Decision Making Under Uncertainty: A Comparison of Simple ... · variance of the uncertain payoffs (denoted ff2), the value of the certain payoff (denoted k\ and the deadline time

$Page 1: Decision Making Under Uncertainty: A Comparison of Simple ... · variance of the uncertain payoffs (denoted ff2), the value of the certain payoff (denoted k\ and the deadline time$
Journal of Experimental Psychology:Learning, Memory, and Cognition1985, Vol. II, No. 3, 538-564

Copyright 1985 by the American Psychological Association, Inc.O278-7393/S5/$OQ.75

Decision Making Under Uncertainty: A Comparison of SimpleScalability, Fixed-Sample, and Sequential-Sampling Models

Jerome R. BusemeyerPurdue University

The purpose of this article is to investigate the learning and memory processesinvolved in decision making under uncertainty. In two different experiments,subjects were given a choice between a certain alternative that produced a singleknown payoff and an uncertain alternative that produced a normal distributionof payoffs. Initially this distribution was unknown, and in the first experiment itwas learned through feedback from past decisions, whereas in the secondexperiment it was learned by observing sample outcomes. In the first experiment,a response deadline was used to limit the amount of time available for making adecision. In the second experiment, an observation cost was used to limit thenumber of samples that could be purchased. The mean and variance of theuncertain alternative and the value of the certain alternative were factoriallymanipulated to study their joint effects on choice probability, choice responsetime (Experiment 1), and number of observations purchased (Experiment 2).Algebraic-deterministic theories developed for decision making with simple gamblesfail to explain the present results. Two new models are developed and tested—fixed- and sequential-sampling models—that attempt to describe the learning andmemory processes involved in decision making under uncertainty.

Decision theorists often find it useful todistinguish three classes of situations—deci-sions made under conditions of certainty,risk, or uncertainty (cf. Luce & Raiffa, 1957,p. 13). Under certainty, each action producesa single (perhaps multidimensional) knownoutcome. For example, deciding which pairof jeans to buy is a decision under certaintybecause you can see what you are buying.Under risk, each action produces a set ofpossible outcomes, and the probability ofeach outcome is known. State lotteries orgambling games are classic examples of riskydecisions, because the probabilities of the

The first experiment was supported by National Insti-tute of Mental Health Grant MH1425706, Larry Jonesprinciple investigator. Douglas Medin generously providedspace and equipment to conduct the first study. Thesecond experiment was supported by a grant awarded byIndiana University-Purdue University at Indianapolis.

The author wishes to thank K. Michael Aschenbrenner,Michael Birnbaum. Douglas Medin, Barbara Mellers,Steve Link, Tom Wallsten, and an unknown reviewer forhelpful comments on this article.

Requests for reprints should be sent to Jerome R.Busemeyer, Psychological Sciences, Purdue University,West Lafayette, Indiana 47907.

outcomes can be calculated. Under conditionsof uncertainty, each action again produces aset of possible outcomes, but the probabilityof each outcome is unknown. For example,deciding to talk with a stranger is an uncertaindecision because you have no way to predicthow the person will react.

More generally, different decision situationscan be ordered in terms of degree of uncer-tainty (see Luce & Raiffa, 1957, p. 299). Onecan only speculate about the frequency ofreal-life situations along this ordering, butone guess is that most decisions are madeunder partial uncertainty, that is, a decisionsituation somewhere between risk and com-plete uncertainty. For example, the decisionto build a nuclear energy plant will not bemade under complete ignorance, but neitherwill it be made with perfect knowledge of theprobabilities of all outcomes.

One way to make decisions under uncer-tainty is to base it on past experience withsimilar situations. For example, suppose thata particular patient has a long history ofadverse reactions to different treatments fora serious medical problem. The patient'sfamily physician would have to rely on this

538

DECISION MAKING UNDER UNCERTAINTY 539

past history to decide how to treat a newemergency call.

Another way to make decisions under un-certainty is to collect new information aboutthe outcomes produced by an action beforemaking a commitment. For example, beforea new drug is introduced on the market, asequence of tests are performed to determineits effectiveness.

Early in the history of behavioral decisiontheory (see Lee, 1971), researchers activelyinvestigated decision making under both riskand uncertainty. Investigations of risky deci-sion making typically used simple gamblesof the form "win 10 dollars with probability.3 or nothing." Investigations of decisionmaking under uncertainty typically used aprobability learning task in which the prob-ability of each payoff was initially unknownand had to be learned through trial by trialfeedback.

