76 8.210.60 112 6.88 76 6.92

0.50 112 6.44 76 6.63

0.40 112 5.82 76 5.64

0.25 112 5.00 74 4.65150/–350 0.75 112 7.76 76 8.00

0.60 112 7.07 76 7.05

0.50 111 6.35 76 6.55

0.40 112 5.76 76 5.710.25 112 5.02 75 4.71

250/–250 0.75 112 6.83 75 6.79

0.60 111 6.03 76 6.20

0.50 112 4.94 74 5.01

0.40 111 4.44 76 4.290.25 110 3.54 74 3.68

350/–150 0.75 112 5.71 75 5.84

0.60 111 5.05 76 5.62

0.50 112 4.19 75 4.320.40 112 3.65 76 3.93

0.25 112 2.92 75 2.85

probability 0.75 and loosing the same amount withthe probability 0.25 is larger than the difference inperceived risk of loosing 150,000 PLN with thesame probabilities. In contrast, this differenceremains constant in the DIM model.

The actual average risk rates were plotted asa function of payoffs such that a separate curvecorresponds to each level of probability of loss.Such plots for 2 experimental groups, dividedaccording to the order of lotteries’ presentation,are shown in two panels of Figure 1. From Fi-gure 1A, which corresponds to average risk rates oflotteries organized by the amount of loss, it can beseen that perceived risk increases with the proba-bility and the amount of loss. However, the diffe-rence in perceived risk of loosing 550,000 PLNwith probabilities 0.75 and 0.25 is almost the sameas the difference in perceived risk of loosing150,000 PLN with the same probabilities. Moregenerally, the curves for different probability levelsare roughly parallel, which is consistent with thehypothesis that ratings reflect an additive combi-nation of probability and amount of loss. Consistentwith the conclusions from the graphical analyses,strong simple effects of probability (F(4, 106)=83.01,p=0.000) and payoffs (F(4, 106)=46.20, p=0.000) onrisk rates were observed in the analysis of variance(ANOVA) for repeated measures with two within-subject factors – probability (5 levels) and payoffs(5 levels). The effect of interactions between payoffand probability on the risk ratings was notsignificant (F(16, 106)=1.55, p=0.074).

Curves in panel B of Figure 1 have a similarpattern to those in panel A. However, somedeviations from parallelism of curves can bedetected. Consequently, the ANOVA for repeatedmeasures showed both very strong simple effectsof payoffs (F(4, 66)=52.82, p=0.000) and probability ofloss (F(4, 66)=92.97, p=0.000) on risk rates as well asmuch weaker but significant effect of interactionsbetween payoffs and probability of loss (F(16, 66)=3.42, p=0.000). However, significant interactionbetween these two factors is not localized in thebilinear interaction component (L�L: F=0.05,p=.819) as required by Anderson’s paradigm.

2. Testing predictions of multiplicative and additivemodels (an extension of Anderson’s paradigm)

Here we present an analysis that can beconsidered an extension of Anderson’s paradigm.In this analysis, we have examined differences inrisk rates for pairs of lotteries with differentprobability levels across all payoffs levels. By doingso we can clearly distinguish between multipli-cative and additive models when lotteries havedifferent both amounts of loss and amounts of gain.This distinction is possible because multiplicativeand additive models imply different predictionsabout the calculated differences. According to BERmodels they depend on the payoffs level. Incontrast, according to DIM models thesedifferences should be the same for all payoff levels.

To test the validity of these two classes ofmodels we exploit the design of the experiment. All

Joanna Soko∏owska, Andrew Pohorille146

Figure 1. Average risk rates due to amount and probability of loss - order 1 and order 2

lotteries are characterized by 5 probability levels(plm pg

m, m=1,…5). For each probability level, thereare 5 levels of payoffs (al

i agi, i=1,…5). In the rest of

the paper we will use a convention that Greekindices m, n abbreviate probability levels and Latinindices i, j abbreviate payoff levels. Then riskratings of each of the 25 lotteries can be defined bya pair of indices (m, i). For BER models risk rating,Rmi is expressed as:

Rmi = wl(plm) �jl(al

i) – wg(pgm) �jg(ag

i) + c (Eq. 3)

and for another probability level, n, but the samepayoff level i:

Rni = wl(pln) �jl(al

i) – wg(pgn) �jg(ag

i) + c. (Eq. 4)

