Effects of Vague Probabilities and of Vague Payoffs on Preference: A Model Comparison Analysis

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Journal of Mathematical Psychology � MP1111

journal of mathematical psychology 40, 130�140 (1996)

Effects of Vague Probabilities and of Vague Payoffs on Preference:A Model Comparison Analysis*

Claudia Gonza� lez-Vallejo

University at Albany, SUNY

Alberto Bonazzi

University of North Carolina, Chapel Hill

and

Andrea J. Shapiro

University of North Carolina, Chapel Hill

Preference orders over gambles having vague probabilities orvague payoffs were investigated. Risk and worth judgments were alsostudied in these two vague conditions. A trade-off and a lexicographicmodel were compared via a new measure of model performance. Thismeasure is defined as the mean proportion of all possible preferenceorders satisfying dominance, D, that are more distant from model, M,than are subjects' preferences, S. Results showed that both modelsdescribe preference data well. Furthermore, the usual aversion to vagueprobabilities was found, and a similar phenomenon occurred withvague payoffs. Analyses of Risk and worth judgments showed thatvague gambles were evaluated as their nonvague counterparts andindicated that less risk and more worth are preferred.. ] 1996 Academic

Press, Inc.

This study focuses on two separate but highly interde-pendent issues of much importance to behavioral decisionresearch. One issue concerns the effects of vagueness onpreference orders; the other concerns the testing of deter-ministic choice models as predictors of behavior. In thefirst case we compare the effects of vaguely representedprobability and payoff information on choices and ratingsof gambles. In the second, we introduce and test two

deterministic choice models via a new evaluation procedure.Because the topics flow out of two different research tradi-tions, we introduce them separately.

1. VAGUENESS EFFECTS

Decision making under vagueness (commonly referred toas ambiguity or uncertainty) has been a topic of concern totheorists at least since Knight (1921) distinguished betweensituations of risk, in which all the relevant event prob-abilities are known, and situations of uncertainty, in whichthey are not known. A clear demonstration of this distinc-tion is due to Ellsberg (1961), who showed that people'schoices over positively valued gambles depend on whetherthe gambles' probabilities are specified precisely (e.g. 1

3) orvaguely (e.g., between 0 and 2

3). Respondents typically avoidvague gambles and therefore produce preference patternsthat are inconsistent with subjective expected utility (SEU).

Following Ellsberg, other studies have shown similarvagueness effects in pricing and choosing lotteries (Becker 6Brownson, 1964; Curley 6 Yates, 1985, 1989; Einhorn 6Hogarth, 1986), judgments of uncertainty based on infer-ences (Einhorn 6 Hogarth, 1985), insurance decisions(Hogarth 6 Kunreuther, 1984), etc. (For a comprehensivereview of this topic, see Camerer 6 Weber, 1992.) Vague-ness effects have also been shown to depend on the size ofthe probability interval and on its location within the prob-ability continuum. For example, vagueness aversion forpositively valued gambles is stronger with high than withlow probabilities (Curley 6 Yates, 1985; Curley 6 Yates,1989), and more aversion occurs with wider ranges (Kahn6 Sarin, 1988).

article no. 0012

1300022-2496�96 �18.00

Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

Correspondence and reprint requests should be sent to ClaudiaGonza� lez-Vallejo, Center for Policy Research, University at Albany,SUNY, Milne 300, 135 Western Ave., Albany, NY 12222. E-mail: ccg15�cnsvax.albany.edu.

* The authors thank Sema Barlas, Colin Camerer, Thomas Stewart,James Townsend, Thomas Wallsten, Elke Weber, and two anonymousreviewers for helpful comments and suggestions, and John Hughes, DavidJohnson, and Elisa Tsao for their programming and�or statisticalassistance. This research was supported by NSF Grant BNS8908554.Alberto Bonazzi was sponsored by the Centro Nazionale Delle Ricerche,Rome, Italy.

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In this study, we extend Ellsberg's paradigm to investigatedecision making when payoffs also constitute vague quantities.In general, vague situations are those in which the distributionof an event's outcomes is not known; in this study, outcomesare probabilities or payoffs. Although by definition probabilityis an index of outcome vagueness, it might be useful todistinguish when a distribution of probabilities is unknownversus when a distribution of payoffs is unknown, as thesetwo situations might lead to different evaluation strategies. Toexemplify, suppose one gamble offers a definite 8.75 with aprobability that lies between .15 and .35, and a second gambleoffers an amount between 8.65 and 8.85 with a definite prob-ability of .25. From a subjective expected utility point of view,the corresponding expected utilities of these two options areequivalent only if one assumes that values within a range aresymmetrically distributed around the midpoints of bothprobability and payoff intervals. Under most evaluationrules, however, the options induce different preferences. Forexample, if one desires to minimize the probability of gettingthe worst possible outcome (80), the second gamble ispreferable because it offers 80 with a fixed probability of .75while the first gamble offers 80 with a probability of up to .85.By similar reasoning, the first gamble is preferable if thecriterion is to maximize the probability of receiving a positiveoutcome. Thus, although a probability is a measure of payoffvagueness (Camerer 6 Weber, 1992), we believe it is useful todistinguish between vague probability and vague payoffsituations in order to learn how people deal with differentkinds of imprecise information. In the following section thisidea motivates the formal development of a choice model.The model treats vagueness as a contextual variable thatdifferentially affects probability and payoff evaluation.