Different theories were developed by re-searchers studying decisions under risk anduncertainty. Deterministic-algebraic theoriessuch as Edwards's (1962) subjective expectedutility theory were developed to describe de-cision making under risk. Stimulus samplingtheories such as Myers and Atkinson's (1964)weak-strong conditioning model were devel-oped to describe decision making under un-certainty.

Recently, Lopes (1983, p. 138) has arguedthat "after 30 years or more of research onrisk, we know a lot about how people makedecisions about simple lotteries, but we knowremarkably little about decision under uncer-tainty, possibly because we have not had agood laboratory model of uncertainty." Ap-parently, this opinion reflects a disenchant-ment with stimulus-response conditioningtheories and a lack of development of cogni-tive theories of decision making under uncer-tainty.

The purpose of this article is to investigatethe cognitive processes involved in decisionmaking under partial uncertainty. Two ex-periments are reported. In the first experi-ment, a probability learning task was used toinvestigate how memory of past outcomesinfluences new decisions. In the second ex-periment, an information purchasing taskwas used to investigate how new informationinfluences decisions.

Experiment 1

On each trial of Experiment 1, subjectswere given a choice between a certain alter-native and an uncertain alternative. If thecertain alternative was chosen, then a knownmonetary payoff was delivered. If the uncer-tain alternative was chosen, then the monetarypayoff was randomly selected from a normaldistribution with a mean equal to zero. Theuncertain alternative was labeled X on sometrials and Y on the remaining trials. Whenthe X cue was presented, the payoff wassampled from a distribution with a smallstandard deviation. When the Y cue waspresented, the payoff was sampled from adistribution with a large standard deviation.Initially, subjects did not know the distribu-tion of outcomes produced by the cues Xand Y. However, following each choice sub-jects were given feedback indicating the payoffthat would have been received if the uncertainalternative was chosen. Choice responses hadto be made within a deadline time limit inorder to avoid a severe penalty. A factorialdesign was constructed by manipulating thevariance of the uncertain payoffs (denotedff2), the value of the certain payoff (denotedk\ and the deadline time limit (denoted L).

In the next section, two decision-makingmodels are developed. The first is a probabi-listic extension of the algebraic-deterministicmodels developed within the risky decision-making paradigm. The second is a probabi-listic model based on the idea of memoryretrieval.

Simple Scalability Models

Suppose subjects kept track of the longrun average utility produced by the cues Xand Y. Then after several hundred trials theirestimates would converge on the mean orexpected utility. If they simply compared theexpected utility of the uncertain alternativewith the utility of the certain alternative, thenthe choice process would be deterministic;for a given condition the same alternativewould always be chosen. In order to introducerandom variation into the choice process,one could hypothesize that the comparisonof alternatives is perturbed by factors ran-domly varying across trials such as the indi-vidual's fluctuating state of wealth, randomly

540 JEROME R. BUSEMEYER

occurring patterns of wins and losses, orlapses of attention. Given these random dis-turbances, this model asserts that the proba-bility of choosing the uncertain alternative isan increasing function of the difference be-tween the expected utility of the uncertainalternative and the utility of the certain alter-native. Becker, Degroot, and Marschak(1963a) proposed this model of choice.

The model just mentioned is one exampleof a general class called "simple scalabilitymodels," which were originally developed asprobabilistic extensions of deterministic-al-gebraic models. (See section 7.1 of Luce &Suppes, 1965). According to simple scalabilitymodels, each alternative produces an inde-pendent utility scale value symbolized asM(UA) for the uncertain alternative, and u{k)for the certain alternative. In general, u(VA)is determined by the probability distributionassociated with the cue X or Y, and u(k) isdetermined solely by the utility of k. Simplescalability models state that the probabilityof choosing the uncertain alternative is givenby />(UA) = F[u(VA), u(k)], where F is astrictly increasing function of w(UA) and astrictly decreasing function of u{k).

The Becker et al. (1963a) model is a specialcase that states that ^(UA) = F[u(VA) -u(k)], where w(UA) was denned as the meanutility of the uncertain alternative. Otherversions are possible by assuming that M(UA)is also determined by the variance, or variouspercentiles of the distribution for the uncer-tain alternative.

