Decisions With Compound Lotteries - INSEAD

Decisions With Compound Lotteries

Yuyu Fan and David V. BudescuFordham University

Enrico DiecidueINSEAD

Violations of the Reduction of Compound Lottery axiom (ROCL) were documented,but they are not fully understood, and only few descriptive models were offered tomodel decision makers’ (DMs) decisions in such cases. This article comprehensivelytests the effects of 6 factors that could influence DMs’ evaluations of compoundlotteries, and models how DMs make decisions in their presence. In an experiment with6 groups of subjects (n � 125), we elicited certainty equivalents of simple andcompound lotteries via a 2-stage choice procedure. We confirmed the existence of thesystematic violations of ROCL. We tested the effect of each factor and their interac-tions, and found that the number of stages and the global probability had prominenteffects. We developed three classes of models to describe the weighting process forcompound lotteries. The best fitting model was the one that assumes that DMs anchoron the lowest stage probability and then apply the weighting function. The “aggregatefirst and weigh second” overall outperformed the “weight first and aggregate second”models, presumably because this process is cognitively easier and more natural.

Keywords: Reduction of Compound Lottery axiom (ROCL), compound lotteries,Prospect Theory, descriptive models

Supplemental materials: http://dx.doi.org/10.1037/dec0000091.supp

Suppose one faces a choice between two lotter-ies: The first lottery offers a 30% chance of win-ning $100 and a 70% chance of winning nothing.The second lottery is carried out in two indepen-dent stages; the first stage involves a 50% chanceof ending without winning or losing anything, anda 50% chance of moving to the second stage, inwhich one has a 60% chance of winning $100 and40% chance of winning nothing. The two lotteriesare structurally equivalent since the product of theprobabilities of winning of the two stages in thesecond lottery equals to that of the first lottery, and

the outcomes are identical. Therefore, a rationaldecision maker should be indifferent betweenthem.

This intuition is captured by the Reduction ofCompound Lottery axiom (ROCL) of ExpectedUtility theory (EU; e.g., French, 1988). Anycompound lottery involving S (S � 2) indepen-dent probabilistic stages can be reduced to anequivalent simple form lottery based on norma-tive probabilistic rules, and the ROCL axiom,states that Decision Makers (DMs) should beindifferent between the simple and compoundform of a lottery, regardless of the number ofstages and the nature of the chance mechanismto resolve the uncertainties. In other words, only“the outcomes and their respective probabilitiesmatter” (Budescu & Fischer, 2001). However,violations of ROCL have been observed in alarge number of studies (e.g., Bar-Hillel, 1973;Budescu & Fischer, 2001; Chung, von Winter-feldt, & Luce, 1994; Miao & Zhong, 2012;Ronen, 1973; Slovic, 1969).

Previous studies identified several factorsthat can lead to violations of ROCL. A primary

This article was published Online First July 12, 2018.Yuyu Fan and David V. Budescu, Department of Psy-

chology, Fordham University; Enrico Diecidue, DecisionSciences, INSEAD.

Special thanks to Professor Andrew Schotter for access tothe facilities of the Center for Experimental Social Sciencesat New York University to run the study.

Correspondence concerning this article should be ad-dressed to David V. Budescu, Department of Psychology,Fordham University, Bronx, NY 10458. E-mail: [email protected]

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Decision© 2018 American Psychological Association 2019, Vol. 6, No. 2, 109–1332325-9965/19/$12.00 http://dx.doi.org/10.1037/dec0000091

109

http://dx.doi.org/10.1037/dec0000091.supp

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1037/dec0000091

factor is the number of stages. For example,Slovic (1969) asked DMs to rate the attractive-ness of one-stage lotteries with winning (losing)probability p against the probabilistically equiv-alent four-stage compound lotteries with stagewinning (losing) probability �4

p. In general,the DMs preferred (disliked) compound lotter-ies in the domain of gains (losses), and overes-timated the probabilities of the compound lot-teries. Bar-Hillel (1973) studied conjunctiveevents—using equal probabilities for all stages,and manipulating the global winning probabili-ties and the number of stages—and also foundthat their subjective probabilities were overes-timated. In Bar-Hillel’s experiment the proba-bilities of each stage in the compound lotterieswere all above .50, the global winning proba-bilities were always below .50, and the numberof stages ranged from 2 to 10. Subjects madechoices between the simple and compoundforms of four lotteries separately, and they werescored based on the number of times they chosecompound lotteries. She found the overestima-tion was stronger for compound lotteries withvery small global winning probabilities andvery high stage-specific probabilities of manystages (e.g., a compound lottery composed of 10stages with stage winning probability .80,whose global winning probability is only .11).

Other researchers tested the effect of unequalstage probabilities, including the ordering of thestage-specific probabilities and the magnitudeof the differences between stage probabilities.Strickland and Grote (1967) designed twothree-stage lotteries with equal global winningprobabilities but one with ascending probabili-ties across the three stages, and the other withdescending probabilities, and the outcome ofeach stage was presented sequentially. Theyfound that subjects were more likely to playoptional trials after the required ones if they sawwinning symbols on early stages, implying thatthe order of the stage-specific winning proba-bilities of a compound lottery affected DMs’behaviors. Ronen (1973) conducted two exper-iments with two-stage lotteries, and found thatDMs systematically preferred lotteries withhigher initial probabilities, and the preferenceswere impacted by the magnitude of the differ-ences between the two-stage probabilities (DMswere indifferent for very small differences).

Budescu and Fischer (2001) established ageneral framework to study the effect of differ-ent chance mechanisms on the valuation of lot-teries. Besides the number of stages and theordering of the stage-specific probabilities (thatthey labeled “sequence of probabilities”), theyalso tested the effects of the complexity ofstructure of lotteries, the sample space size,simultaneous versus sequential resolution of thelotteries, transparent verses opaque resolutionprocesses, and equal and unequal time distribu-tion of the various stages. They examined theseven factors separately using direct choiceswithin pairs of properly matched options, andfound that the two most prominent factors werethe number of stages and the order of stage-specific probabilities. Contrary to Slovic (1969)and Bar-Hillel’s (1973) conclusions, Budescuand Fischer found that subjects systematicallypreferred the single-stage lotteries to compoundlotteries in both domains of gains and losses. Apossible explanation is that Budescu andFischer’s manipulation of the number of stagesdid not directly make the stage winning proba-bilities seem higher. For example, in a pair oflotteries they used to test the effect of number ofstages, the one-stage lottery promised a gain if afair coin lands on “Head,” and the two-stagelottery offered the same prize if two coinstossed land on the same side. In each event, thestage probability was .50, seemingly the sameas the one-stage lottery. In terms of sequencesof probabilities, subjects preferred compoundlotteries with higher probabilities in the earlierstages in the domain of gains, and preferredthose with lower probabilities in the earlierstages in the domain of losses, consistent withStrickland and Grote’s (1967) and Ronen’s(1973) studies.

Brothers (1990) and Chung et al. (1994),however, obtained mixed results regarding theorder of stage probabilities. They tested eventcommutativity, the expectation that DMs shouldbe indifferent to the ordering of the stage prob-abilities. Brothers (1990) found that aroundone-quarter of the subjects preferred lotterieswith higher initial winning probabilities whenthey were asked to specify their degree of pref-erences within pairs of lotteries in one experi-ment, but subjects in the second experimentassigned higher mean Certainty Equivalents(CEs) to lotteries with lower initial probabilitieswhen they were asked to directly state the CEs.

110 FAN, BUDESCU, AND DIECIDUE

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In contrast, Brothers’ third experiment foundthat event commutativity held for the majorityof subjects when they used the choice-proce-dure (Bostic, Herrnstein, & Luce, 1990) to elicitsubjects’ CEs of the lotteries, and this findingwas replicated by Chung et al. (1994). Thus,this preference reversal may be because of thedifferent elicitation mechanisms (in Chung, etal.’s words, “response-mode bias”), a phenom-enon that has been studied extensively (Slovic,1995; Tversky, Sattath, & Slovic, 1988).

Aydogan, Bleichrodt, and Gao (2016) alsoobtained mixed results regarding the order ofstage probabilities. They used a choice-basedprocedure to elicit CEs, and found that—in twocases—subjects preferred the compound lotter-ies with lower initial stage probabilities to thematched compound lotteries with higher initialstage probabilities, but preferred another com-pound lottery with higher initial stage probabil-ity to the matched compound lottery with lowerinitial stage probability. Clearly, their findingsviolated event commutativity and cannot besimply attributed to response-mode bias. Thus,there are no conclusive results on the effect oforder of stage probabilities.