Reference to prior research is necessary at this point toclarify the notion of vague payoffs. Preferences over gambleswith outcome ranges have been studied by other researchers(Lopes, 1984; Schoemaker 1989), but the concept of outcomevagueness has not been addressed in the same manner as thenotion of vague probabilities. Lopes' work, for example,focused on preferences over gambles having payoff ranges, butthe options were not vague because the outcome distributionswere precisely specified. Similarly, Schoemaker studied aver-sion for uncertain outcomes versus uncertain probabilities, butagain there was no vagueness, because he provided subjectswith probability distributions of payoffs and of probabilities.Our study distinguishes vague probability and vague payoffinformation, defining both situations in terms of unknowndistributions of probability and of payoff, respectively.

2. MODELING CHOICE BEHAVIOR

2.1. The Delta Choice Model

Curley 6 Yates (1989) studied several choice (ranking)models in which vagueness was operationally defined as the

FIG. 1. A decision maker would have to trade off the higher probabil-ity of receiving the larger gold bar against the lower probability of receivingthe smaller gold bar.

width of a probability interval. They tested 6 three-factoralgebraic models1 and concluded that none was anadequate descriptor of performance. Their conclusiondepended in particular on the finding that considerablevagueness seeking occurred with low probabilities, whilevagueness aversion was present with high probabilities.That is, probability behaved as a sign-dependent factor notassumed by any model.

In the present study, we introduce a model of choicebehavior that can account for the described vaguenesseffects without assuming sign dependency. The rule is aspecial case of the $-strategy developed by Bonazzi (1991),which is modeled via an algebraic trade-off preferencestructure. We do not derive the properties of a trade-offpreference structure here (see Bonazzi, 1991, for deriva-tions), but we note that the structure (A, �), where A is aset of two-dimensional objects and � is a relationship in A,satisfies the following axioms: asymmetry, independence,one-dimensional transitivity, monotonicity, restrictedsolvability, interval independence, and one-dimensionalrepresentability in the set of Real numbers (Re). This systemof axioms is sufficient to represent preferences via two func-tions $i : Ai_Ai � Re, each monotone along its sections.

The delta choice model is a special case of the $-strategyand assumes that people make proportional trade-offsbetween probabilities and payoffs. An example described byBonazzi exemplifies this idea in a situation without vague-ness. Imagine you must decide between the options in Fig. 1,where two bars of gold are offered with probabilities definedby the shaded areas of two spinners. Because the larger spin-ner area is associated with a smaller piece of gold, while thesmaller spinner area is associated with the larger gold bar,you want to determine the trade-off. An easy strategy is to

131VAGUE PROBABILITIES AND PAYOFFS

1 The three factors were labeled P (for probability), U (for utility), andA (for ambiguity). The models tested were as follows: statistical decisiontheory (P V U); lexicographic (P V U, A); distributive ([P+A] V U); dis-tributive (P V [U+A]); dual-distributive (P V U+A); and multiplicative(P V U V A).


estimate how many times the smaller gold bar is containedin the bigger one, or, equivalently, estimate the relativeproportional difference between the bars. You can do thesame with the spinners. The difference of these two com-parisons is a measure of the trade-off.

Stating the model formally, let Ai=(mi , pi) andAj=(mj , pj) be two gambles, where mi and mj are payoffs tobe obtained with probabilities pi and pj , respectively. Letmi>mj>0 or mi<mj<0, and pj> pi ; then Ai is weaklypreferred to Aj (i.e., Ai � Aj) iff

f (mi)& f (mj)f (mj)

&%g( pj)&g( pi)

g( pi)�0, (1)

where f (m) and g( p) are the subjective values of the numeri-cally expressed dimensions, and % is the weight given to theprobability trade-off relative to the payoff trade-off. For%=1, the model reduces to SEU. The model never violatesdominance, but does allow realistic SEU violations, such asintransitivity when %{1 (see Appendix). In vague situa-tions (i.e., either vague probabilities or vague payoffs), themodel is transformed in two different ways. First, considerthe case where gambles' probabilities are ranges of probabil-ity values, and let ( p, v) represent a probability intervalcentered at p with width v. Equation (1) becomes2


&%gvj( pj)&gvi( pi)

gvi( pi)�0. (2)

Next, consider the case where gambles' payoffs constitutevague quantities, and let (m, v) represent a payoff intervalcentered at m with width v. In this case, Eq. (1) becomes

fvi(mi)& fvj(mj)

fvj(mj)&% V

g( pj)&g( pi)g( pi)

�0; (3)

gv( p) and fv(m) are general function forms, monotonicallyincreasing within a given level of vagueness.

An important feature of this model is that it does not treatvagueness as a factor, but rather incorporates vaguenessinto the evaluation or judgment process. Vagueness maybe conceived of as a context that changes the evaluationof probabilities or utilities, not as another independentvariable affecting responses. An advantage of this view isthat it can allow for effects of probability level withoutassuming a signed probability function. On psychologicalgrounds, it seems natural to think that rather than changingstrategies, decision makers evaluate gambles differently inprecise and vague situations.

2.2. The Lexicographic Model

We agree with Curley 6 Yates (1989) that the three-factor models they tested are inadequate descriptors ofbehavior, but we think that their data did not trulyinvalidate the lexicographic rule. Briefly, in their first experi-ment subjects rank ordered two vague and two precise gam-bles. Precise gambles were as follows: lotteries N1 and N2

offering 810.01 and 810.00, respectively, with a 500 chance,and vague lotteries A1 and A2 offering 810.01 and 810.00,respectively, with a chance between 300 and 700. Curley6 Yates reasoned that because there exists a payoffdominance relation within precise and vague pairs, thechoices are made only with respect to vagueness and thuseither ranking N1 , A1 , N2 , A2 (i.e., vagueness avoidance) orranking A1 , N1 , A2 , N2 (i.e., vagueness seeking) is consis-tent with a lexicographic rule. Nevertheless, this is a narrowview of a lexicographic strategy. Suppose, for example, thatprobability is the most important dimension of a gamble(Tversky, Sattath, 6 Slovic, 1988) and that payoffs deter-mine choice only after probabilities are considered. Further,suppose that precise probabilities are preferred to probabil-ity intervals when neither dominates the other (i.e., whenthe precise probability is contained within the interval of thevague probability). In this situation, N1 , N2 , A1 , A2 rankingis consistent with a lexicographic rule that puts preciseprobabilities over vague ones and that orders according topayoffs only after the first sorting has been performed. Ifunder the same conditions vague probabilities are preferred,then A1 , A2 , N1 , N2 . Hence, we reconsider the lexico-

graphic rule and compare it to the delta choice model in aranking and a choice experiment. More importantly, wecompare these models via a new measure of model perfor-mance.