Predictions, There is one general propertythat all special cases of simple scalabilityobey: independence between alternatives (cf.Tversky & Russo, 1969). The direction of theeffect of manipulating the distribution for theuncertain alternative should be the same forall values of the certain alternative. For ex-ample, the pattern of results illustrated inboth Figures la and lb conform to theindependence between alternatives property.In both figures, the *probability of choosingthe uncertain alternative is plotted as a func-tion of the certain value, denoted k. (k isplotted from positive to negative so that thecurves increase from left to right.) Each linerepresents a different distribution for theuncertain alternative, one with low varianceand another with high variance. Figure la

represents an uncertainty seeking type ofindividual—increasing the variance increasesthe tendency to choose the uncertain alter-native for all k. Figure lb represents anuncertainty averse type of individual—in-creasing the variance decreases the tendencyto choose the uncertain alternative for all k.

The a X k crossover interaction shown inFigure lc violates the independence betweenalternatives property (see Busemeyer, 1982,Appendix A). Contrary to independence, in-creasing the variance either increases or de-creases P(UA) depending on the certain value,k. However, caution is needed because anunambiguous test of the independence prop-erty requires an analysis of individual perfor-mance. To see why, suppose we average acrossthe uncertainty seekers illustrated in Figurela and the uncertainty avoiders illustrated inFigure lb. Although neither type of individualproduces a crossover pattern, the averageshown in Figure lc artificially produces thecrossover. The present study provides a testof independence at the individual level ofanalysis.

Fixed-Sample Model

According to the fixed-sample model, thedecision process for each choice trial beginsby retrieving a fixed number of recent tracesfrom memory. Each memory trace is a recordof the numeric value of a past outcomeassociated with the cue X or Y presented onthe current trial. The number of traces re-trieved from memory is limited by the dead-line time limit. The utilities of the retrievedoutcomes are averaged, and this moving av-erage is compared with the utility of thecertain alternative. Only if the sample meanis greater will the uncertain alternative bechosen.

Choice probability. The probability ofchoosing the uncertain alternative, denoted.P(UA), is an increasing function of the ratio[fi(u) — u(k)]/<j(u)y where u denotes the samplemean, fi(u) is the population mean utility ofthe uncertain payoffs, u(k) is the utility ofthe certain payoff, and cr̂ (w) is the varianceof the sample mean.

Figure 2 illustrates the distribution of thesample mean under the low- and high-vari-ance conditions. The three vertical lines in-


dicate u(k) when the certain value is negative(k < 0), zero (k = 0), and positive (k > 0)..P(UA) is represented as the area above eachvertical line.

Figure 2 illustrates the effect that variancehas on choice probability. When k is positive,the area above the vertical line is greaterunder the high variance curve. When A: isnegative, the area above the vertical line is

o*= low— - a = high

01

01

1b

— 1C

K>0 K<0Figure 1. Fictitious results indicating different possiblepatterns of the uncertain alternative variance (a2) bycertain value (k) interaction. [/'(UA) equals the probabilityof choosing the uncertain alternative.]

greater under the low variance curve. In sum,the direction of the eifect of variance onP(\JA) depends on the sign of the certainvalue, producing a crossover interaction sim-ilar to that shown in Figure 1c. Thus thefixed-sample model asserts that the indepen-dence between alternatives property shouldbe violated for individual subjects.

This crossover interaction can be derivedmathematically as follows. Define n as thenumber of recalled outcomes (i.e., the samplesize of u), and let <r2(w) be the variance ofthe utility of the uncertain payoifs. Then thevariance of the sample mean equals ^(u) =a*(u)fn. In the present study, the mean of theuncertain payoffs was always zero so that themean difference [M(W) - u(k)] was determinedby -u(k). Given these assumptions, P(\JA)is an increasing function of the productiri[-u{k)fu{u)]. Note that P(UA) increases ask decreases from positive to negative values,and that the slope is determined by thevariance. Low variance produces sharp dis-crimination of the mean difference, yieldinga steep slope. High variance produces poordiscrimination yielding a flatter slope.

Now consider the effect of increasing thetime limit, L. This allows an increase in thenumber of recalled outcomes, «, which thenmagnifies all of the slopes shown in Figurelc. When k is positive, then increasing thetime limit increases the probability of cor-rectly1 choosing the certain alternative. Whenk is negative, then increasing the time limitincreases the probability of correctly choosingthe uncertain alternative. More concisely, ac-curacy always increases as the deadline timelimit increases.