If we subtract these two risk ratings thedifference d i

mn = Rmi – RnI is expressed as follows:

d imn = (wl (pl

m) – wl (pln)) �jl(al

i) – (wg (pgm) – wg (pg

n)) �jg (agi),

(Eq. 5)

For DIM models the equivalent equations areas follows:

Rmi = wl(plm) + jl(al

i) – wg(pgm) – jg(ag

i) + c (Eq. 6)

Rni = wl(pln) + jl(al

i) – wg(pgn) – jg(ag

i) + c (Eq. 7)

and then:

d imn = wl (pl

m) – wl (pln) – (wg (pg

m) – wg (pgn)), (Eq. 8)

The comparison of Eq. (5) with Eq. (8) illustra-tes that predictions regarding the behavior of d i

mn

based on BER and DIM models are quite different.Specifically, according to DIM models thedifference in risk ratings between two differentprobability levels – m, n should not depend on thepayoff level. For example, difference in risk ratingsbetween probability levels 0.75 and 0.25 shouldbe approximately the same, no matter whetherthe payoffs are, for example, 450,000/–550,00 PLNor 150,00/–350,000 PLN. This simple relation,however, does not hold for BER models.

To obtain Eq. (5) and (8) no assumptions aboutthe functional form of perception of payoffs andprobabilities are necessary. However, to clarifythe difference between these equations we willuse some weak, commonly accepted assumptions.First, we consider probability weighting function.In some studies (Soko∏owska and Pohorille, 2000),it was found that this function can be welldescribed by a linear function of probabilities, atleast in the range of probabilities studied here. Inother words, w(p) = cp � p, where cp is a constant.Considering that pl

m + pgm = 1 for two-outcome

lotteries, the sum wl (plm) + wg (pg

m) for a givenprobability level m is clearly a constant equalto cp. In other studies, wl (pl ) and wg (pg ) were

represented as two different, nonlinear functionsof probabilities. For example, in CumulativeProspect Theory (Tversky and Kahneman, 1992)wl (pl ) for pl levels of 0.75, 0.6, 0.5, 0.4 and 0.25,considered in this work, are 0.63, 0.52, 0.45, 0.39and 0.29 respectively. The values of wg(pg ) for thecorresponding levels of pl are 0.29 and 0.37, 0.42,0.47 and 0.57. The sum wl (pl

m) + wg (pgm) varies

slightly with the probability level, but it deviatesfrom the average value of 0.88 by no more than4%. Thus, with good accuracy we can assume thatfor commonly accepted probability weightingfunctions wl (pl

m) + wg (pgm) is equal to a constant

(that we abbreviate cp). Consequently

wg (pgm) = cp – wl (pl

m), wg(pgn) = cp – wl(pl

n), Eq. (9)

This relation reflects psychological intuitionthat people do not perform drastic transformationson probabilities in the range studied here.Substituting Eq. (9) to Eqs. (5) and (8) we obtain

d imn= (wl(pl

m) – wl(pln)) � (jl(al

i) + jg(agi)), (Eq. 10)

for BER models and

dimn= 2(wl(pl

m) – wl(pln)), (Eq. 11)

for DIM models.The two equations differ by the term jl (al

i) +jg (ag

I) that is present in BER models but not inDIM models. Recall that jl (al ) and jg (ag ) arefunctions reflecting perception of payoffs, thusthey are proportional to utilities of negative andpositive outcomes. Previous research has shownthat utility of payoffs does not increase linearlywith their objective values (Bernoulli, 1954;Galanter, 1962). Moreover, this function hasa different shape for gains and losses (Stevens,1959; Kahneman and Tversky, 1979; Coombs andLehner, 1981, 1984; Luce and E.U. Weber, 1986).For example in CTP (Tversky and Kahneman,1992), utility of losses is defined as l|al|

a andutility of gains is defined as ag

b, where l=2.25 anda=b=0.88. If we order the payoff levels accordingto the amount of loss, such that i=1 for the lotterywith payoffs –550,000/450,000 PLN and i=5 forthe lottery with payoffs –150,000/350,000 thenjl (al

i) + jg (agI), i=1,…5 calculated from CPT are,

respectively, 796.5, 615.3 472.1, 418.9 and 358.2.Thus, they decrease with payoff level i. Thedifference between the extreme values is largerthan a factor of 2.6

Risk judgment 147

6 Other utility functions have also been proposed. For example,

it was suggested that β might be smaller than α (e.g.