3. MODEL EVALUATION

3.1. Model Comparisons via the Measure ?(S, M; D)

Deterministic models of behavior, such as the two dis-cussed in the previous sections, always pose an evaluationproblem because departures from the models' predictions,i.e., errors, have no place in the psychological theories.3

Error theories, assuming different psychological mecha-

nisms, can be added to deterministic models. A recentdevelopment in this direction is the one by Harless 6Camerer (1992), who define model performance in terms ofboth systematic and unsystematic errors to a theory. Themost common evaluation strategy, however, has been to use

132 GONZA� LEZ-VALLEJO, BONAZZI, AND SHAPIRO

2 To compare Eqs. (1), (2), and (3), use the convention g( p)=g0( p) andf(m)=f0(m).

3 Another way of dealing with variability of behavior is to developtheories that are intrinsically stochastic (e.g., see Falmagne, 1976;Busemeyer 6 Townsend, 1993). We defer discussion of evaluation of thesemodels, however, as our measure is intended to deal with the problem oferror in deterministic rather than in stochastic rules.


a known statistic's sampling distribution to judge deviationsfrom preference orders assumed by a model's axioms (forexample, see Coombs 6 Huang, 1970; Curley 6 Yates,1989). Let us discuss this latter method.

Association measures such as Spearman rank-ordercorrelation coefficient (rs), or Kendall rank-order correla-tion coefficient (T), can be computed between observedrank orders and those predicted by a model (or its axioms).The sampling distribution of these statistics, under the nullhypothesis of random responding, can then be used toevaluate the models' predictions. Random responding isdefined as a strategy such that all orders are equally likely,i.e., predicted, and subjects' orders are not associated. Levelsof the correlation are then used for inferential purposes.

One problem with the correlational approach resides inthe definition of random responding. Under the nullhypothesis, random or chance levels are determined byassuming that a subject's ranking is equally likely associatedwith any ranking of the total set of objects present in theexperiment. This assumption is generally incorrect, as someorders, such as those that respect dominance, are almostcertainly more likely than others to occur. If this is so,models that assume dominance, as most do, will appear topredict data well. This will be true even if subjects providerandom orders under dominance, because the orders ofboth the model and the subjects are constrained by the sameprinciple. Furthermore, predictions of dominance assumingmodels are correlated, and thus it is difficult to make modelcomparisons.

Finally, the correlational approach does not take intoaccount the fact that different models predict different num-bers of preference orders, and thus less restrictive models(i.e., ones consisting of greater numbers of preferenceorders) are more likely to fit subjects data trivially (thisissue was also considered by Harless 6 Camerer, 1992). Asan extreme example, suppose that a model predicts virtuallyall possible rankings that can be obtained from a set of Nobjects. In this case, it is easy to see that it is very likely thatthe degree of association between a model's ranking and asubject's ranking (both drawn at random from their respec-tive populations) will be high, but only because the modelis not sufficiently restrictive.

With these issues in mind, we develop a new measure ofmodel performance, ?(S, M; D), that is defined within theset of choice patterns that respect dominance. This statisticis the mean probability that a model's predictions are closerto the subjects' preferences than to random data, withhigher values indicating better model performance. UnlikeHarless 6 Camerer (1992), we do not add error to thealgebraic models in order to account for variability, butinstead use errors (or switches) to define a measure of dis-crepancy. The measure is a true distance, such that modelslying closer to subjects' data are better than those lyingfurther away. The measure also penalizes models that

predict many patterns making explicit a trade-off betweenfit and parsimony (similar to Harless and Camerer'snotion).

3.2. Derivation of ?(S, M; D)

Let the n=N(N&1)�2 pairs of objects (gambles) that canbe formed from the N individual ones be representedby [(O1 , O2)1 , (O3 , O4)2 , ..., (O(2N&1) , O2N)n], and letchoices between pairs be represented by a string c=[c1 , ..., cn] of n numbers ci # [0, 1] such that ci=1 iffOj>Ok for (Oj , Ok) i and ci=0 otherwise. A choice patternis then denoted by any c. We will refer to the set of allpossible patterns of choices as C. Let a, b, c, be n-tuples inC, and for each a # C, let ai denote the i th element of a.

Our measure of model performance is based on the``taxicab'' or ``city-block'' metric4 from metric topology(Croom, 1989, p. 58) which is defined, for any two a, b # C, as

\(a, b)= :n

i=1

|ai&bi |.

The metric \(a, b) measures the amount of disagreementbetween choice patterns a and b.

Because our ultimate interest is in model performance,and models represent sets of choice patterns (i.e., subsets,M�C) rather than single choice patterns, we borrow theformal concept of distance from metric topology. Distanceis defined for any a # C, M�C as (Croom, 1989, p. 61)

d(a, M)=min[\(a, m) | m # M].

The distance d(a, M) measures the minimum amount of dis-agreement between a single choice pattern, a, and all thepatterns of choices in M.