Kahneman and Tversky did not study com-pound lotteries per se, but the experimentalquestions they used to demonstrate the isolationeffect in Prospect Theory (Kahneman & Tver-sky, 1979) suggest violations of ROCL. Themajority of subjects preferred the lottery (4000,.20) over (3000, .25), but the preference wasreversed for the two-stage lotteries (.75, 0; .25,[1, 3,000]) and (.75. 0; .25 [.80, 4,000]; .20, 0)that can be reduced to the two simple ones,based on ROCL. Obviously, transforming sin-gle stage lotteries to compound lotteries canlead to preference reversals.

There are also numerous studies with otherfoci in which ROCL plays an important role.For example, Carlin (1992) found that violationof ROCL was one of the main drivers for sys-tematic violations of the expected utility theoryassociated with the Allais paradox (Allais,1953, 1979) and the common-ratio version ofthis paradox used by Kahneman and Tversky(1979). Another thread of studies examinesROCL in Ellsberg-like contexts that involveimprecise and/or ill-specified probabilities(Ellsberg, 1961). Segal (1987) was the first tosuggest that Ellsberg paradox could be consid-ered as a two-stage lottery. Systematic viola-

tions of ROCL were also confirmed in thesesituations (see, Abdellaoui, Klibanoff, &Placido, 2015; Bernasconi & Loomes, 1992;Halevy, 2007; Miao & Zhong, 2012). We focuson the canonical case of binary lotteries withprecise probabilities, so we do not review thesestudies in detail.

Despite considerable evidence of violationsof ROCL, there is not yet a full understandingof the factors driving them. In addition, there isno sound descriptive model proposed to accom-modate the violations. Aydogan et al. (2016)tested the reduction invariance assumption (fora detailed discussion of reduction invariance,refer to Luce, 2001) in terms of Prelec’s two-parameter weighting function (Prelec, 1998,2000). Their results were consistent with reduc-tion invariance, but not fully with ROCL.Around 40% of their subjects violated ROCL,and most of them preferred compound lotteriesover simple ones.

The first aim of this article is to comprehen-sively test the effects of different factors thatcould impact DMs’ evaluations of lotteries. Thesecond aim is to model how DMs make simpledecisions in the presence of compound lotteries.To this end we elicit monetary CEs of multiplecompound lotteries and model them in theframework of a canonical model of choice. Un-der this model, the value V of a two outcomelottery L � (x, p; y), is given by

V(L) � U(x)W(p) � U(y)�1 � W(p)�, (1)

where x and y (x � y � 0) are monetary prizes,p is the probability of getting prize x, U is autility function (mapping prizes into real num-bers) and W(p) is a probability weighting func-tion (monotonically increasing in p, with theconstraints that W(0) � 0 and W(1) � 1). If y �0 and U(0) � 0, the model simplifies to:

V(L) � U(x)W(p). (2)

This canonical model called Binary RDU (orBRDU for short; Luce, 1991; Marley & Luce,2002; Wakker, 2010) consists of the sum of theutilities weighted by decision weights and, in itsgenerality, embraces RDU, CPT (Tversky &Kahneman, 1992), original PT (Kahneman &Tversky, 1979), Birnbaum’s TAX, and even EUin the case where the probability weighting

111DECISIONS WITH COMPOUND LOTTERIES

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function is the identity function. This generalevaluation of lotteries poses an interesting chal-lenge because all of the above mentioned theo-ries are silent with respect to compound lotteriesand do not account for how the stage-specificprobabilities are aggregated and weighted. Athird aim of the article is to shed new light onthis question. We will test, for the first time,whether DMs aggregate the probabilities firstand use this aggregated probability as the argu-ment in the weighting function

V(L) � U(x)W(p*), (2a)

where p� is the probability aggregated over theS stages; or, alternatively, weight the stage spe-cific probabilities separately and then aggregatethe resulting weights,

V(L) � U(x)W*(p), (2b)

where W�(p) is the aggregated weight.We focus on DMs’ preferences over two-

outcome compound lotteries in the domain ofgains with precise probabilities in atemporalsettings (i.e., the time between the stages isimmaterial). We denote the winning probabili-ties of each stage of a compound lottery as pi(i � 1 to S). Players have pi chance of enteringinto the next stage, and (1 � pi) chance ofending the game with payoff $0. In the laststage, players either win the payoff x with prob-ability ps, or end the game with payoff $0. Theprobabilistically equivalent simple form (i.e.,the one-stage lottery) for such a compound lot-tery is (p, x; 1 � p, 0) with global winningprobability, p � �pi.

The study contributes to the existing litera-ture in several ways. First, unlike previous stud-ies that only tested one or two factors in oneexperiment, we systematically test the main ef-fects of six factors and their interactions that canlead to violations of ROCL in a fractional fac-torial design. This design leads to new insightsabout ROCL. Second we contribute to the liter-ature on modeling and evaluating the decisionson compound lotteries. Budescu and Fischer(2001) speculated on a potential model bywhich compound lotteries would be evaluated,but they did not have the data to test it. UnlikeBudescu and Fischer who used choices, we usedCEs, allowing us much more flexibility and

power in modeling and testing the fit of severalmodels (alluded to, but not tested by, Budescuand Fischer). We tackle the issue of propermodeling of compounding by embedding ourmodels in the framework of the general BinaryRank Dependent Utility (BRDU) model andmaking some reasonable assumptions aboutfunctional forms and obtaining the necessarydata to estimate individual parameters. This al-lows us to develop and compare three classes ofmodels seeking to explain the aggregation/weighting process described earlier.

We consider the modeling of this aggrega-tion/weighting process to be the primary andmost important contribution of our work to theliterature on ROCL. Ours is the first article toexplicitly consider the probability weightingprocess in an attempt to better understand thecognitive process underlying the decision mak-ing with compound lotteries.

Experimental Method

Experimental Design

Following the seminal work by Tversky andKahneman (1992) we used choice-based cer-tainty equivalents (CEs) to elicit DMs’ valua-tions of the lotteries. In each lottery playerswere informed that they would get the statedpayoff if they win (with the displayed probabil-ity); otherwise they would not win or lose any-thing. DMs provided CEs both for simple one-stage lotteries and compound lotteries.

One-stage lotteries. We created 16 one-stage lotteries with various winning probabili-ties and various payoffs. The winning probabil-ities had six levels (.09, .18, .25, .50, .75, and.90) and the payoffs had seven levels ($2, $5,$10, $15, $20, $25, and $28). They werecrossed as shown in Table 1 for the purpose ofestimation of the utility and weighting functionparameters of Equation 1. We used seven lot-teries with p � .50 to estimate the utility func-tion and six lotteries with an outcome of $20 toestimate the probability weighting function. Theother values were chosen to match the com-pound lotteries to be described in the next sec-tion.

Compound lotteries. We considered sixfactors that can influence DMs’ evaluation ofcompound lotteries and created 32 compoundlotteries by crossing these factors. The factors


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were chosen based on previous work outlined inthe introduction and they are:

(1) The number of stages S (S � 2 or 3; e.g.,Slovic, 1969), and the winning probabil-ity of the ith stage was denoted as pi, i �1, 2 or i � 1, 2, 3.

(2) The magnitude of the global winningprobability p (p � �pi) with two levels(.09 and .18) and the amount of payoffswith two levels ($20 and $10, respec-tively; e.g., Bar-Hillel, 1973). The low pwas paired with the high payoff and thehigh p was paired with the low payoff tomaintain equal EV ($1.8).

(3) The resolution mechanism of the com-pound lotteries (simultaneous or sequen-tial; e.g., Budescu & Fischer, 2001). Insimultaneous compound lotteries all un-certainties are resolved concurrently andDMs find out the results in a one-timeplay. In sequential compound lotteriesthe uncertainties are resolved one at atime, meaning that if a DM wins at stagei, he or she will continue to stage (i � 1);otherwise the game ends. DMs win thepayoff if, and only if, they win at allstages.

(4) Equality, or inequality, of the winningprobabilities of the ith and jth stages (pi,pj) of a compound lottery (e.g., compar-ing the study by Bar-Hillel (1973) andthe study by Ronen (1973)). For theequal cases pi � pj, and for the unequalones, pi � pj, for all i � j.

(5) The order of the probabilities of the var-ious stages in sequential lotteries with

unequal probabilities (ascending or de-scending; e.g., Chung et al., 1994;Ronen, 1973).

(6) The differences between the probabilitiesof adjacent stages with two levels (lowand high; e.g., Ronen, 1973). For exam-ple, the two-stage lotteries with globalwinning probability .18 can be expressedas p1 � .30 and p2 � .60, or p1 � .20 andp2 � .90. The difference between p1 andp2 in the former is lower than that of thelatter.