Stevens, 1959; Luce and E.U. Weber, 1986) or even that

utility of gains might be a logarithmic function of ag

(Soko∏owska and Pohorille, 2000). Under these assum-ptions the values of jl(al

i) + jg(agI) change but their order

remains the same. In fact, the ratio of the extreme values

will be even larger than the ratio in CPT.

At this point we can clearly define the testand the predictions arising from the two classes ofmodels considered here. We calculate thedifference in the average risk ratings between twodifferent probability levels m and n. For five levelsof probabilities there are 10 such pairs. For eachpair we obtain five differences, d i

mn, eachcorresponding to a different level of payoffs. Forexample, if m=1 and n=5 these are five differencesbetween risk ratings of lotteries with pI=0.75 andpl=0.25, each corresponding to one of five levelsof payoffs (al=–550,000, –450,000, –350,000,–250,000 and –150,000 PLN). According to DIMmodels, these five values should be very similar. Incontrast, predictions from BER models are thatthese values should decrease by approximatelya factor of 2–3 for the order of lotteries definedabove.

The results obtained from the experiment arelisted in Table 2. Again, according to DIM modelsthe values in one row of Table 2 should be similar,whereas according to BER models they shoulddecrease by a factor of 2–3. The results are alsoshown graphically in Figure 2. For clarity, onlyvalues from rows 1, 5 and 9 of Table 2 are shownin Figure 2.

For these pairs of probability levels we alsoshow a straight line that corresponds to thearithmetic average of d i

mn for the five levels ofpayoffs. For order 1, the root mean squaredeviations (rmsd) for the 10 pairs of probabilitylevels vary between 0.05 and 0.25 and the globalrmsd (for all pairs) is 0.17. This is in a very goodagreement with the predictions arising from DIMmodels. For order 2, rmsd vary between 0.13 and0.40 and the global rmsd is 0.26. These values aresomewhat larger that the corresponding valuesfor order 1, but they are still in good agreementwith DIM models. This agrees well with the resultsof ANOVA presented in the previous section,which yielded strong simple effects for both ordersand some interaction effects for order 2.

In contrast, the agreement between the dataand predictions from BER models is unsatisfac-tory. Visual inspection of Figure 2 is sufficient todetermine that there is no systematic change ofdimn, with the index i (or, equivalently, the amount of

loss), which is expected in these models. As anexample, we calculated the values of di

mn, predictedby CPT by choosing the scaling constant in front ofthe utility functions to minimize rmsd between themodel and the data. The model fits the data poorly.Not surprisingly, global rmsd for order 1 and 2are 0.64 and 0.71, respectively. This is markedly(three to four-fold) worse than the correspondingvalues for DIM models. On this basis we concludethat the data are not consistent with any BER-type model.

The above conclusion is formulated based onthe analysis of averages. As mentioned in theIntroduction, such an analysis raises concernsthat the additive combination found for averagerisk rates can result from averaging individualstrategies, in which participants pay attention toonly one dimension, either probability or payoff.Thus, the conformity of risk rates to BER and DIMmodels has also been tested with risk rates ofindividual participants.

3. Testing predictions of multiplicative and additivemodels in the analysis of risk rates of individualparticipants

For each participant, d imn for the five levels of

payoffs were calculated. Next it has been checkedwhether they are independent of the index i foreach pair of probability levels. As the measure ofagreement with predictions from DIM models weused the global rmsd, i.e. the standard deviation ofthe d i

mn, as predicted by DIM models. Taking intoaccount that participants rated risk usinga discrete scale with the intervals equal to 1, ithas been assumed that rmsd in the ranges of 0–0.5,0.5–1 and 1–1.5 correspond to very good, goodand fair agreement with DIM models. If the global

Joanna Soko∏owska, Andrew Pohorille148

Table 2

The average differences in risk rates of a pair of lotteries with the same payoffs but different probabilities of loss

Pair of Payoffs

probabilities 450/–550 250/–450 150/–350 250/–250 350/–150

of loss Group 1 Group 2 Group 1 Group 2 Group 1 Group 2 Group 1 Group 2 Group 1 Group 2