Finally, we define a measure of model performance. LetS�C be the set of all subjects' choice patterns with elementss # S, M�C be choice patterns allowed by a model, D/Cbe the orders that respect dominance, and Card[ } ] referto the cardinality (i.e., number of elements) of a set. Fora single subject s # S, ?(s, M; D) � Re is defined byCard[[c # D: d(c, M)>d(s, M)]]�Card[D]. Over all s # S,a mean value, ?(S, M; D) is defined as

?(S, M; D)=:s # S card[[c # D: d(c, M)>d(s, M)]]

card[D] card[S],

0�?(S, M; D)�1, and is called the performance of modelM relative to S.


4 A metric on a set C is a function r: C_C � Re which satisfies thefollowing for all a, b, c # C:

(1) r(a, b)�0, and r(a, b)=0 only when a=b;(2) r(a, b)=r(b, a);(3) r(a, c)�r(a, b)+r(b, c).A metric generalizes the everyday concept of length or distance.

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FIG. 2. A situation in which two models fit subjects' data perfectly, but?(S, M; D) is better for the more restrictive model, M1 .

Notice that for each s # S, ?(s, M; D) can be thought of asthe probability that the distance to the model of a randomlyselected choice pattern in D is greater than the distance froma subject to the model (i.e., probability that a random pat-tern is farther away from the model than the subject's data);?(S, M; D) is the average of these probabilities over subjectsand is unique for a given set of choice alternatives and data.Some properties of ?(S, M; D) are the following:

(1) ?(S, M; D) is defined only within a dominance setD. This allows models to be differentiated within D.

(2) ?(S, M; D) is defined in terms of d(a, M), a mini-mum distance between a choice pattern and a set of pat-terns. This allows for a precise description of model fit interms of distance of data to model's predictions.

(3) The numerator of ?(S, M; D) counts the number oftimes patterns in D, but not in M, are further away from Mthan a subject's pattern. This counting procedure ensuresthat very general models are penalized for lack of precision.To see this, suppose that two models, M1 and M2 perfectlyfit a set of data S; that is, S�Mi , and d(s, Mi)=0 for all Sand Mi . Assume further that M1 is contained in M2 . As wecount the number of times a pattern in D is further awayfrom a model than a pattern in S, we see that all patternsoutside a model will be further away because d(s, Mi)=0.More patterns will be outside M1 than M2 , however, andthus ?(S, M; D) will be greater for M1 than for M2. Figure 2demonstrates this situation with a Venn diagram:?(S, M1 ; D)= 14

20 and ?(S, M2 ; D)= 920 .

4. METHOD

4.1. Subjects

Forty volunteers participated as paid subjects in each oftwo experiments. In each case, 20 were randomly assignedto each of two vagueness conditions. They were paid con-tingent on performance and received course credit for anintroductory psychology course at the University of NorthCarolina, Chapel Hill.

FIG. 3. The probability of a gamble corresponds to the probability astar has of passing through a barrier (which moves in and out at a specifiedrate) and reaching the opposite end.

4.2. Procedure

We conducted ranking and choice experiments to studythe effects of vague payoffs and vague probabilities onpreferences. Because the procedures and materials were vir-tually identical for the two experiments, we will describe onegeneral procedure and note differences between experimentswhere appropriate.

A set of four positively valued gambles of the form (m, p)was used to generate all the stimuli in this study, and a videogame was used to represent gambles' probabilities. Specifi-cally, gambles' probabilities corresponded to the chances ofa star reaching the end of a slot. The star reaches the endwith a probability p. Figure 3 shows the display correspond-ing to the event before the star was released. By pressing anykey on the computer's keyboard, the star began to movethrough the slot and either was stopped by the barrier orreached the end.

Both experiments had two vague conditions. In the PVcondition, gambles' probabilities were given as intervals ofprobability values, while in the MV condition payoffs weregiven as intervals. In either case, the vague dimension wasgiven as an interval centered at one of two values (.25 and.75), and the width of the intervals (v) varied. Moreprecisely, in each vagueness conditions, two levels of prob-ability (.25 and .75), two levels of payoff (80.25 and 80.75),and three interval widths (0, .20, and .50) were combined toyield a set of 12 gambles of the form (m, p, v). Note that inone condition, the intervals were applied to p and in theother they were applied to m. Gambles, not events, werepresented to subjects during the experiment. An example ofa vague probability gamble can be seen in Fig. 4.

The subjects' tasks were to rank order gambles in termsof preference in Experiment 1 (ranking experiment) and tochoose among all possible pairs of gambles (66) in Experi-ment 2 (choice experiment). After ranking or choosing, sub-jects rated each individual gamble in terms of risk and worthon a 0 to 10 scale (with 0 and 10 representing least and mostof each attribute, respectively).

In the ranking task, subjects were instructed to rankorder the gambles, from most to least preferred, by placinga number of squares (1 to 12) next to each gamble. Themaximum number of squares, 12, was to be placed next to

FIG. 4. A vague probability gamble as shown to a subject.



the most preferred gamble, 11 next to the second mostpreferred gamble, etc., where the least preferred gamblereceived 1 square. At the end of the experiment, one gamblewas randomly selected and played for real money. Subjectswere encouraged to place squares in accordance with theirpreferences because for each gamble, if the star reached theend when released, the gamble's payoff was multiplied bythe number of squares. Everything else being equal, thisprocedure motivates individuals to maximize the payoffs oftheir most preferred options (because it increases payoffs asa function of preferences). Because it is possible that thisincentive system may reveal something other than subjects'preferences for the simple gambles, the choice experimentserves as a further experimental test of the generality of ourresults in this task.5 In the choice task, subjects chose themost preferred gamble of a pair by placing a single squarenext to it. At the end of the experiment, subjects played 10randomly selected decisions for real monetary rewards.Note that in both experiments subjects do not know whichdecisions will played for real money until the end of theexperiment. This is done to motivate individuals to makegood decisions all throughout a session.