Choice response time. According to thefixed-sample model, increasing the numberof outcomes recalled, n, increases the numberof operations required to compute the samplemean. As a consequence, the time requiredto estimate the sample mean and ultimatelymake a choice response is an increasing func-tion of the sample size. However, increasingthe sample size also increases the probabilityof choosing the correct alternative. In sum, a

1 For this binary choice task, the term correct alternativeis a label for the alternative producing the larger expectedutility. The term accuracy is equivalent to the probabilityof choosing the correct alternative.


basic property of the fixed-sample model isthe speed-accuracy trade-off, that is, fasterresponses to the short deadline time limityield lower accuracy (cf. Swensson & Thomas,1974).

Method

Subjects. Six psychology students (4 seniors and 2graduate students, 2 males and 4 females) from theUniversity of Illinois-Champaign volunteered to partic-ipate for 15 1-hr daily sessions. Subjects were paidaccording to their performance, and they earned anaverage of $3.25 per session.

Apparatus. Each subject was tested individually in aquiet room. The experiment was computer controlled,stimuli were presented on a video terminal driven by anApple microcomputer, and response times were recordedin milliseconds using a Mountain Hardware clock.

Design and procedure. Monetary payoffs for theuncertain alternative were generated according to a normaldistribution with a mean of zero, and a standard deviationof 5 or 50 units (in units of $.01) depending on thestimulus condition. The payoff for the certain alternativeequaled either - 3 , 0, or +3 units, and the deadline timelimit was either 1, 2, or 3 s. The variance of the uncertainalternative, the payoff values for the certain alternative,and the time limit conditions were combined accordingto a 2 X 3 X 3 factorial design. Failure to beat thedeadline on any trial produced a loss equal to 25 units.Pilot research indicated that a simple reaction timerequired .25 s.

The protocol for a typical trial is illustrated as follows:

Deadline = 1 s

(.5-s delay)

Ready

(,5-s delay, then screen clears)

Choose X or - 3

(response timing begins)(subject types a response and selects the certain alternative)(response timing ends)

X = -45 on this trialYour pay equals - 3 on this trial

(subject initiates new trial by typing the return key)(screen clears)

As the display illustrates, each trial started with a messageindicating the deadline requirement, after a delay thescreen cleared, and then the choice stimulus was presented.In the display, X is the cue for the uncertain alternativeand the certain value equals —3. The low- or high-variance condition was indicated by the cues X or Y,where X signaled the high variance and Y the lowvariance for half the subjects, and the reverse pairing wasused for the other half. Feedback followed each response,which indicated both the payoff that would have beenreceived if uncertain alternative was chosen (e.g., X =-45), and the actual payoff received based on the selectedalternative (e.g., —3, because the certain alternative waschosen in the example).

The actual distributions associated with the cues Xand Y were strictly stationary and independently distrib-uted across trials. However, subjects were given no priorinformation about the payoffs produced by each cue.After the experiment, many subjects remarked that themeans seemed to vary across sessions in an unsystematicfashion. In fact, the sample means did vary acrosssessions, even though the population means were alwayszero.

Each subject received six blocks per session. Thedeadline requirement was constant within a block of 60trials, and varied across blocks in a counterbalancedorder. The values of a and k were selected at randomwith equal probabilities for any given trial. The spatialposition of the alternatives remained constant within agiven session, but alternated from left to right acrosssessions.

Initially, subjects were told that their final pay wouldbe determined by their accumulated wins and lossesacross all sessions. Because it was possible under thehigh-variance condition to win or lose a dollar on each

f(U)

k=o looFigure 2. Illustration of the effect of uncertain alternative variance (a2) on the probability of choosing theuncertain alternative, according to the fixed-sample model.


trial, there was considerable incentive to perform well.They were not told their accumulated wins until the endof the experiment. However, after the experiment wasconcluded, they were paid their cumulative wins andlosses, and they were also given a bonus for completingthe experiment.

After completing the experiment, subjects were askedto provide preference j udgments for each of the six choicepairs produced by combining the two uncertainty cues,X and Y, with the three certain values, k. The preferencescale was a 10 cm line anchored at the teft, center, andright with the labels "uncertain alternative is 10 unitsbetter," "equal," or "certain alternative is 10 units better."Subjects placed a tick mark anywhere along the scale toindicate the direction and strength of preference. Subjectswere also asked to construct separate relative frequencyhistograms for each of the uncertainty cues X and Yusing five categories of their own choosing.