We created 32 different compound lotteriesbased on all combinations of these factors (seefull list in Table 2). All of them had the sameEV ($1.8). Half of those compound lotterieswere equivalent (i.e., had the same expectedvalue under the assumption of ROCL) to theone-stage lottery with winning probability .18and payoff $10, and the other half were equiv-alent to the one-stage lottery with winning prob-ability .09 and payoff $20 (see Table 1).

Thus, overall there were 48 lotteries (16 one-stage � 32 compound). To alleviate experimen-tal fatigue and increase internal validity, wereduced the number of evaluations for eachDM. All subjects evaluated the one-stage lotter-ies and only half of the compound lotteries. Thecompound lotteries were classified into four cat-egories based on the number of stages and theglobal probability of winning. This generatedsix experimental groups as shown in Table 3.Group 1 evaluated two-stage lotteries withglobal winning probability .18 and payoff $10and with global winning probability .09 andpayoff $20; Group 2 evaluated two-stage andthree-stage lotteries with global winning prob-ability .18 and payoff $10; Group 3 evaluatedtwo-stage lotteries with global winning proba-bility .18 and payoff $10 and three-stage lotter-ies with global winning probability .09 and pay-off $20; Group 4 evaluated two-stage lotterieswith global winning probability .09 and payoff$20 and three-stage lotteries with global win-ning probability .18 and payoff $10; Group 5evaluated two-stage and three-stage lotterieswith global winning probability .09 and payoff$20; and Group 6 evaluated three-stage lotterieswith global winning probability .18 and payoff$10 and with global winning probability .09 andpayoff $20.

Table 1Expected Values (EV) and Parameters of the 16One-Stage Lotteries

Payoff

Probability of winning

.09 .18 .25 .50 .75 .90

$2 — — — 1.00 — —$5 — — — 2.50 — —$10 — 1.80 2.50 5.00 — —$15 — — 3.75 7.50 11.25 —$20 1.80 3.60 5.00 10.00 15.00 18.00$25 — — — 12.50 — —$28 — — — 14.00 — —

Note. The bold values are the two one-stage lotteries usedas references for all the multi-stage lotteries.


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To assess the reliability of the DMs’ re-sponses, we replicated two one-stage lotteriesand two compound lotteries of each category.Overall, each subject evaluated 38 lotteries (16one-stage � 16 compound � 6 replications)presented in randomized order.

Subjects

We recruited 125 students at New York Uni-versity (NYU; 76 male and 49 female) via e-mail announcement. Subjects were from variousmajors and their mean age was 22 years old. Allsessions were run using Z-tree (Fischbacher,2007) at the NYU Center for Experimental So-cial Science in June 2015. Subjects were paid a$5 show-up fee plus what they won during theexperiment. Their average payment was $9.45.

Procedure

Subjects were run individually using Z-leafs(Fischbacher, 2007). After signing the informedconsent the subjects received detailed instruc-tions (see Appendix A in the online supplemen-tal material). They were also provided with ahard copy of the instructions for their reference.

At the beginning of each experimental ses-sion, each subject was randomly assigned toone of the six groups described in Table 3 andthe order of all lotteries he or she was about toface was randomized. We prevented repli-cated lotteries to appear next to each other(there were at least three other lotteries be-tween replicates). Our goal was to have atleast 20 subjects in each group, and westopped data collection when this goal wasachieved (see Table 3).

The experiment consisted of four stages. Inthe first stage, subjects evaluated half of thelotteries. In the second stage they completed adecision-making style questionnaire (Allin-son & Hayes, 1996; Scott & Bruce, 1995). Inthe third stage they evaluated the other half ofthe lotteries. In the fourth stage subjects com-pleted a numeracy questionnaire in a multiplechoice format (Cokely et al., 2012). The twoquestionnaires were included to make the ex-periment more engaging and less monotonousfor the subjects, and also to test the possibilitythat the individual differences measured bythese variables could be related to the CEs.

Each lottery was accompanied by both ver-bal and graphical representations. The verbalT

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part described the rules and the procedure, theprobability of winning and the possible priz-es. The graphical part represented the lotteryusing chance wheels with two colors (Greenand White) and a Red pointer. The probabilityof winning and the possible prizes wereshown on the wheel. For example, a one-stagelottery with .50 probability of winning $10 isshown in Figure 1; a two-stage sequentiallottery with .60 winning probability in thefirst stage, .30 winning probability in the sec-ond stage, and payoff $10 is shown in Figure2; a three-stage simultaneous lottery with .45winning probabilities in each stage and payoff$20 is shown in Figure 3.

We used a two-step procedure to elicit theCEs. In the first step, a subject was presentedwith a series of binary choices between alottery and sure amounts of money ranging

from 0 to the highest possible payoff of thelottery in intervals of $2 listed in ascendingorder. After identifying an interval of sureamounts matching the lottery, the subject wasasked to refine his or her choice to the nearest$0.1 by entering an amount in the range se-lected in the first step. To ensure the subjectsfully understood the evaluation process, theypracticed two trials of lottery evaluation be-fore starting the first stage.

To motivate subjects to provide their trueevaluations, they were informed that we se-lected four lotteries and paired them ran-

Table 3Lotteries Evaluated by the Six Groups

Group

Number of stages

One

Two Three

Samplesize

p � .18 andpayoff � $10




1 16 8 8 — — 212 16 8 — 8 — 213 16 8 — — 8 214 16 — 8 8 — 215 16 — 8 — 8 216 16 — — 8 8 20

Sample size 125 63 63 62 62 125

Note. Numbers in each cell (except the marginal) represent the number of unique lotteries.

Figure 1. Representation of the one-stage lottery with .50probability of winning $10. See the online article for thecolor version of this figure.

Figure 2. Representation of the two-stage sequential lot-tery with .60 winning probability in the first stage, .30winning probability in the second stage, and payoff $10. Seethe online article for the color version of this figure.


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domly, and that at the end of the experimentthey would play (for real money) the one towhich they assigned a higher value in eachpair.

Results

Quality of the Data

We first examined the quality of subjects’responses by calculating the reliability of eachsubjects’ evaluations of the six repeated lotter-ies, checking whether monotonicity was satis-fied, and inspecting three special patterns ofCEs.

Reliability was evaluated by three indices.First, we calculated the differences between thetwo CEs for the same lottery and the meanabsolute differences across the six repeated lot-teries for each subject. The smaller the value,the more consistent the subject’s responseswere. The mean of the individual mean absolutedifference between the repeated CEs was $1.57(SD � 1.43).1 Only 12 subjects’ mean absolutedifferences exceeded $3.6.2 Second, we calcu-lated the signs of these differences and summedthem within subjects. The absolute sum of thesigns ranges from 0 to 6, and the closer thevalue is to 0, the more consistent the subject’sresponses were. The mean of the absolute sumof the signed differences was 1.36 (SD � 1.33).Only three subjects’ absolute sum of signs was

5 or 6. Third, we calculated the chance-corrected coefficient of identity (Zegers, 1986).3

The coefficient ranges from �1 to 1. The closerit is to 1, the more similar the repeated CEs forthe same lottery were. The median within sub-ject coefficient of identity was .96 (SD � .11),with 45 subjects having coefficients of identitybelow .30. Generally, reliabilities of most sub-jects’ responses are good, and we use the meanof the two repeated CEs in the following anal-ysis.

To examine monotonicity, we calculated twosets of Kendall’s rank correlations, �, for theone-stage lotteries: one for the six lotteries withfixed payoff ($20) and one for the seven lotter-ies with fixed winning probability (.50). Mono-tonicity implies that the CEs increase as a func-tion of the winning probabilities, and as afunction of payoffs, so � should be positive and,ideally, close to 1. We used .60 as the thresholdfor Kendall’s rank correlations �. This valuecorresponds to a probability of concordance of.80. The mean � value for the one-stage lotterieswith winning probability .50 was .93 (SD �.11), and only two subjects’ � values were be-low .60; the mean � value for the one-stagelotteries with payoff $20 was .74 (SD � .33),and 33 subjects were below .60.

We considered certain CEs to be of specialinterest because high proportions of those val-ues may indicate that the subjects were strategicin their performance. For example, to completethe experiment quickly, a subject could alwaysuse the payoffs of the lotteries as CEs, a heu-ristic that DMs would rarely employ in real-life.Similarly, the minimal value allowed ($0.1) andthe maximum value allowed (Payoff � $0.1; ifa subject always prefers to play the lottery) werealso considered as special CEs. Twenty-four

1 We counted the number of risk attitude reversals—cases where two CEs assigned to that same lottery implydifferent risk attitudes—among replicated lotteries for eachsubject. The rate of risk attitude reversal is very low: 96 ofthe 125 subjects (77%) had identical risk attitudes for allreplicated lotteries, and the mean number of reversals was0.37 out of 3 (12.2% rate).