0.75 vs. 0.25 3.13 3.28 2.78 3.54 2.74 3.27 3.32 3.07 2.79 2.99

0.75 vs. 0.40 2.19 2.43 1.96 2.57 2.00 2.29 2.41 2.51 2.05 1.93

0.60 vs. 0.25 2.50 2.00 1.88 2.27 2.05 2.32 2.51 2.49 2.13 2.750.75 vs. 0.50 1.55 1.81 1.34 1.60 1.45 1.47 1.89 1.82 1.52 1.59

0.60 vs. 0.40 1.57 1.16 1.05 1.28 1.31 1.34 1.58 1.91 1.41 1.68

0.50 vs. 0.25 1.58 1.50 1.44 1.96 1.33 1.81 1.42 1.35 1.27 1.46

0.75 vs. 0.60 0.63 1.27 0.90 1.26 0.69 0.94 0.79 0.59 0.65 0.24

0.60 vs. 0.50 0.92 0.53 0.43 0.34 0.74 0.53 1.09 1.20 0.88 1.380.50 vs. 0.40 0.65 0.67 0.62 0.99 0.58 0.84 0.54 0.73 0.53 0.36

0.40 vs. 0.20 0.94 0.84 0.82 1.00 0.74 0.96 0.89 0.59 0.73 1.04

rmsd for a participant was larger than 1.5 it hasbeen concluded that DIM models were notappropriate to describe risk ratings given by thisparticipant. Inspection of the data and numericalexperiments, in which a small number ofdeviations from the “perfect” risk ratings weregenerated at random, confirmed that ourassumptions about different ranges of rmsd werereasonable.

Among 106 participants who evaluated riskof all lotteries in order 1, 54 gave risk ratings thatwere in very good or good agreement with DIMmodels. Ratings of another 33 participants were infair agreement with the model. However, in thisgroup there were seven participants whose riskratings were consistent with BER models. Thiswas the case because for these parcitipants di

mn,tent to decrease with the lottery index i for mostpairs of probability levels. Another seven suchparticipants who followed a BER model wereamong 19 participants whose risk ratings wereclearly inconsistent with DIM models. Takentogether, 80 participants (75%) rated riskconsistently with DIM models, 14 participants(13%) rated risk consistently with BER modelsand no model was identified for the remaining 12participants (11%). Among 67 participant whoevaluated risk of all lotteries in order 2, 47 (70%)and 10 (15%) gave ratings consistent with DIMand BER models, respectively. Another 14participants (15%) appeared not to use either ofthese two models.

4. Relative contributions to individual risk ratesfrom payoffs and probabilities

One way to analyze relative contributions torisk rates from payoffs and probabilities is tocompare for each participant the relativevariability of risk rates due to changes inprobabilities with such variability due to changesin payoffs. As can be seen from Eq. (2), risk ratingin dimensional models is described as a sum ofcontributions from the amounts of loss and gainand the corresponding probabilities. Exploitingthe experimental design used in this study it ispossible to eliminate contributions from payoffsfrom the model by taking appropriate differencesof risk ratings. This leads to Eqs. (8) and (11),which depend only on probabilities. For DIMmodels it is also possible to do the opposite – toconstruct differences in risk judgments such thatthey depend on payoffs but not on probabilities. Todo so we return to Eqs. (6) and (7), but this timewe write a DIM description of risk ratings at twodifferent levels of payoffs, i and j, instead of twodifferent levels of probabilities:

Rmi = wl(plm) + jl(al

i) – wg(pgm) – jg(ag

i) + c (Eq. 12)

Rmj = wl(plm) + jl(al

j) – wg(pgm) – jg(ag

j) + c (Eq. 13)

Subtracting Rmj from Rmj we obtain:

dijm= jl(al

i) – jg(agi) – (jl(al

j) – jg(agj)) (Eq. 14)

Risk judgment 149

Figure 2. Average differences in risk rates for pairs of lotteries with the same payoffs and different probabilities - order 1 and order 2

where dijm = Rmi – Rmj. In contrast to Eq. (8), this

equation depends only on payoffs. From Eq. (14) itfollows that the difference in risk rates betweentwo different levels of payoffs (and there are 10such pairs) should be the same for all fiveprobability levels.

Eqs. (11) and (14) allow us to gaugecontributions to risk ratings from probabilitiesand payoffs separately by calculating di

mn and dijm.