Results

Choice probability.2 The proportion oftrials that the uncertain alternative was cho-sen, denoted />(UA), was estimated for eachsubject by pooling across Sessions 3 to 15.The first two sessions were treated as practice,and there were no significant training effectsfollowing the first two sessions.

Figure 3 is a plot of P(XJA) averaged acrosssubjects, as a function of the certain value,/c, with a separate curve for each level ofvariance, cr2, and a separate panel for eachtime limit, L. The expected total number ofchoice responses per data point in Figure 3equals 1,560.

First consider the a X k interaction whichis illustrated within each panel of Figure 3.This is precisely the interaction predicted bythe fixed-sample model, and it directly con-tradicts the independence between alternativesproperty implied by the simple scalabilitymodels. All 6 subjects produced this crossoverpattern, and so the crossover shown in Figure3 is not the result of averaging across differenttypes of subjects. A repeated measures anal-ysis of variance (ANOVA) indicated that thea X k interaction was significant, F(2, 10) =14.98, p < .001.

Next consider the effect of the time limit,L, on the a X k interaction. This three-wayinteraction can be seen by comparing theslopes across panels in Figure 3. Recall thatthe fixed-sample model asserts that increasingthe time limit should increase all slopes.Contrary to this prediction, increasing thetime limit actually decreased the slope for

the high-variance condition, although theslope for the low-variance condition increasedas predicted. The three-way interaction pat-tern shown in Figure 3 was statistically sig-nificant, F(4, 20) - 8.3, p < .001, and thispattern was consistent across all 6 subjects.

Table 1 illustrates the effects of the deadlinetime limit and variance on accuracy. Theobserved proportion correct within each cellof Table 1 was calculated by averaging P(VA)when k was negative, with 1 - P(UA) whenk was positive. The table shows that underthe low-variance condition, accuracy consis-tently increased as the time limit increased.But under the high-variance condition, ac-curacy consistently decreased as the timelimit increased.

Judged preference. Figure 4 is a plot ofthe mean preference as a function of thecertain value, k, with a different curve foreach level of variance, a2. Negative scalevalues indicate increasingly stronger prefer-ences favoring the certain alternative, whereaspositive scale values indicate increasinglystronger preferences favoring the uncertainalternative. Figure 4 is a replication of thea X k interaction shown in Figure 3, butbased on a single judged preference fromeach subject rather than a proportion pooledacross several hundred choice trials. All 6subjects produced the crossover interactionindicated by Figure 4, and the <r X k inter-action effect was statistically significant, F{2,10) = 10.72, MSe = 22.35, p < .01.

Mean choice response time.3 Mean choiceresponse times (CRT) for each subject wereobtained by averaging the latencies acrossSessions 3 to 15. Figure 5 is a plot of theselatencies in seconds averaged across subjects,as a function of the sure thing value, k, witha separate curve for each level of variance,<r2, and a separate panel for each deadlinetime limit, L. Note that CRT represents thelatencies pooled across both responses.

Recall that the fixed-sample model assertsthat mean choice response time is an increas-

2 The analysis of variance was computed on the arcsinetransformed proportions in order to achieve homogeneousvariance.

3 The analysis of variance was computed on choicespeed (the reciprocal of the latencies) in order to achievehomogeneous variance.


L-2

1

.9

.8 -

.7

.6 -«

A ~

.3

.2 -

.1

0 -

<r 5

I- 3 - 3

1 i . Probability of choosing the uncertain alternative, [#(UA)] plotted as a function of the certainvalue (k), with a different curve for each level of variance (<^) and a different panel for each level ofdeadline time limit (L).

ing function of sample size, n, and samplesize increases with the time limit, L. Figure5 clearly supports this expectation, and theeffect is statistically significant, F{2, 10) =44.9, p < .001.

The fixed-sample model also asserts that aspeed-accuracy trade-off relation should al-ways occur when the deadline time limit isincreased, i.e., faster responses to the shortdeadlines should produce lower accuracy.Note that this speed-accuracy trade off rela-tion failed to occur under the high-variancecondition—faster responses produced greateraccuracy when performance to the short andlong time limits are compared. The predictedspeed-accuracy trade-off relation was con-firmed for the low-variance condition.