2 We used $3.6 as the threshold for mean absolute dif-ference, because it was twice of the EV ($1.8) of all thereplicated lotteries. Although it was a little arbitrary, it is agood limit in the absence of recognized gold standard.

3 This coefficient is more appropriate (and more strin-gent) than the Pearson correlation coefficient, because allthe CEs are on a ratio scale with a common 0.

Figure 3. Representation of the three-stage simultaneouslottery with .45 winning probabilities in each stage andpayoff $20. See the online article for the color version ofthis figure.


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subjects had at least half of their CEs of theone-stage lotteries equal to one of the specialCEs and 20 subjects had at least half of theirCEs of the compound lotteries equal to one ofthe special CEs.

We counted the number of criteria (i.e., thethree indices for reliability, two sets of Kend-all’s rank correlations for monotonicity, and thethree types of special CE responses) with valuesbelow the selected thresholds for each subject.Most of the subjects’ responses were consistentin most respects, but there were 39 subjects whohad at least two criteria below the thresholds,and 17 subjects who had at least three criteria.To test the robustness of our analyses we ran allthe following analyses on the full sample andtwo subsamples, one excluding the 17 subjectswho failed at least three criteria and one ex-cluded the 39 subjects who failed at least twocriteria. The results on the two selected sub-samples were consistent with the results on thefull sample, so we present the results for the fullsample in the text.

Validations of ROCL

We first validated the existence of the sys-tematic violations of ROCL. Let CEmk representthe CE of the mth compound lottery (m � 1 to32) as evaluated by subject k (k � 1 to 125), andlet CEone_stagek

represent the CE of a one-stagelottery evaluated by subject k. Depending ongroups, CEone_stagek

either refers to the one-stagelottery with winning probability .18 and payoff$10, or the one-stage lottery with winning prob-ability .09 and payoff $20. Let

Dif fmk � CEmk � CEone_stagek(3)

be the signed difference between the DMs’ CEsof the compound lotteries and their CE(s) of thecorresponding one-stage lottery (i.e., the onewith the same global winning probability andpayoff). This difference compares the com-pound lotteries of every DM to his or her singlestage CE that already reflects one’s baseline riskattitude, so all nonzero differences reflect thenet effects of the compounded format. Table 4presents the distribution of the signs of thesedifferences for each of the 32 compound lotter-ies. A negative difference indicates preference(higher CE) of one-stage lotteries, a positive

difference indicates preference of compoundlotteries, and zero indicates indifference.4

We tested the significance of differencebetween the proportion of preferences ofcompound lotteries and the proportion ofpreferences of the one-stage lotteries. Theindifference cases (i.e., equal CEs) were ex-cluded from these tests. Overall, more thanhalf (56%) of the CEs were higher for the com-pound lotteries, and only about one-fifth (19%)had higher CEs for the equivalent one-stagelotteries, indicating systematic violations ofROCL. In addition, 18 of the 32 tests (56%)were significant at the .002 significance level(we use the Šidák adjustment that preserves anoverall 0.05 level). The preference for com-pound lotteries was more pronounced for thethree-stage lotteries (12/16 compared with 6/16for the two-stage lotteries) and for those lotter-ies with p � .18 to win $10 (13/16 comparedwith 5/16 for those with p � .09 to win $20). Infact the difference is significant for all eighttests involving the three-stage lotteries offeringp � .18 to win $10 but only one case fortwo-stage lotteries promising p � .09 to win$20. In addition, compound lotteries with equalstage probabilities were more preferred than thecompound lotteries with unequal stage proba-bilities (6/8 compared with 12/24).

Testing the Factors Affecting Violationsof ROCL

We tested the significance of the six factorsand their interactions using repeated measuresanalysis of variance (ANOVA), assuming com-pound symmetry and with the Kenward-Rogercorrection for degrees of freedom. We fit themodel to all lotteries with EV � $1.8 (includingboth one-stage lotteries and all compound lot-teries), and specified 33 unique contrasts re-flecting the effects of the various relevant fac-tors and all their interactions. The contrastmatrix is shown in Appendix B in the onlinesupplemental material. We set the significancelevel of individual contrasts at .002, accordingto the Šidák adjustment.

4 If the absolute value of diff was smaller than or equal to$0.1, we inferred that the DM was indifferent between thecompound lotteries and the one-stage lotteries. This thresh-old was based on the fact that the CEs were only determinedto $0.1 resolution.


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Tab

le4

Dis

trib

utio

nof

the

Sign

sof

Dif

fere

nces

Bet

wee

nC

Es

ofC

ompo

und

and

One

-Sta

geL

otte

ries

and

Sign

ifica

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Tes

tfo

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nsby

Fac

tors

Num

ber

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ages

and

payo

ffR

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(equ

ality

and

orde

r)D

iffe

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twee

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ilitie

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1(p

refe

ron

e-st

age)

0(i

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nt)

1(p

refe

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und)

Tes

tif

prop

ortio

n�

.50

Tw

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with

payo

ff$1

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alN

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1339

3.96

�

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qual

Low

1016

373.

94�

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h14

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2.89

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entia

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qual

No

1014

394.

14�

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Hig

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2225

1.41

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cend

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621

364.

63�

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3.13

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1815

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33�

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We found six significant contrasts: (a) themean CE of the compound lotteries (meanCE � 7.11) was significantly higher than themean CE of the one-stage lotteries (mean CE �5.78; F(1, 2092) � 61.21, p � .001; d � 0.52);5

(b) the mean CE of the two-stage lotteries(mean CE � 7.14) was significantly higher thanthe mean CE of the one-stage lotteries (meanCE � 5.78; F(1, 2097) � 40.99, p � .001; d �0.43); (c) the mean CE of the three-stage lotter-ies (mean CE � 7.09) was significantly higherthan the mean CE of the one-stage lotteries(mean CE � 5.78; F(1, 2097) � 66.46, p �.001; d � 0.54); (d) the mean CE of all thelotteries with global winning probability .09 andpayoff $20 (mean CE � 8.65) was significantlyhigher than the mean CE of all the lotteries withglobal winning probability .18 and payoff $10(mean CE � 5.42; F(1, 2172) � 546.28, p �.001; d � 1.56); (e) the mean CE of the com-pound lotteries with global winning probability.09 and payoff $20 (mean CE � 8.74) wassignificantly higher than the mean CE of thecompound lotteries with global winning proba-bility .18 and payoff $10 (mean CE � 5.48;F(1, 2128) � 503.45, p � .001; d � 1.50); (f)the three-way interaction between global win-ning probability/payoff, the resolution mecha-nism of the compound lotteries and the equalityof the various stages of a compound lottery wassignificant (F(1, 2093) � 20.83, p � .001; d �0.31). The relationship is displayed in Figure 4.Under both conditions, the mean CE of thesimultaneous lotteries was higher than the meanCE of the sequential lotteries. The CEs for thehigher payoffs/lower probabilities were consis-tently higher than their lower payoffs/higherprobabilities counterparts. The differences be-tween the CEs of the lotteries with equal andunequal stage probabilities are negligible inmost cases. The interaction can be attributed tothe unique pattern of CEs for lotteries with lowglobal winning probability and high payoffwhere the mean CE of the equal simultaneouslotteries (mean CE � 9.88) was higher than themean CE of the unequal simultaneous lotteries(mean CE � 8.89).

Three additional contrasts were significant atthe nominal level (.05) and we list them here toprovide a full picture of the results: (a) the meanCE of the two-stage lotteries (mean CE � 7.14)was significantly higher than the mean CE ofthe three-stage lotteries (mean CE � 7.09; F(1,

2126) � 5.79, p � .02; d � 0.16); (b) the meanCE of the simultaneous lotteries (mean CE �7.31) was significantly higher than the mean CEof the sequential lotteries (mean CE � 7.00;F(1, 2092) � 6.98, p � .01; d � 0.18);6 (c) thethree-way interaction among the compound lot-teries between the number of stages, the globalwinning probability/payoff, and the order of theprobabilities (ascending, or descending) was sig-nificant (F(1, 2092) � 5.54, p � .02; d � 0.16).