One way to do so is to combine these quantitiesinto a single index variable. For each pair ofprobability levels we calculate the value of d i

mn

averaged over different payoff levels (i.e. theaverage over numbers in a row of Table 2). Thenwe selected the largest of these differences. Thiscorresponds to the difference in probability levels0.75 and 0.25 (the first row in Table 2) andrepresents the maximum variability of risk ratesdue to probabilities. We abbreviate this quantity asDp. Similarly, we identify the maximum dij amongall pairs of payoff levels and abbreviate it as Da

(this corresponds to the difference between payoffs450/–550 and 350/–150). Da measures the maxi-mum variability of risk rates due to payoffs. Then,the single index variable, V, is constructed asfollows:

V = (Dp – Da)/(Dp + Da) (Eq. 15)

This quantity can be calculated for eachparticipant. If a participant is strongly focused onprobabilities and pays little attention to payoffs

Dp is large and Da is small. Then V is close to 1.Conversely, if a participant considers only payoffsand does not differentiate lotteries with respectto probabilities than Da is large and Dp is small. Asa result, V approaches –1. Finally if a participantpays approximately equal attention to both types ofdimensions Vp and Va are nearly the same and V isclose to zero. This, however, does not imply that ifparicipants consider both probabilities and payoffsin rating risk the most likely value of V must bezero. Participants may take into account bothdimensions but consider one of them somewhatmore important than the other. This would shiftthe value of the index away from zero in eitherthe positive or the negative direction.

The analysis was performed for participantswhose risk ratings were best described by the DIMmodel (i.e. 73% of all participants). The fractionsof participants in different ranges of V are given inFigure 3. These results show that more paricipantsfocused their attention on probabilities than onpayoffs (48% and 33%, respectively). 19% ofparticipants considered both dimensions asapproximately equally important. This findingpoints to individual differences in weightingdifferent characteristics of a risky situation.Despite these differences, however, all theseparticipants rated risk consistently with anadditive model. Furthermore, the results do notsupport doubts that the additive relation betweenprobabilities and payoffs observed for average riskratings is an artifact of combining individual

Joanna Soko∏owska, Andrew Pohorille150

1.00 0

0.75

0.50

0.25

0.00

–0.25

–0.50

–0.75

–1.00

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Valu

e of

ind

ex V

Ratio of subject

0.12

0.18

0.18

0.19

0.06

0.02

0.1

0.15

Figure 3. The relative importance of probabilities and payoffs in risk judgment.

strategies, in which participants pay attention toonly one of these dimensions. Then, V should beclose to either 1 or –1. However, only 22% ofparticipants evaluated risk such that the absolutevalue of V was larger than 0.8.

Conclusions

The first goal of the experiment was tocompare how well BER and DIM models accountfor actual risk ratings. The graphical analyses, aswell as the results of ANOVAs indicate that thelinear combination of information about payoffsand probabilities, assumed in DIM models, isa good approximation of the process of riskjudgment. Moreover, the analysis of differencesin risk rates of two lotteries with the same payoffsbut different probability levels have shown clearlythat small deviations from parallelism in Figure 2cannot be explained by the multiplicativecombination of these factors, assumed in BERmodels.

The above conclusions based on the analysisof averages have also been confirmed forindividual risk rates. A majority of participants(73%) rated risk consistently with DIM models,and only 13% of participants rated riskconsistently with BER models.

Our findings are consistent with severalprevious empirical studies, which have shown thatprobability and utility combine additively to yieldrisk ratings (Joag et al., 1990; Mellers et al., 1992;Mellers and Chang, 1994; Soko∏owska andPohorille, 2000). Moreover, the results forindividual paricipants as well as the analysis ofrelative contributions into risk judgment fromamount and probability of loss also demonstratethat the additive relation between these dimen-sions applies not only to average risk rates butalso to individual strategies focused on eitherprobabilities or payoffs.

A limitation of the proposed dimensionalmodel is that, in the current form, the model isvalid only for two-outcome situations. The modelcould be easily extended to multiple outcomesonly under the assumption that people diminishcognitive load by clustering outcomes into positiveand negative subcategories. Using the languageof the original prospect theory, people might editsuch situations in order to obtain two-outcomeprospects. Thus, they might neglect outcomes withlow payoffs and/or low probabilities, focus on themost probable and/or the most extreme outcome,on outcomes close to the aspiration level, and soon. Nevertheless, even if people translate multi-outcome prospects into two-outcome prospects,knowledge of how such editing is done is requiredto predict risk rates using the dimensional model.However, it is also possible that a psychological

representation of multi-outcome risky eventsinvolves other dimensions. This might be, forexample, the relative probability and severity ofthe worst outcomes in the distribution, asproposed by Lopes (1984, 1987). Thus, this issueneeds further empirical investigation.