Table iProportion Correct* for Each Combination ofDeadline Time Limit and Variance Condition

Variance condition

a = 5a= 50

1

.885

.676

Deadline time limit

2

.920

.633

3

.937

.593

* Proportion correct for each cell was calculated by av-eraging P(UA) when k = - 3 , with 1 - f{VA) whenk = +3. Each cell is based on approximately 3,120 ob-servations.

Another interesting finding in Figure 5 isthat mean choice time increased as the certainvalue, k, decreased from positive to negative.Both the main effect of the certain value andthe Certain Value X Time Limit interactioneffect were statistically significant, F\2, 10) =9 3 1 , p < ,01; F(4, 20) - 5.19, p < .05. Noneof the effects due to the variance manipulationproduced statistically significant effects onmean choice response time.

+3 0 - 3K

Figure 4. Mean preference rating plotted as a functionof certain value with a different curve for each level ofvariance. (Negative ratings indicate preference for thecertain alternative, and positive ratings indicate preferencefor the uncertain alternative.)

DECISION MAKING UNDER UNCERTAINTY

L-1 L=2 1-3

545

. 9 -

5.7

1 I\ 0

K

1•» 1

310K

\- 3

13

I0K

1- 3

. 5 -

Figure 5. Mean choice response time (CRT) in seconds plotted as a function of the certain value (fe), witha different curve for each level of variance {(P) and a different panel for each level of deadline timelimit (L).

Judged cumulative probabilities. The are shown in Figure 6, and they are comparedjudged cumulative probabilities averaged with the cumulative normal distribution. Ifacross subjects for each variance condition the judged distributions were perfectly accu-

normal

lity

obab

i

a.

>

I3

o

1

.9

.8

.7

.6

.5

.4

.3

.2

- 5 -4 - 3 - 2 - 1

Standard Score

0

2Figure 6. Mean judged cumulative distributions for the iow- and. high-variance conditions, as comparedwith the normal distribution.


rate, then all three curves would coincide.The means of the judged distributions equal.185 and —3.36 for the low- and high-varianceconditions, which in standard units equals.037 and -.067, respectively. The standarddeviations of the judged distributions equal7.36 and 67.58 for the low- and high-varianceconditions, respectively.

Discussion

Summary of major results. Probability ofchoosing the uncertain alternative increasedas the certain value decreased, and the mag-nitude of this effect was inversely related tothe variance, producing a a X k crossoverinteraction (see Figure 3). This effect wasreplicated with judged preference (see Figure4). Increasing the time limit increased accu-racy under the low-variance condition, but itdecreased accuracy under the high-variancecondition (see Table 1).

Mean choice response time also increasedas the certain value decreased (see Figure 5).Increasing the deadline time limit produceda large increase in mean response time. Vari-ance did not have a reliable effect on meanresponse time.

The finding that accuracy decreased as thetime limit increased under the high-variancecondition may have important practical im-plications. Sensitivity to the safe alternativeseems higher under short deadlines. Appar-ently, when discrimination is very poor, ad-ditional time to think about the choice de-creases sensitivity to the sure thing, and ac-curacy suffers.

Relation to prior research. An earlier studyby Myers, Suydam, and Gambino (1965)reported a a X f c interaction similar to thatshown in Figure lc. However, they used abetween subjects design, and only reportedaverages across subjects. Thus their resultscould have been artificially produced by av-eraging across uncertainty seekers and uncer-tainty avoiders, as shown in Figures la and1 b. The present study resolved this difficultyby demonstrating that the choice proportionsand single preference judgments of individualsproduced the crossover interaction pattern.

The decrease in accuracy with increasedtime limit for the high-variance conditionwas a surprising result. However, it is inter-esting to note that a similar finding occurred

in an earlier study by Irwin, Smith, andMayfield (1956), although they did not em-phasize this result in their report. Subjects inthat study were asked to decide whether themean of a deck of cards was less than orgreater than zero, based on sample sizes of10 or 20 cards. Their results indicated thatwhen discriminability was low (because of asmall mean difference and a large varianceas in the present study), accuracy in detectingpositive mean differences decreased with in-creased sample size. Together the results ofthe present study and Irwin et al. suggest thatthis finding is reliable and robust across tasks.