In both experiments, subjects worked individually at aPC and had a practice session before the experiment began,in which ranking (choosing) and rating, as well as the payoffschema, were demonstrated. Subjects started the experimentonly when they felt comfortable with the task. In choice,gamble pairs were randomly presented throughout a ses-sion, and the position of each gamble on the screen was alsorandomly determined. All rating questions were randomlypresented to subjects and were given after the ranking(choice) task.

5. RESULTS

We will present results of both experiments (i.e., rankingand choice) together and make note of differences asanalyses are being discussed. First, we check for vaguenessaversion to determine whether the usual results arereplicated. Next, we present model-free comparisons ofpreference orders under the two vagueness conditions,followed by model comparisons performed via ?(S, M; D).Finally, we consider the rating data.

5.1. Vagueness Aversion

We define a measure of vagueness aversion�preference asthe number of times a more vague gamble was preferred toa more precise one and compute it for each subject in eachgroup. Comparisons were limited to those pairs in whichboth gambles matched in m and p (i.e., mi=mj and pi=pj)and differed only in v, for a total of 12 comparisons. In the

ranking experiment, a MANOVA having condition as abetween-subjects factor, and center of the interval (i.e., .25,.75) as a within-subjects factor, resulted in no condition bycenter interaction and no main effect of condition. The maineffect of center was significant (F(1, 36)=6.92, p<.05),showing more aversion at .75 than at .25, similar to Curley6 Yates' (1989) results (means over groups were equal to3.78 at .25, and 2.81 at .75). The overall mean of 3.29 issignificantly different from the indifference point of 6(t(37)=9.36, SE=1.78, p<.01), demonstrating the vague-ness aversion effect. In the choice experiment, theMANOVA resulted in no interaction of condition and cen-ter and no main effects. Over groups and centers, the meanvagueness preference 2.27 is significantly different from 6(t(37)=21.33, SE=1.07, p<.01), demonstrating vaguenessaversion as well. Thus, we replicate the standard pattern ofresults, with both vague probabilities and vague outcomes.

5.2. Ranking and Choice Model-Free Tests

Because a basic assumption in virtually all theories ofdecision making is that subjects' preferences agree with thedominance principle, we checked the number of dominanceviolations for each subject, per group. Note that a total of 45dominance violations are possible with these gambles(ignoring vagueness). In the ranking experiment, the meannumber of violations for the PV group was 4.68, and themean for the MV group was .83. The difference between thegroups is significant (t(38)=2.27, p<.05). In the choiceexperiment, means were 3.04 and 2.34, respectively, not asignificant difference. The overall means of 2.75 and 2.69constitute 6.1 and 5.90 violations, respectively. Theseviolations are due to a few subjects: 22 violations were madeby each of two subjects in the ranking experiment; 22 and 28were made by two subjects in the choice experiment.Without these subjects, the overall percentages ofdominance violations drop to 3.3 and 3.730 for rankingand choice experiments, respectively. These numbers aresmall, corresponding to an average of 1.5 violations, but arenot predicted by any model.

A second principle of many decision making models istransitivity. Unlike dominance, however, the models we testin this study do not assume transitivity. Because a rankingtask necessarily produces orders that are transitive, wecould look for violations of transitivity only in the choiceexperiment, where a maximum of 72 violations werepossible. Mean numbers of intransitivities were 5.35 and 8.6for PV and MV groups, respectively. The difference betweenthe groups is not significant (t(19)=.9, p=.37), and theoverall mean of 6.97 (9.680) is small compared to propor-tion of violations found in studies designed to produceintransitivities (see Tversky, 1969).

We then looked at the degree of association between eachsubject's preference order, the order obtained by ranking


5 See Erev 6 Wallsten (1993) and Gonza� lez-Vallejo, Erev 6 Wallsten(1994) for other applications of this procedure.


gambles according to probabilities (ignoring all othergambles' dimensions) and the order obtained by rankinggambles according to payoffs (also ignoring the otherdimensions). In this and subsequent analyses the four sub-jects who greatly violated dominance were omitted, butsimilar results were obtained when they were included.

A measure of association between two rank orders is theV statistics (Nelson, 1984), which is defined for tworankings, x and y, as V(x, y)=a

��(a

�+d

�), where a

�is the

number of pairs ranked in the same direction by x and y,and d

�is the number of pairs ranked in the opposite direc-

tion. Let cp be the choice pattern resulting from orderinggambles by probabilities (using the midpoints of intervals),cm be the choice pattern resulting from ordering gambles bypayoffs (also ignoring interval widths), and s be a subject'spattern. We computed V(s, cp) and V(s, cm) for each subjectand looked for differences between groups MV and PV.Note that ordering according to probabilities does notdetermine a payoff order, and conversely ordering accord-ing to payoffs does not imply a probability order (i.e.,V(cp , cm)=.5; thus, V(s, cp) and V(s, cm) are independent).Note also that ties are not counted, and thus a total of 36comparisons (encompassing all orders) are made to com-pute V.