The last two columns in Table 5 include themean CEs of the lotteries with EV $1.8, listedseparately for the $10 and $20 payoffs. Clearlythese CEs are highly sensitive to the payoffs re-flecting differences between individual risk atti-tudes (see more on this later in the article), and thevariance is higher in the CEs of lotteries withpayoff $20.

To better describe the overall pattern of eval-uations induced by the various factors, as theypertain to compound lotteries in a way that isindependent of the outcomes (in $), we fitted theBradley-Terry-Luce (BTL)model to the DMsresponses to these lotteries. This is a logit modelfor pairwise comparisons (Agresti, 2002; Brad-ley & Terry, 1952):

log��mm�

�m�m�� m � �m�. (4)

On the left side of the equation, mm= denotesthe probability of lottery m being preferred overlottery m= (CEm CEm=), and m=m denotes theprobability of lottery m= being preferred overlottery m (CEm= CEm). On the right side, �mand �m= denote nonnegative scale values of lot-teries m and m=, respectively, which are esti-mated when fitting the model. When two lotter-ies have equal CEs and they are chosen withequal probability (i.e., log(0.5/0.5) � 0), theirscale values coincide. The more likely is one ofthe lotteries to be preferred over the other by the

5 The standardized mean difference (effect size) d was

computed using d � �2F�1�r�K , where r is the common

within-subject correlation (.72), F is the F values of therelevant test, and K is the sample size (125).

6 A minor difference between the results of the full sam-ple and the results of the two subsamples is that the contrastbetween the simultaneous lotteries and the sequential lot-teries were significant at the nominal � level .05 on the fullsample, but significant at the Šidák adjusted � level .002 onthe two subsamples.


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DMs, the larger the difference between the cor-responding scale values.

To fit the model we inferred the DMs’ binarypreferences and created pseudo-pairwise-choices among these lotteries by comparing theCEs for each DM.7 Then we fit the model andestimated the scale values for these 34 lotterieswith the one-stage lottery of low payoff (.18probability to win $10) as the reference cate-gory (that is assigned a value of 0 in the esti-mation). The patterns of the estimated scalevalues between high and low payoffs acrossfactors were very similar. To achieve a clearand concise graphical representation, we com-bined proportions of choice across the two pay-offs, and re-estimated scale values for the 17combined lotteries. In this case, the combinedone-stage lottery was used as the reference cat-egory. Table 5 shows the estimated scale valuesfor the combined lotteries and for the originallotteries with payoff $10 and payoff $20. ThePearson correlations between them were high(.91 between the scale values for the combinedlotteries and for the lotteries with payoff $10,and .81 between the scale values for the com-bined lotteries and for the lotteries with payoff$20).

We present the scale values of the combinedlotteries, separately by the various manipulatedfactors, in Figure 5 using different colors. Thefirst panel shows that the three-stage lotterieswere assigned higher values than the two-stage

lotteries, and both were higher than the one-stage lotteries. Scale values of the simultaneouslotteries were higher than scale values of thesequential lotteries, conditioned on the numberof stages. Scale values of the lotteries withequal stage probabilities were higher than scalevalues of the lotteries with unequal stage prob-abilities conditioned on the number of stages.Scale values of the sequentially descending lot-teries were higher than scale values of the se-quential ascending lotteries conditioned on thenumber of stages. There is no discernable pat-tern of scale values of the lotteries with variousmagnitudes of differences between the proba-bilities of adjacent stages.

The estimated scale values from the BTLmodel are mostly consistent with the findingsbased on the repeated measures ANOVA, withone noticeable exception: the three-stage lotter-ies have higher mean scale value (mean scalevalue of the combined estimates � 1.00) thanthat of the two-stage lotteries (mean scale valueof the combined estimates � 0.57), but themean CE of the two-stage lotteries (mean CE �7.14) was higher than the mean CE of three-stage lotteries (mean CE � 7.09). To explain

7 If the absolute value of the difference between two CEswas smaller than or equal to $0.1, we inferred that the DMwas indifferent between the two lotteries and assigned .50 toeach lottery. This threshold was based on the fact that theCEs were only determined to a $0.1 resolution.

Figure 4. Three-way interaction between global winning probability/payoff, the resolutionmechanism of the compound lotteries, and the equality of the various stages of a compoundlottery. See the online article for the color version of this figure.


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the inconsistency, we examined the differencesbetween CEs of the three-stage lotteries andCEs of the two-stage lotteries (CEthree_stage –CEtwo_stage) for all the matched cases and plot-ted their distribution in Figure 6. Positive (neg-ative) values indicate that a three-stage lotterywas assigned a higher (lower) CE than amatched two-stage lottery. The modal differ-ence is 0, but the figure shows that the three-stage lotteries were preferred to the two-stagelotteries 2,228 times (57%), while the two-stagelotteries were preferred to the three-stage lotter-ies only 1,672 times (43%). Therefore, the BTLscale values of the three-stage lotteries werehigher than the two-stage lotteries. However,absolute mean of the negative differences (4.07)was higher than the mean of the positive differ-ences (3.75), which suggests that the prefer-ences for the two-stage lotteries were strongerthan the preferences for the three-stage lotteries.Thus the mean CE of the two-stage lotteries washigher than the mean CE of the three-stagelotteries, as repeated measures ANOVA resultsshowed.

Parametric Estimates of Utility andWeighting Function for One-Stage Lotteries

We used the CEs of the one-stage lotteries toestimate each subject’s parameters for the value

function and weighting function. We use PT asa particular instantiation of the BRDU model(Equation 1), and we fit functional forms thatare widely used in applications of this model.8

More specifically, we use a power utility func-tion (Equation 5) and the one-parameter Prelecprobability weighting function (Equation 6).We adopted these two function forms becauseStott (2006) has shown that they “maximize theinformation extraction from participant data andthereby increase the chances of detecting a re-lationship with other measures of behavior,”and because of their simple and parsimoniousparameterization (only one parameter for eachfunction).

Utility function: U(x) � x�, (5)

Weighting function: w(p) � exp(�(�ln p)r),

(6)

The full model is shown in Equation 7 andEquation 8 (after substituting Equations 5 and 6into Equation 7). We estimated � and inEquation 8 simultaneously by least squares and

8 PT and CPT are indistinguishable for binary lotteries.

Table 5Bradley Terry Model Scale Values and Mean CEs of the Lotteries

Stages ResolutionPattern (equality

and order)Differences between

probabilities

Estimated scalevalues Mean CEs

Combined $10 $20 $10 $20

One-stage — — — 0 0 1.09 4.48 7.07Two-stage Sim Equal No .77 .75 1.97 5.09 9.22

Unequal Low .64 .94 1.61 5.35 8.82High .66 .73 1.94 5.33 9.96

Seq Equal No .54 .83 1.71 5.30 8.84Ascending Low .47 .41 1.83 4.81 9.27

High .39 .36 1.68 4.79 9.20Descending Low .56 .70 1.67 5.19 8.85

High .49 .64 1.64 5.44 8.73Three-stage Sim Equal No 1.26 1.19 2.17 6.02 9.02

Unequal Low 1.18 1.17 2.04 5.98 8.87High 1.10 1.24 1.81 6.09 7.94

Seq Equal No .90 .73 2.11 5.47 8.39Ascend Low .76 .92 1.62 5.56 8.04

High .77 .87 1.55 5.74 7.68Descend Low 1.01 1.03 1.96 5.96 8.22

High 1.05 .90 2.08 5.60 8.88

Note. CE � Certainty Equivalents.


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the estimation procedure is shown in AppendixC in the online supplemental material.9

U(CE) � U(x)w(p), (7)

CE� � x� � exp(�(�ln p)r). (8)

The joint distribution of the two parameters isplotted in Figure 7. Twenty-three subjects’ pa-rameter values (either �, or , or both) wereextreme; thus, were not plotted in Figure 7.10

The mean estimated � was 1.57 with SD �0.83, and the median estimated � was 1.28; themean estimated weighting parameter was 0.80with SD � 0.45, and the median estimated was 0.71. The two gray lines (� � 1 and � 1)divide the plane into four quadrants. There are14 subjects who have concave value functionsand overweight (underweight) small (large)winning probabilities (� � 1 and � 1), threesubjects who have concave value functions andunderweight (overweight) small (large) winningprobabilities (� � 1 and 1), and 20 subjectswho have convex value functions and under-weight (overweight) small (large) winningprobabilities (� 1 and 1). The majority(55) of subjects have convex value functionsand overweight (underweight) small (large)winning probabilities (� 1 and � 1). Sevensubjects have linear value functions and applyno weighting on probabilities (� � 1 and �1). Three more subjects have linear value func-tions (� � 1); 2 of them overweight (under-

weight) small (large) winning probabilities( � 1) and the third one underweight (over-weight) small (large) winning probabilities( 1). Of interest, we did not find any mean-ingful correlations between the individual pa-rameter estimates and the numeracy and deci-sion-making style scores (see Appendix E in theonline supplemental material).