A possible difference between decisionmaking under risk versus partial uncertaintyis illustrated by comparing the present resultswith an earlier study by Ben Zur and Breznitz(1981). Subjects in the latter study were re-quired to choose between two gambles withina deadline time limit. Each gamble producedone of two possible outcomes, and the prob-ability of each outcome was displayed upona request from the subject. The two gambleswithin a pair had equal expected values, butthe variance differed. Ben Zur and Breznitzreported a decrease in the probability ofchoosing the high variance gamble within apair as the time limit decreased from 32to 8 s.

The present study did not replicate thiseffect. In order to compare the present resultswith those reported by Ben Zur and Breznitz(1981), the expected value of the alternativesmust be equated, which is true only for thecondition with k = 0. As can be seen inFigure 3 for k = 0, P(UA) remained at .5across all deadline conditions.

Ben Zur and Breznitz (1981) concludedthat their results were caused by increasedattention to the displayed loss probabilityunder the short time limit. In the presentstudy, outcome probabilities were not dis-played, and subjects had to rely on recall ofpast outcomes. Apparently, the tendency torecall negative or positive outcomes was notdifferentially influenced by the deadline timelimit.

Simple scalability models. Deterministic-algebraic models of risky decision making(e.g., subjective expected utility theory) cannotbe used to describe the quantitative aspectsof choice probability without making addi-


tional assumptions. The most natural way toextend these models is the class of probabi-listic models called simple scalability models.Simple scalability models continue to be pop-ular among applied researchers (business,marketing, and consumer behavior) becauseeach choice alternative can be assigned asingle scale value independent of other alter-natives, which can easily be estimated bycommonly available scaling programs.

Despite the popularity of these models,there are strong reasons for rejecting thisentire class as a representation of decisionmaking under certainty, risk, or uncertainty.Tversky (1972) has shown that this class isinadequate for describing decision under cer-tainty because the similarity between theoutcomes produced by each alternativestrongly influences choice. Becker, DeGroot,and Marschak (1963b) have shown that thisclass is inadequate for describing choice underrisk because the known correlation amongoutcomes produced by each alternative influ-ences choice. Finally, the present study hasshown that this class is inadequate for de-scribing choice under uncertainty because thevariance of the difference among outcomesproduced by each alternative influenceschoice. In all three cases, independence be-tween alternatives was violated, which is afundamental property of simple scalabilitymodels.

Fixed-sample model. Violations of theindependence property are evidenced by thecrossover interactions within each panel ofFigure 3. This crossover interaction can easilybe explained by a memory retrieval model.Presumably, subjects base their decision on asample of recalled outcomes, and comparethis sample mean (rather than a populationmean) with the certain value. The variabilityof this difference influences discriminabilitysimilar to a signal detection task. Formally,the variance of the sample mean differencedivides the mean utility of both alternatives,which causes them to be interlocked ratherthan independent.

Of course there are other possible expla-nations for the Uncertain Variance X CertainValue crossover interaction. The expectedloss ratio model (Fishburn, 1976; Myers etal., 1965) and the weak-strong conditioningmodel (Myers & Atkinson, 1964) are possi-

bilities. However, both of these models areinadequate for different reasons. The expectedloss ratio model provides no explanation forthe dynamics of the underlying decision pro-cess, and cannot explain the systematic effectsof deadline and sure thing value on choiceresponse time. Although the weak-strongconditioning model does provide a mecha-nism for describing mean choice responsetime, it does not provide any mechanism forexplaining the effects of information process-ing variables such as the deadline time limit.The fixed-sample model generates predictionsfor choice probability and response time as afunction of deadline time limit, but it occa-sionally makes incorrect predictions.

A basic property of the fixed-sample modelis the predicted speed-accuracy trade-off re-lation—increasing the time limit increasesthe number of recalled outcomes (i.e., thesample size), which then causes an increasein mean response time and accuracy. Al-though the predicted speed-accuracy trade-off was obtained under the low-variance con-dition, it was violated under the high-variancecondition.

One could argue that sample size does notnecessarily increase with the time limit, in-stead the time limit simply increases themean time to execute each of the operationsinvolved in computing the sample mean andthe comparison with the sure thing value.However, this explanation does not save thefixed-sample model. In order to explain thespeed-accuracy trade-off effect obtained underthe low variance, it would be necessary toassume that slower mean execution timesproduce fewer computational errors. But thereduction of computational errors should alsooccur for the high-variance condition.