A MANOVA was performed on V having condition (PVand MV) as a between-subjects factor and dimension (prob-ability or payoff) as a within-subjects factor. In the rankingexperiment, there was a marginal condition by dimensioninteraction, and main effects of both condition and dimen-sion (F(1, 36)=5.37, MSE=.001, p<.5, for condition;F(1, 36)=45.43, p<.01, for dimension). Means of V(s, cp)were .96 and .93 for PV and MV groups, and means ofV(s, cm) were .71 and .79, respectively. That is, choices wereprimarily based on probabilities for both groups, and payoffordering was stronger for the MV group than the PV group.In the choice experiment, means of V(s, cp) were .92 and .90for PV and MV groups, and means of V(s, cm) were .74 and.77. The MANOVA analysis showed no condition bydimension interaction and no main effect of condition. Themain effect of dimension was significant (F(1, 36)=26.28,p<.01), replicating the ranking experiment's result thatchoices were primarily based on probabilities. In bothexperiments, these model-free results suggest that subjectsin both groups are using a similar decision strategy thatemphasizes the probability dimension (Tversky et al., 1988).

5.3. Model Comparisons via ?(S, M; D)

We used the measure ?(S, M; D) to test (i.e., fit to data)the delta and lexicographic models and thereby examinesubjects' decision strategies in a more precise fashion. Weobtained ?(S, M; D) exactly, without statistical approxima-tions, because we were able to generate all the choice pat-terns in D as well as all the patterns predicted by each

model. The patterns in D correspond to choice patterns, c,that include the 45 choices in which a dominance relation isdefined, as well as all possible choices for the 21 gamblepairs in which no dominance relation exists (i.e., 221

r2million possibilities). The patterns of choices for each modelwere generated by finding feasible solutions to the set of 21inequalities that each choice pattern in D imposes on agiven model structure. For each model, we only assumedmonotonicity over probabilities and payoffs for a givenvagueness level; for the delta model, theta is constant for agiven choice pattern, but can assume any values acrosschoice patterns as long as a solution to the set of inequalitiesexists. Therefore, without assuming specific functionalforms, the models were tested in their most general versions.Because the computations involved in generating those setsare heavily dependent on the gambles used, we do notdescribe them here.6

Both the delta and lexicographic models allow two sets ofchoice patterns for each set of stimuli. In the case of the deltamodel, the two sets result from applying Eqs. (2) and (3) tothe gambles used by the PV and MV groups, respectively.Similarly, two sets are allowed by the lexicographic rule bytreating the precise dimension of a gamble as primary. Thatis, gambles are ordered first by payoffs when probabilitiesare vague and by probabilities when payoffs are vague.

We computed ?(S, M; D) for each of the four submodelsapplied to each group in each experiment, using the 21 sub-jects in the ranking experiment and the 18 in the choiceexperiment who exhibited zero dominance violations. Werestricted our data to these subjects because the models aredefined within D, and measures of model performance aremeaningful only for people who respect this criterion.Table 1 shows these results. Measures ?(S, M; D) for eachversion of a model (two submodels in each case) are labeledvp and vm, indicating the set of choice patterns generatedwith vague probability and with vague payoff gambles.Groups are PV and MV, referring to subjects in either thevague probability or vague payoff condition. Note that allthe models were tested on all groups of subjects irrespectiveof the stimuli actually seen by subjects. For example,


6 Those interested may write to the first author to obtain the programused to generate these sets. In general, we discarded choice patterns in Dwhich did not satisfy the set of inequalities imposed by a preference pattern.For example, assume that gv=0(.75)>gv=.5(.25) for the delta model withvague probabilities. The preferences: (80.25, .75,0) � (80.25, .25,.5) and(80.75, .75,0) � (80.75, .25,0.5) lead to a contradiction for any value of %.That is, substituting these gambles in to Eq. (2), we obtain thatg(.75)v=0>g(.25)v=.5 in the first case, but gv&0(.75)<gv=.5(.25) in thesecond case. Any pattern having the above two preferences is thus dis-carded. In this manner, we checked for all possible orders allowed by themodel when all possible ordinal relationship among the gv=0(.25),gv=.2(.25), gv=.5(.25), gv=0(.75), gv=.2(.75), and gv=.5(.75) were consideredin combination with different % values (i.e., for % less than, equal to, orgreater than zero). The same principles applied when generating patternsfor the delta vague payoff model and the lexicographic models.


TABLE 1

?(S, M ; D ) for Each model, Group, and Experiment

Model

Delta Lexicographic

Experiment N Group vp vm vp vm

Ranking 16 MV 0.83 0.85 0.12 0.875 PV 0.88 0.80 0.20 0.80

Choice 11 MV 0.83 0.85 0.12 0.837 PV 0.90 0.80 0.33 0.58

preferences of subjects who saw only gambles with vagueprobabilities (PV groups) were tested with models thatassume vague probabilities (vp columns), but also withmodels that assume vague payoffs (vm columns). We usedboth types of submodels with each group to test how muchbetter the appropriate model fitted the data in each case.For example, PV groups are expected to be closer to themodel described by Eq. (2) than to that described by Eq. (3)(i.e., ?(S, M; D) in column vp of the delta model should belarger than those in the vm column for the PV groups).Suppose, however, that the opposite pattern occurs (i.e., PVsubjects are closer to the model described by Eq. (3) than tothe model described by Eq. (2)). This result would suggestthat subjects interpreted the vagueness of one dimension asvagueness in the other dimension. Because we used a sym-metric design of payoffs and probabilities in which bothvariables have values in the [0, 1] interval, it was worthexploring this possibility.

First, let us examine the means in Table 1 without the aidof statistical tests. Looking at the delta models, ?(S, M; D)for the MV and PV groups are higher when the appropriatedelta model is used in each case (i.e., for MV groups, meansunder column vm are higher than means under the vpcolumn, and the reverse is true for the PV groups). The dif-ferences are, nevertheless, relatively small and the two sub-models of the delta rule seem to do as well for both groupsin both experiments. A different pattern is observed for thelexicographic model and we see that both groups in bothexperiments do much better with the model that orders firstaccording to probabilities, regardless of where vagueness islocated (i.e., means under column vm are larger than meansunder column vp in all cases).