References

Allais, M. A. (1979). The foundation of a positive theory of

choice involving risk and a criticism of the postulatesand axioms of the American School. In: M.A. Allais

and O. Hagen (ed.), Expected utility hypothesis and the

Allais paradox. Dordrecht, The Netherlands: Reidel.

Anderson, N.H. (1982). Methods of Information Integration

Theory. New York: Academic Press.

Anderson, N.H. (1974). Algebraic models in perception. In:

E.C. Carterette and M.P. Friedman (eds), Handbook of

Perception, (pp. 215–298). New York: Academic Press.Bernoulli, D. (1954). Exposition of a new theory on the

measurement of risk. Econometrica, 22, 23–26.

Bettman, J.R., & Kakkar, P. (1977). Effects of information

presentation format on consumer information acquisitionstrategies. Journal of Consumer Research, 3, 233–240.

Brachinger, H.W., & Weber, M. (1997). Risk as a primitive:

A survey of measures of perceived risk. OR Spektrum,

19, 235–250.

Camerer, C., & Kunreuther, H. (1989). Decision processes forlow probability events: Policy implications. Journal of

Policy Analysis and Management, 8, 565–592.

Casey, J.T., & Scholz, J.T. (1991). Boundary effect of vague

risk information on taxpayer decisions. Organizational

Behavior and Human Decision Processes, 50, 360–394.

Coombs, C. H., & Lehner, E. P. (1981). Evaluation of two

alternative models of a theory of risk: I. Are moment of

distributions useful in assessing risk? Journal of

Experimental Psychology: Human Perception and

Performance, 7, 1110–1123.

Coombs, C. H., & Lehner, E. P. (1984). Conjoint design analysis

of the bilinear model: an application to judgments of risk.Journal of Mathematical Psychology, 28, 1–42.

Coombs, C. H., Donnell, M. L., & Kirk, D. B. (1978). An

experimental study of risk preferences in lotteries. Journal

of Experimental Psychology: Human Perception and

Performance, 4(3), 497–512.Fishburn, P. C., (1982). Foundations of risk measurement: II.

Effects of gains on risk. Journal of Mathematical

Psychology, 25, 226–242.

Fox, C. R., & Levav, J. (2003). Partition-edit-count: Naïveextensional reasoning in judgment of conditional

probability. Unpublished manuscript.

Galanter, E. (1962). The direct measurement of utility and

subjective probability, American Journal of Psychology,75(2), 208–220.

Hsee, Ch. (1996). The evaluability hypothesis: An explanation of

preference reversals between joint and separate

evaluations of alternatives. Organizational Behavior and

Human Decision Process, 46, 247–257.

Hsee, Ch., Blount, S., Loewenstein, G., & Bazerman, M. (1999).

Preference reversals between joint and separate

evaluations of options: A review and theoretical analysis.Psychological Review, 125, 576–590.

Huang, L. (1971). The expected risk function. Michigan

Mathematical Psychology Program Report, 71–76.

Huber, O., Wider, R., & Huber, O.W. (1997). Active information

search and complete information presentation innaturalistic risky decision tasks. Acta Psychologica, 95,

15–29.

Risk judgment 151

Jarvenpaa, S.L. (1990). Graphic display in decision making:

The visual salience effect. Journal of Behavioral Decision

Making, 3, 247–262.

Jarvenpaa, S.L. (1989). The effect of task demands and

graphical format on information processing strategies.

Management Science, 35, 285–303.Jia, J., Dyer, J. S., & Butler, J.C. (1999). Measure of perceived

risk. Management Science, 45(4), 519–532.

Joag, S. G., Mowen, J. C., & Gentry, J.W. (1990). Risk

perception in a simulated industrial purchasing task: Theeffects of single versus multi-play decisions. Journal of

Behavioral Decision Making, 3, 91–108.

Kahneman, D., & Tversky, A. (1979). Prospect theory: An

analysis of decision under risk. Econometrica, 47(2),263–291.

Keller, L.R, Sarin, R.K., & Weber, M. (1986). Empirical

investigation of some properties of the perceived riskiness

of gambles. Organizational Behavior and Human Decision

Process, 38, 114–130.