Modeling CEs of the Compound Lotteries

In this section we use the individual levelparameters estimated from the one-stage CEs topredict the CEs of compound lotteries for eachsubject. These predictions are based on the as-

9 To validate our parameter estimation, we also fit modelsusing exponential utility function and Prelec probabilityweighting function, power utility function and symmetricneo-additive weighting function (see Chateauneuf, Eich-berger, & Grant, 2007), and exponential utility function andlinear probability weighting function. The parameters fromthe different utility and weighting functions are not on thesame scale and cannot be not be compared directly, thus wedid two analyses. First, we compared the correlations be-tween predicted CEs of the one-stage lotteries using differ-ent sets of parameters estimates as well as the root meansquared errors of those predictions, the results were highlycorrelated. Second, we conducted Fisher’s exact tests toexamine whether the proportions of subjects who were riskseeking were significantly different across the four sets ofparameter estimates, and no test was significant. Details ofthese analyses are shown in Appendix D in the onlinesupplemental material.

10 The 102 cases whose � and were within the range[0.1, 4.75] were plotted and the other 23 cases are omitted.

Figure 5. Scale values from Bradley-Terry-Luce Model (based on all lotteries) as a functionof the manipulated factors. See the online article for the color version of this figure.


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sumption that compound lotteries are exten-sions of the one-stage ones and evaluated ac-cordingly, that is, using the same functions andwith identical individual-level parameter, � and . We use this approach to compare multipleaggregation models across the S stages. Let

CEmk represent the predicted CE of the mth(m � 1 to 32)11 compound lottery by subject k,where �k and k are the estimated parametervalues for subject k based on his or her one-stage CEs. The winning probability of stage i isdenoted as pi (i � 1 to S, S � 2 or 3). Wedeveloped three classes of models based on PT.

Combine and weight. A DM k, first com-bines the stage-specific probabilities of a com-pound lottery m to obtain its effective globalwinning probability, Pm

�, and then weigh Pm� as

shown in Equation 9.

CEmk� � exp ��(�ln Pm

* )k� � xm�k, (9)

With simple algebra we can re-express CEmk:

CEmk � exp � �(�ln Pm* )k

�k� � xm. (10)

We compared six ways to calculate the effec-tive global winning probabilities, Pm

�, from thestage-specific ones. They reflect a variety ofaggregation rules:

M1: The product of the stage probabilities,Pm

� � �pmi;

M2: The arithmetic mean of the stageprobabilities, Pm

* � 1S�pmi;

M3: The geometric mean of the stage prob-

abilities, Pm* � �

S �pmi;

M4: The median of the stage probabilities,Pm

� � median{pm1, . . . , pmS};

M5: A weighted average of the stage prob-abilities that overweighs the early stagesimplying a pronounced primacy effect. Werepresent, somewhat arbitrarily, this familyof models by the simplest (linear) weight-ing form: Pm

� � (3pm1 � 2pm2 � pmk)/6 for

11 Because subjects were assigned to different groups,each subject evaluated a unique subset of the compoundlotteries.

Figure 6. Certainty Equivalents (CEs) of three-stage lotteries subtracted by CEs of two-stage lotteries. See the online article for the color version of this figure.


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three-stage lotteries, and Pm� � (2pm1 �

pm2)/3 for two-stage lotteries;

M6: A weighted average of the stage prob-abilities that overweighs the last stages,inducing a pronounced recency effect. Asin the previous section, we invoke, some-what arbitrarily, the simplest weightingform, Pm

� � (pm1 � 2pm2 � 3pm3)/6 forthree-stage lotteries, and Pm

� � (pm1 �2pm2)/3 for two-stage lotteries.

Weight and combine. A DM k, weighs thestage-specific probabilities of lottery m first(Equation 11),

w(pmi)k � �(�ln pmi)k, (11)

and then combines the weighted stage probabil-

ities into a global weight wmk. CEmk can becomputed as shown in Equation 12,

CEmk � exp � ln(wmk)

�k� * xm. (12)

We compared six ways of combing theweighted stage probabilities that mirror the ag-gregation models used in the first class:

M7: The product of the stage weights,wmk � �w(pmi)k;

M8: The arithmetic mean of the stageweights, wmk � 1

S�w�pmi�k;

M9: The geometric mean of the stage

weights, wmk � �S �w�pmi�k;

M10: The median of the stage weights,wmk � median{w(pmi)k, . . . , w(pmS)k};

M11: A weighted average of the stageweights that overweights the early stages,wmk � (3w(pm1)k � 2w(pm2)k � w(pm3)k)/6for three-stage lotteries, and wmk �(2w(pm1)k � w(pm2)k)/3 for two-stage lot-teries;

M12: A weighted average of the stageweights that overweighs the last stages,wmk � (w(pm1)k � 2w(pm2)k � 3w(pm3)k)/6for three-stage lotteries, and wmk �(2w(pm1)k � w(pm2)k)/3 for two-stagelotteries.

Anchoring. A DM, k, anchors on one ofthe stage probabilities and then applies theweighting. Unlike the first two classes thesemodels do not involve aggregation acrossstages but, instead, anchor on one of the Sprobabilities and use it as the effective prob-ability of winning. Therefore, there is nomeaningful distinction between “combine andweight” and “weight and combine.” The four

Figure 7. Distributions of � and parameters estimated from the one-stage lotteries.


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anchors we considered are based on resolu-tion order and on magnitude:

M13: The highest stage probability, Pm� �

max{pm1, . . . , pmS};

M14: The lowest stage probability, Pm� �

min{pm1, . . . , pmS};

M15: The first stage probability, Pm� � pm1;

M16: The last stage probability, Pm� � pmS,

S � 2 or 3.

Finally, we fit three baseline models that con-strain the parameters used in predicting the CEof the compound lotteries:

BM1: The EV of the lotteries (� � 1, and � 1);

BM2: The EU of the lotteries (i.e., � 1, theproduct of the stage probabilities without anyprobability weighting); and

BM3: Probability weighting (weighted prod-uct of stage probabilities) and linear utility(� � 1).

We predicted CEs of the compound lotteriesunder the 19 models and we compared the qual-ity of the models’ predictions at three levels.First, we calculated the root mean squared error(RMSE) for each DM across all the lotteries:

RMSEk � �MSEk

� 1

16* �m�1

16 (CEmk � CEmk)2 ,

(13)

and examined the boxplots of RMSEs acrossthese models.

Second, we calculated RMSE for each lotteryacross all the subjects who evaluated it, where:

RMSEm � �MSEm

� 1

N* �k�1

N (CEmk � CEmk)2,

N � 62 or 63, (14)

and examined the boxplots of RMSEs with lot-teries features (i.e., number of stages andamount of payoffs) taken into consideration.

Finally, we compared the 19 models directly,using the pairwise competition approach devel-oped by Broomell, Budescu, and Por (2011),across all 1,904 predicted CE values for eachmodel, across all DMs and lotteries.

The results of the model comparisons werenot fully consistent across the three levels ofanalysis. However, several models performedworst by all measures. For example, the EVbaseline model was always the worst or secondworst model. In the interest of brevity, we onlypresent results of 10 models representing thethree classes: Models M1, M3, M5 from theclass “first combine and then weight,” modelsM7, M9, M10 from the class “first weight andthen combine,” and models M14 and M15 fromthe “anchoring” class. We also include resultsfor two baseline models (BM2 and BM3).

Figure 8 includes the boxplots of these sub-ject-level RMSEs. Clearly, the top models wereM1, the product of the stage probabilities aseffective global winning probabilities and M14,the lowest stage probabilities as an anchor. Al-though the median RMSE of M14 (medianRMSE � 2.58) was higher than the medianRMSE of M1 (median RMSE � 2.10), thedispersion of M14 was much smaller than M1.Twenty-three DMs were best described by M1,the product of the stage probabilities as effec-tive global winning probabilities, 20 subjects bymodel M14, the lowest stage probability as ananchor, and 14 subjects by the baseline modelwith power utility function and no weighting.Frequencies for all other models were less than10. Note that none of the “first weight and thencombine” models fits many individuals, al-though M7, the product of the weighted stageprobabilities, does quite well overall.