How do these results compare to the previous onesobtained with the model-free tests ? Earlier results on Vscores showed that people used similar decision strategiesindependent of which gamble's dimension was vaguelyspecified. Our model-dependent analyses confirm thissuggestion by showing that both PV and MV groupsare equally described by a single model. Within thelexicographic model's framework, it is clear that the strategy

of ordering first according to probabilities and then accord-ing to payoffs is more descriptive. We see this because?(S, M; D) in the vm column are always larger than thecorresponding ?(S, M; D) in the vp column of this model,across groups and experiments (as a reminder, 0�?(S, M; D)�1, with higher numbers describing betterperformance).

In the case of the delta model, the appropriate submodeldoes better in each group (vp better describes PV, and vmbetter describes MV, in both experiments), but both sub-models yield very similar performance (e.g., in the rankingexperiment, ?(S, M; D)=0.85 for the vm model tested onthe VM data, but the vp model does almost as well,?(S, M; D)=0.83 for this group; similar results hold inchoice). We might understand these results by noting thatEqs. (2) and (3) differ only if vagueness is introduced intothe choice process; however, if vagueness is ignored then thetwo models are identical (i.e., they reduce to Eq. (1)).Hence, if subjects in both conditions were influenced byvagueness only in some cases (e.g., when pi=pj andmi=mj), but not in others, then their patterns would besimilarly described by both submodels, as is the case.

To compare the delta and the lexicographic models, notein Table 1 that the delta model performs better than the bestversion of the lexicographic model in all but one group(i.e., the vm model tested with MV data in the rankingexperiment). The differences are, nevertheless, very small.Statistical tests, MANOVAs on subjects' ?(s, M; D), withexperiment and condition as between-subjects factors, andmodel and submodel as within-subjects factors support theabove conclusions. The only significant difference foundwith these data is that ?(S, M; D) of the lexicographicmodel, column vp, are significantly smaller than the othermeans (F(1, 37)=30.53, p<.01).

5.4. Ratings

We performed MANOVAs with condition as a between-subjects factor and gamble as a within-subjects factor on thesubjects' judgments of risk and worth. Regarding riskjudgments, there was no interaction of condition and gam-ble, and only a significant main effect of gamble was presentin both experiments (F(11, 26)=9.31, p<.01, in ranking;F(11, 26)=6.32, p<.01, in choice). That is, vagueness didnot alter risk perception.

In terms of worth judgments, the pattern of means is quitesimilar for both experiments. In the ranking experiment,there is only a main effect of gamble that is traced to changesin expected value (EV) F(11, 26)=29.46, p<.01). In thechoice experiment, there is more change, but the generaltrend is similar to that of the ranking experiment. There isa condition by gamble interaction (F(11, 26)=2.19,p<.05), but simple tests do not reveal any significantchange of mean as a function of both condition and gamblelevel.



An interesting question addressed with these data was towhat degree do judgments of risk and worth systematicallyagree with the preference selections in the ranking andchoice experiments. One can obtain measures of riskattitudes independently of utility functions by assessingwhether individuals select those gambles judged as less ormore risky, as Weber 6 Bottom (1989) did in their study.Individuals who consistently select the member of the pairthey judge to be riskier (less risky) are labeled as ``risk seek-ing'' (``risk averse''), while no systematic preference for riskindicates ``risk neutrality.'' Similarly, one can address con-sistency between judgments of worth and preferences. In thiscase, high agreement is expected only in the direction ofmore worth being preferred to less worth.

The relationships of risk and preference and worth andpreference were assessed with a binomial test. Under thenull hypothesis of no judgment-preference consistency, theprobability of selecting a gamble given a judgment is .5. Foreach individual, proportions of less risky preference (i.e., theproportion of times the gamble judged as less risky wasselected) and of more worthy preference were obtained.Pairs of gambles receiving identical judgments on anattribute were not included in the analyses. In the rankingexperiment, 6 subjects are ``risk neutral'' ; the rest of the sub-jects show high consistency, typically preferring the lessrisky gamble of the pair: 32 are ``risk averse'' and 2 are ``riskseeking.'' Proportions of .67 and .81 are the first and thirdquartiles of the distribution of subjects' scores. More con-sistency is found with the worth judgments. Only 3 subjectsshow judgment-preference proportions that are close to .5;the first and third quartiles are .78 and .90, respectively.

Similar results are obtained in the choice experiment. Inthe case of risk-preference agreements, 23 subjects are ``riskaverse,'' 5 are ``risk seekers,'' and 12 are ``risk-neutral.'' Thefirst and third quartiles of this distribution are .59 and .81.In the case of worth-preference consistencies, only 3individuals give proportions close to .5; the first and thirdquartiles of this distribution are .78 and .90.

6. DISCUSSION

Many studies have examined how decisions are made invague situations, but vagueness has been treated as a factoradding probabilistic uncertainty rather than outcomeuncertainty. The present study made explicit the conceptualdistinction between these two classes of vague conditionsand explored preferences when probabilities versus payoffswere vaguely described.

Our results replicated the well-known vagueness aversionphenomenon and generalized it to situations in whichpayoffs were vague. A surprising result emerging from ourexperiments, however, was that subjects were affected byvagueness only when mi=mj and pi=pj , but otherwise,

contrary to our expectations, their preferences did notdepend on whether probabilities or payoffs were vague.Ratings of risk and worth further demonstrated that percep-tions of gambles were marginally affected by vagueness.This was true even with risk ratings, which are typicallyexpected to increase as a function of gamble variance (seeWeber 6 Bottom, 1989, for a discussion of various theoriesin which more variability is equivalent to more risk.)Because preferences and ratings agreed at both group andindividual levels, we find further support for the notion thatsubjects behaved similarly under vague probability andvague payoff situations.