Kuhn, K.M., & Budescu, D.V (1996). The relative importance of

probabilities, outcomes, and vagueness in hazard risk

decisions. Organizational Behavior and Human Decision

Process, 68, 301–317.Kunreuther, H., Meszaros, J., Hogarth, R.M., & Spranca, M.

(1995). Ambiguity and underwriter decision processes.

Journal of Economic Behavior and Organization, 26,

337–352.Kunreuther, H., Noemsky, N., & Kahneman, D. (2001). Making

low probabilities useful. Journal of Risk and Uncertainty,

23, 102–120.

Lopes, L. L. (1987). Between hope and fear: The psychology ofrisk. Advances in Experimental Social Psychology, 20,

255–295.

Lopes, L. L. (1984). Risk and distributional inequality. Journal

of Experimental Psychology: Human Perception and

Performance, 10, 465–485.

Luce, R. D. (1981). Correction to ‘Several possible measures of

risk’. Theory and Decision, 13, 381.

Luce, R.D. (1980). Several possible measures of risk. Theory

and Choice, 12(3), 217–228.Luce, R. D., & Weber, E. U. (1986). An axiomatic theory of

conjoint, expected risk. Journal of Mathematical

Psychology, 30(2), 188–205.

Markowitz, H. M. (1959). Portfolio Selection. New York: Wiley.

Mellers, B.A., & Chang, S. (1994). Representations of risk

judgment. Organizational Behavior and Human

Performance, 57, 167–184.Mellers, B.A., Chang, S., & Birnbaum, M. H., Ordonez, L.D.

(1992). Preferences, prices and ratings in risky decision

making. Journal of Experimental Psychology: Human

Perception and Performance, 18, 347–361.Payne, J. W. (1976). Task complexity and contingent processing

in decision making: An information search and protocol

analysis. Organizational Behavior and Human Perfor-

mance, 16, 366–387.Payne, J. W. (1975). Relation of perceived risk to preferences

among gambles. Journal of Experimental Psychology:

Human Perception and Performance, 104, 86–94.

Quiggin, J. (1982). A theory of anticipated utility. Journal of

Economic Behavior and Organization, 3(4), 324–345.

Russo, J.E. (1977). The value of unit price information. Journal

of Marketing Research, 14, 193–201.

Sarin, R. K., & Weber, M. (1993). Risk-value models. EuropeanJournal of Operational Research, 70, 135–149.

Shantau, J. (1974). Component processes in risky decision

making. Journal of Experimental Psychology, 103,

680–691.Shapira, Z. (1994). Risk Taking: A managerial perspective. New

York: Russell Sage Foundation.

Slovic, P. (1967). The relative influence of probabilities and

payoffs upon perceived risk of a gamble. Psychometric

Science, 9(4), 223–224.Slovic, P., & Lichtenstein, S. (1968a). Importance of variance

preferences in gambling decisions. Journal of Experi-

mental Psychology, 78, 646-654.

Slovic, P., & Lichtenstein, S. (1968b). Relative importance ofprobabilities and payoffs in risk taking. Journal of

Experimental Psychology Monograph, 78 (3, Pt. 2).

Slovic, P., Fischhoff, B. S., & Lichtenstein, S. (1977). Behavioral

decision theory. Annual Review of Psychology, 28, 1–39.Soko∏owska, J., & Pohorille, A. (2000). Models of Risk and

Choice: Challenge or danger. Acta Psychologica, 104,

339–369.

Stevens, S.S. (1959). Measurement, psychophysics and utility,[in:] C. W. Churchman, P. Ratoosh (ed.). Measurement:

Definitions and theories. (pp. 18–63). New York: Willey.

Tversky, A., & Kahneman, D. (1992). Advances in prospect

theory: Cumulative representation of uncertainty. Journal

of Risk and Uncertainty, 5(4), 297–323.Weber, E.U., & Bottom, W.P. (1990). An empirical test of the

transitivity, monotonicity, accounting, and conjoint axioms

for perceived risk. Organizational Behavior and Human

Decision Processes, 45, 253–275.Weber, E.U., & Bottom, W.P. (1989). Axiomatic measures of

perceived risk: some tests and extensions. Journal of

Behavioral Decision Making, 2, 113–131.

Weber, E. U., Anderson, C., & Birnbaum, M. H. (1992). Atheory of perceived risk and attractiveness. Organizational

Behavior and Human Decision Processes, 52, 492–523.

Joanna Soko∏owska, Andrew Pohorille152