Figure 9 shows the boxplots of RMSE oflotteries of four classes of lotteries, defined bythe number of stages and their payoffs, sepa-rately, only for the 4 best models based onanalysis of RMSE across subjects.12 Overall

12 We also plotted RMSE across the four classes oflotteries for all the 19 models and for the selected subset of10 models. In both plots, the best-fitting model was M14that relies on the lowest stage probability, and the worstmodel was the baseline model with Prelec weighting func-tion and linear utility function. The performances of othermodels were very similar and also depended on the types oflotteries. Thus, for simplicity and clear comparison, we onlyplotted the four best models in Figure 8.


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(based on median RMSE and the dispersion ofRMSE) the best-fitting model was M14 thatrelies on the lowest stage-specific probability asan anchor, followed by M1 that uses the productof stage probabilities as effective global proba-bilities. In all cases, the CEs of the lotteries withlow payoff ($10) and higher probabilities (p �.18) were better predicted than CEs of the lot-teries with high payoff ($20) and lower proba-bilities (p � .09), and CEs of the two-stagelotteries were better predicted than CEs of thethree-stage lotteries.

The full pairwise model “tournament” reliesonly on those cases where the models makedistinct predictions and excludes cases where apair of models made identical predictions.13

The proportion of distinct predictions for eachpair of models (row model and column model)is shown in Table 6.

For each pair of compound lotteries wecounted the number of times the predicted CEby model t was closer to the observed CE thanthe predicted CE by model t’, as well as thenumber of reversed patterns. Each cell in Table7 (except the last column) is the ratio of thenumber of cases of better predictions by the rowmodel compared with the number of better pre-dictions by the column model. If the ratio islarger than 1, the row model “beats” the columnmodel; if the ratio is 1 the two models fit equallywell; otherwise, the column model beats therow model. The last column in Table 7 repre-

13 Two predicted CEs of the same compound lottery in apair of models were counted as “identical” if the differencebetween them was equal to, or less than, $0.1, because theresolution of the CEs was $0.1.

Figure 8. Box plots of the root mean square error (RMSE) of various models based onindividual subjects. The models compared: M1: the product of the stage probabilities; M3: thegeometric mean of the stage probabilities; M5: a weighted average of the stage probabilitiesthat overweighs the early stages; M7: the product of the stage weights; M9: the geometricmean of the stage weights; M11: a weighted average of the stage weights that overweights theearly stages; M14: the lowest stage probability; M15: the first stage probability; BM2: the EUof the lotteries; BM3: probability weighting. See the online article for the color version of thisfigure.


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sented the weighted geometric means (WGMs)of the ratios across rows. Each ratio wasweighted by the number of unique predictionsmade by the row and column models, so thepairs that make many (few) distinct predictionsare over (under) weighted in this calculation.14

Based on WGM, the top four models were:M14 (lowest stage probability as effectiveglobal winning probabilities), M1 (product of

the stage probabilities as effective global win-ning probabilities), M3 (geometric mean of

14 We presented results for 10 models in Table 6. How-ever, the WGM was calculated across all the 19 models:

WGMt � ��119 rt

nt�1⁄�119nt, where rt is the ratios (t � 1 . . . 19)

in a given row of the model comparison matrix and nt is thenumber of distinct predictions for each ratio.

Figure 9. Box plots of the root mean square error (RMSE) of lotteries classified by payoffand number of stages for four selected models. M1: the product of the stage probabilities; M7:the product of the stage weights; M14: the lowest stage probability; BM2: the EU of thelotteries. See the online article for the color version of this figure.


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stage probabilities as effective global winningprobabilities), and M11 (overweighting proba-bilities of early stages). The direct pairwisecomparisons of the four models are consistentwith the global ranking of the first three models.However, overweighting stage probabilities ofearly stages model (M11) outperformed geo-metric mean of stage probabilities (M3) in pair-wise comparison (the ratio of the former to thelatter is 1.05).

To complement this descriptive table of ef-fect sizes, we conducted one-tailed sign tests totest whether the row model was significantlybetter (or worse) than the column model. Weused the Šidák adjustment to preserve an overall0.05 error rate, so the significance level for eachpairwise test was set at a very conservative.0003. In Table 8, the row models are sorted in

a descending order based on their WGMs. Theperformance of M14 and M1 was not signifi-cantly different (p � .25), but both models weresignificantly better than all other competingmodels (all p � .0001).

In general, the “first combine and thenweight” outperformed the “first weight and thencombine” models: the pairwise model compar-isons showed that the three selected modelsfrom the first class outperformed the three se-lected models from the second class in sevenout of the nine comparisons (five of them beingsignificant). Furthermore, the mean ratios(weighted by the number of comparisons)across the nine comparisons (1.24) favors “firstcombine and then weight.”

According to all approaches M14 and M1are the best fitting models. M14 assumes that

Table 6Proportion of Distinct Predictions Made by the Row Model Compared With the Column Model

Model M1 M3 M5 M7 M9 M11 M14 M15 BM2 BM3

M1: PProd 1.00 .97 .98 .85 .98 .97 .92 .94 .84 .88M3: PGeoMean .97 1.00 .57 1.00 .42 .76 .73 .73 1.00 .99M5: POverWEarly .98 .57 1.00 1.00 .67 .64 .73 .73 1.00 1.00M7: WProd .85 1.00 1.00 1.00 1.00 1.00 .98 .99 .70 .88M9: WGeoMean .98 .42 .67 1.00 1.00 .81 .73 .73 .99 1.00M11: WOverWEarly .97 .76 .64 1.00 .81 1.00 .84 .85 .99 .99M14: Min .92 .73 .73 .98 .73 .84 1.00 .25 .98 .99M15: FirstStage .94 .73 .73 .99 .73 .85 .25 1.00 .99 .99BM2: BaseU .84 1.00 1.00 .70 .99 .99 .98 .99 1.00 .87BM3: BaseW .88 .99 1.00 .88 1.00 .99 .99 .99 .87 1.00

Note. Color Coding of comparisons: “first combine and then weight” vs. “first weight and then combine”; “first combineand then weight” vs. “anchoring”; “first weight and then combine” vs. “anchoring.”

Table 7Ratio of the Number of Cases Where the Predictions of the Row Model Were Closer to the Observed CEand the Number of Cases Where the Column Model Was Closer

Model M1 M3 M5 M7 M9 M11 M14 M15 BM2 BM3 WGM

M1: PProd 1.00 1.38 1.52 1.79 1.42 1.22 .97 1.31 1.82 4.36 1.64M3: PGeoMean .72 1.00 1.16 1.05 2.40 .95 .48 1.03 1.02 1.25 1.50M5: POverWEarly .66 .86 1.00 1.00 1.12 .69 .43 1.04 .92 1.16 1.12M7: WProd .56 .95 1.00 1.00 .97 .79 .65 .88 .64 2.04 1.04M9: WGeoMean .71 .42 .90 1.03 1.00 .77 .47 1.01 .97 1.20 1.29M11: WOverWEarly .82 1.05 1.44 1.26 1.30 1.00 .73 1.40 1.14 1.52 1.40M14: Min 1.03 2.07 2.34 1.53 2.11 1.37 1.00 4.57 1.40 1.93 1.97M15: FirstStage .76 .97 .96 1.14 .99 .71 .22 1.00 1.04 1.46 1.10BM2: BaseU .55 .98 1.08 1.55 1.03 .88 .71 .96 1.00 2.25 1.16BM3: BaseW .23 .80 .86 .49 .84 .66 .52 .68 .45 1.00 .77

Note. CE � Certainty Equivalents; WGM � Weighted Geometric Mean across the comparisons involving the row modelwith all 18 other models. Color Coding of comparisons: “first combine and then weight” vs. “first weight and then combine”;“first combine and then weight” vs. “anchoring”; “first weight and then combine” vs. “anchoring.”