Using ?(S, M; D), we confirmed that subjects used asimilar decision strategy irrespective of what was vague.Data were well described by either a delta model or alexicographic model whose main focus is in probabilitiesand payoffs, in that order. In contrast, we observed thatsubjects' choice patterns were very far from the predictionsspecified by a lexicographic model that emphasizes payoffs,even when gambles' payoffs were precise.

The good fit of both lexicographic (vm model) and deltamodels points to the interesting fact that both noncompen-satory and compensatory strategies can, in some situations,closely describe behavior. This is an important issue becauseit shows that it is not necessary to assume that individualsignore relevant information to produce behavior that isdescribed by a noncompensatory model. This point was alsoraised by Tversky (1969), who favored an additive dif-ference model over a lexicographic semiorder strategy toaccount for intransitive behavior. Similarly, we favor thedelta model over the lexicographic rule, because it is hard tobelieve that individuals ignore important information indeciding among two-dimensional objects. Further researchis needed, nevertheless, to give full support to the deltastrategy.

On another perspective, our study demonstrated the useof ?(S, M; D) in testing competing parallel psychologicalhypotheses. The main advantages of this measure are asfollows. First, ?(S, M; D) is based on a metric that has anatural interpretation of disagreement between choice pat-terns that allows us to talk in terms of distances among sets.Second, ?(S, M; D) rests on a specific definition of randomperformance which is relative to the set of choice patternsrespecting dominance, and to M, the subset predicted by amodel. This makes model comparisons meaningful andstraightforward. Models can be compared by testing dif-ferences among population means, making use of the well-known properties of the sampling distribution of means,without requiring models to be algebraically related (i.e.,nested). The problem of models having different number ofparameters is solved by testing the models in their mostgeneral form, which provides an upper bound of perfor-mance. Third, ?(S, M; D) incorporates the size of the modelas an important factor in measuring model performance, by



which bigger models are penalized for not making moreprecise predictions.7

Finally, we think our procedure provides a better way oftesting deterministic models than those which add error tothe rules. In a sense, procedures that add error (e.g., Harless6 Camerer, 1992) transform the original models into newstochastic versions. It is not clear that the new versions arethe best stochastic representations of the original models. Inprinciple, many different error assumptions could be intro-duced, yielding families of stochastic versions that mightdiffer in predictive ability. Which one should be used as theprototype of the original model is not a trivial question; ourprocedure avoids this dilemma.

In summary, we made a distinction between situations inwhich uncertainty regarding payoffs is described vaguelyversus situations in which payoffs themselves are vaguelyspecified. Within this distinction, individuals behaved as ifall information was known except when only imprecisiondistinguished two options. Vagueness affected behavior inthe usual manner when gambles of equal expectation, butdiffering in precision, were compared, but induced similarpreferences across vague probability and vague payoffsituations. Vagueness aversion is then a local effect, andthis speaks to the singularity of theories that explain thisphenomenon without more generally addressing the ques-tion of how decision strategies are affected by lack of infor-mation. Our study is a step in this direction.

The fact that a similar decision strategy was employed inthe two vague situations is intriguing and deserves furtherattention. A recent study designed to extend results fromthe present work has shown that preferences are similarlyaffected by the two vague context in a real world decisionmaking task. Briefly, Dhar, Gonza� lez-Vallejo, and Soman(1995) studied the effects of vague price promotions onactual sales of a candy bar. Promotions significantlyincreased sales as expected, but sales did not depend onwhether promotions had vague probability or vague dis-count information. Interviews of shoppers leaving the storeshowed that customers knew of the promotions and wereaware of the lack of information on the relevant dimension.This implies that individuals perceived vagueness, butbehaved similarly in the two vague situations.

How is vagueness treated? We believe that results of thepresent study support the notion that vagueness behaveslike a context which affects behavior only when intercontextcomparisons are required. Our rating data showed that in

an absolute sense, vagueness did not affect worth of alter-natives, and thus decision makers treated vague gambles astheir precise counterparts in all vague�vague comparisons.Further research is needed to specify the extent to whichdecision strategies remain constant as amount of informa-tion of the alternatives vary. Of particular interest is toexplore this contextual effect of vagueness in multiattributedecision making situations.

APPENDIX

Let Ai , Aj , Ak be three simple gambles where|mi |>|mj |>|mk | and pi<pj<pk . The intransitive patternAi>Aj , Aj>Ak , and Ak>Ai leads to the inequalities


&% Vg( pj)&g( pi)

g( pi)>0

f (mj)& f (mk)f (mk)

&% Vg( pk)&g( pj)

g( pj)>0

f (mi)& f (mk)f (mk)

&% Vg( pk)&g( pi)

g( pi)<0.

The set of inequalities reduces to the expression

( f (mj)&f (mk)) V (g( pj)&g( pi)) V (1&%)>0.

Assuming monotonicity, we know that f (mj)>f (mk) andg( pj)>g( pi) by definition; hence the inequality holds for%<1.

In the same manner, the intransitive pattern Ak>Aj ,Aj>Ai , and Ai>Ak leads to the set of inequalities


&% Vg( pj)&g( pi)

g( pi)<0

f (mj)& f (mk)f (mk)

&% Vg( pk)&g( pj)

g( pj)<0

f (mi)& f (mk)f (mk)

&% Vg( pk)&g( pi)

g( pi)>0

that reduces to ( f (mj)&f (mk)) V (g( pj)&g( pi)) V (1q)<0,which holds for %>1.

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Received: December 3, 1994


Documents

Effects of Vague Probabilities and of Vague Payoffs on Preference: A Model Comparison Analysis