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the DMs anchor on the lowest stage probabil-ity and then apply the weighting function, andM1 assumes that DMs first calculate the prod-uct of the stage probabilities and then use thisas the argument in the weighing function.Figure 10 presents the deviations of the pre-dictions of M1 and M14 from the actual CEs(judged CEs subtracted from model predic-

tions) across all the subjects. The box-plot atthe top of the graph shows that, in most cases,the CEs were too high, compared with theobserved CEs, whereas the box plot at theright of the graph shows that prediction basedon the lowest probability are, approximately,symmetrically distributed around 0. Recallthat these results are individualized, that is,

Table 8p-Value of the One-Tailed Sign Tests Comparing the Row Model and the Column Model

Model M14 M1 M3 M11 M9 BM2 M5 M15 M7 BM3

M14: Min — .250 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001M1: PProd .250 — �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001M3: PGeoMean �.0001 �.0001 — .190 �.0001 .370 .010 .290 .160 �.0001M11: WOverWEarly �.0001 �.0001 .190 — �.0001 .002 �.0001 �.0001 �.0001 �.0001M9: WGeoMean �.0001 �.0001 �.0001 �.0001 — .280 .030 .470 .270 �.0001BM2: BaseU �.0001 �.0001 .370 .002 .280 — .040 .220 �.0001 �.0001M5: POverWEarly �.0001 �.0001 .010 �.0001 .030 .040 — .260 .470 �.0001M15: FirstStage �.0001 �.0001 .290 �.0001 .470 .220 .260 — .003 �.0001M7: WProd �.0001 �.0001 .160 �.0001 .270 �.0001 .470 .003 — �.0001BM3: BaseW �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 �.0001 —

Note. WGM � Weighted Geometric Mean. The models are listed according to their overall fit, as determined by theirWGMs (see last column of Table 7). Color Coding of comparisons: “first combine and then weight” vs. “first weight andthen combine”; “first combine and then weight” vs. “anchoring”; “first weight and then combine” vs. “anchoring.”

Figure 10. Comparison of deviations (judged Certainty Equivalents, CEs, subtracted frommodel predictions) of the two best fitting models M1 (the product of the stage probabilities)and M14 (the lowest stage probability).


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they adjust for individual differences in riskattitudes and probability weighting, so theyconfirm that people underestimate the effectof compounding.

Discussion

Consistent with the literature, we observedsystematic violations of ROCL. About one thirdof our subjects were indifferent between theone-stage lotteries and their compounded coun-terpart, but the majority of the subjects sethigher CEs for the compound lotteries suggest-ing that they preferred them. Repeated measuresANOVA and scale value estimates of the BTLmodel confirmed the conclusion that the major-ity of subjects favored the compound lotteries.This finding is consistent with Slovic’s (1969)and Bar-Hillel’s (1973) studies, and conflictswith Budescu and Fischer (2001). Because thelotteries being compared had equal outcomesthe differences can be attributed to the proba-bilities and in our design, the stage-specificprobabilities of the compound lotteries werealways higher than the probabilities of the sim-pler one-stage lotteries.

To maintain equal EV ($1.8), we paired thelow p with the high payoff and the high p withthe low payoff. The CEs of high payoff lotteries(p � .09 to win $20) were significantly higherthan the CEs of high probability lotteries (p �.18 to win $10). Clearly, the payoffs had aprominent effect on DMs’ evaluations. This pat-tern is well documented and typically attributedto the scale compatibility hypothesis (Slovic,1995; Slovic, Griffin, & Tversky, 2002; Tver-sky et al., 1988), and it is remarkable that itpersists even when using the (two-step) choiceprocedure. At the same time, overweightingsmall probabilities may also exacerbate the CEdifferences between them.

The repeated measures ANOVA contrastsshowed that the mean CE of the two-stage lot-teries was higher than the mean CE of thethree-stage lotteries (at the significance level of.05), but the BTL models showed that the meanscale value of the three-stage lotteries washigher than the mean scale value of the two-stage lotteries. This can be explained by exam-ining the differences between the CEs of thethree-stage lotteries and two-stage lotteries ofall matched cases: we found that there weremore preferences of the three-stage lotteries to

the two-stage lotteries, but the magnitude of thepreference of the two-stage lotteries over thethree-stage lotteries were higher, which ex-plains the inconsistency. In addition, because ofthe constraint of equal EV across all compoundlotteries, the difference between the stage prob-abilities of the two-stage lotteries and the three-stage lotteries were not very large. For example,the probabilities of the two-stage lottery (p1 �.30, p2 � .60, payoff � $10) are quite similar tothe equivalent three-stage lottery (p1 � .45,p2 � .50, p3 � .80, payoff � $10), and the twoare hard to differentiate.

Consistent results were found for the resolu-tion mechanism of the compound lotteries. TheCEs and the scale values of the BTL model forthe simultaneous lotteries were higher thanthose of the sequential lotteries conditioned onnumber of stages, and the repeated measuresANOVA contrast was significant at nominallevel of .05. Simultaneous lotteries were lesscomplex than their probabilistically equivalentsequential lotteries, and the DMs seem to preferthe simple resolution mechanism (see also Bu-descu & Fischer, 2001).

Two other factors—the (in)equality of thestage-specific probabilities and their ordering—have relatively clear patterns, in terms of theirCEs and the scale values under the BTL model,but the corresponding contrasts were not signif-icant. Overall, the DMs tend to prefer lotterieswith equal probabilities at all stages, over lot-teries with unequal probabilities. This couldalso be interpreted as a preference for a simplerresolution process over a more complicated one.It is not surprising that a majority of DMsassigned higher evaluations to these lotteries,but the magnitude of the preference is not largeenough to reach significance. Finally, there is aweak preference of the descending sequentiallotteries to ascending sequential lotteries, whichis consistent with the conclusions of Budescuand Fischer (2001); Ronen (1973); Stricklandand Grote (1967); and the typical violations ofevent commutativity (Chung et al., 1994).

We did not find any systematic patterns orsignificant effects for the magnitude of differ-ences between the stage-specific probabilities.The constraint that EVs of all compound lotter-ies be equal limited our ability to manipulatethis factor, so we cannot rule out the possibilitythat larger differences involving very high


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and/or very low probabilities, would affectDMs’ behavior.

We estimated parameters for the utility andthe weighting functions, at the individual level,based on the CEs of the simple one-stage lot-teries. A unique feature of our study was the useof these parameters to predict the CEs of thecompound lotteries under a large number ofmodels that varied with respect to the way inwhich the stage-specific probabilities are inte-grated in the process of forming the eventualCEs. Much of the empirical work on ROCLprecedes the publication of PT and, interest-ingly, the impact of nonlinear weighting ofprobabilities was never considered in this con-text. Thus, a contribution of this article is theincorporation of the weighting process as part ofthe explanation of the observed empirical regu-larities. The models we considered can be clas-sified according to the implied process—firstcombine and then apply the weighting function,first apply the weighting function and then com-bine, or simply anchor on a single stage-specificprobability—and the specific combination func-tion within each class.

We compared the quality of all models inthree levels: RMSE for each DM across all thelotteries, RMSE for each lottery across all thesubjects, and the pairwise competition ap-proach. Among all the models used to predictDMs’ CEs of compound lotteries, models M14(that assumes that DMs anchor on the loweststage probability and then apply the weightingfunction) and M1 (that assumes that DMs usethe product of the stage probabilities as effec-tive global probabilities and then apply theweighting function) in general are the best-fitting models based on the three levels of anal-ysis. Note that there is no single dominantmodel that can predict most of the subjects’evaluations of compound lotteries.

Model M14 was superior to all the othermodels in the pairwise competitions, and wasthe best model for 20 individuals. The successof this model implies that even DMs with min-imal knowledge of probability theory recognizethat the global winning probability cannot ex-ceed any of the stage-specific probabilities, andthey use the lowest stage probability as an ap-proximate anchor that does not require anycomputational effort.

Among the models that assume a computa-tional aggregation across the S stages, the “first

combine and then weight” family overall out-performed the “first weight and then combine”models, presumably because it is cognitivelyeasier, requires fewer mental operations(Shugan, 1980), and more “natural.” In retro-spect one could claim that, although PT makesno explicit reference to compound lotteries, thisclass of models is better aligned with PT. Kah-neman and Tversky (1979) suggest that weight-ing can be inferred from choices among pros-pects, and that the weights measure the impact(not only the likelihood) of the events on theglobal desirability of the prospect. In otherwords, the weights are necessary and invokedonly when they are associated with specificoutcomes, so there is no need to calculateweights for the stage-specific probabilities thatare associated with intermediate, predecision,stages. It is interesting that of all the models inthis class, the best one (M1), which also hap-pens to fit the largest number of individualsubjects, relies on the normative aggregationrule of the S probabilities, their product.

We used a fixed EV for all compound lotter-ies. Future experiments should consider variousEVs, so that the effect of global winning prob-abilities and the effect of payoffs can be testedseparately. We used a choice-based CE elicita-tion. Future experiments can compare differentelicitation approaches, such as binary choicesand rating. Systematic comparisons across dif-ferent elicitation approaches are necessary toestablish the robustness of the effects.

We focused on binary lotteries in the domainof gains. The study can be extended into thedomain of losses, and lotteries with impreciseprobabilities. It is interesting to test whether ourfindings (i.e., the effects of the six factors, an-choring is the best fitting model for computingcompound CEs, and “combine and weight” willoutperform “weight and combine”), will hold inthe domain of losses, and for lotteries withimprecise probabilities.

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