Another problem for the fixed-samplemodel is the fact that mean choice time wasslower for negative than for positive certainvalues. This could be explained by assumingthat subjects waited for the choice stimulusto appear, and then selected a larger samplesize (i.e., recalled more past outcomes) whenthe certain value was negative. This impliesthat the factors influencing the choice ofsample size are more complicated than orig-inally assumed.

Correlated bias. One way to revise thefixed-sample model is to assume that subjects


choose the uncertain alternative only if thesample mean difference, « — u(k), is greaterthan some bias factor, b. This bias factor maybe positively correlated with the certain valueso that b > 0 when h > 0, and b < 0 whenk < 0. Without any further assumptions thepredictions remain unchanged, but if the biasis assumed to decrease in magnitude towardszero as the deadline time limit increases,then the fixed-sample model can explain thepattern of results in Table 1.

Briefly, the accuracy obtained under thehigh-variance condition may be entirely dueto the correlated bias because discriminabilityis so very poor. As the deadline increases andthe bias decreases in magnitude, then accuracywould decrease for the high-variance condi-tion. This does not happen under the low-variance condition because discriminabilityis very high, and the increase in discrimina-bility produced by the increased sample sizedominates the effect of the decrease in bias.

One problem with this explanation is thead hoc assumption that the bias decreases inmagnitude with increasing time limit. Duringthe introduction to Experiment 2, a sequen-tial-sampling model is proposed which pro-vides an explanation for the decreasing mag-nitude of the bias.

Conclusion. The results of Experiment1 indicate that the deterministic-algebraicmodels used to describe decision makingunder risk with known outcome probabilitiescannot be directly applied to decision makingunder uncertainty with outcomes learnedfrom past experience. The reason is thatwhen individuals rely on past experience,they do not assign a utility scale value inde-pendently to each alternative and comparethese values. Instead, it seems that they com-pare the estimated value of each alternative,and these estimates are based on a smallnumber of outcomes retrieved from memory.Because of the probabilistic nature of thememory retrieval process, the estimated valueof each alternative fluctuates from one deci-sion to the next. This variability influencesdiscriminability which in turn influenceschoice behavior.

Experiment 2

The second experiment used an informa-tion purchasing task to investigate decision

making under uncertainty. Subjects wereasked to choose between an uncertain alter-native and a certain alternative. The payofffor the uncertain alternative was generatedby randomly selecting a lottery ticket from afictitious urn, and the values of the tickets inthe urn were normally distributed. Prior toeach choice trial, a new urn was randomlyselected, so the population mean and varianceof the urn was initially unknown. Informationabout this distribution was learned by asequence of requests for sample observations.On each request, the computer sampled aticket from the urn, displayed the value ofthe ticket, and then returned the ticket to theurn. Although an unlimited number of ob-servations could be requested, each one costa fixed amount. In sum, a single choice trialconsisted of a sequence of sample observa-tions, followed by a terminal choice of eitherthe certain or the uncertain alternative.

One can think of the present task as adiscrimination problem—the subject has todetermine on the basis of sample observationswhether the mean of the uncertain alternativeis greater or less than the certain value. Fromthis perspective, the present task is similar tothe "expanded judgment" task investigatedby Irwin and Smith (1956; 1957).

The second experiment was designed todetermine whether or not the decreased ac-curacy with increased sample size found inExperiment 1 (see Table 1) could be replicatedby manipulating observation cost rather thana deadline time limit. A four-way factorialdesign was constructed by manipulating themean of the uncertain alternative, denotedn(x)> the variance of the uncertain alternative,denoted (^(x), the certain value, denoted /c,and the cost of purchasing observations, de-noted C.

Two decision-making models are describednext—a fixed-sample model similar to thatdescribed in Experiment 1, and a sequential-sampling model. The sequential-samplingmodel is worth considering for three reasons.One is that under certain ideal conditions, itis the optimal strategy for minimizing ex-pected losses (see Edwards, 1965). The secondis that sequential-sampling models are usefulfor describing psychophysical discrimination(see Laming, 1968; Link & Heath, 1975;Stone, 1960; Vickers, 1979). The third is that


sequential-sampling models have been appliedto decision making under certainty by As-chenbrenner, Albert, and Schmalhofer (1984),to decision making under risk by Petrusicand Jamieson (1978), and to decision makingunder uncertainty by Busemeyer (1982). Themodel developed by Busemeyer (1982) wasrestricted to binary outcomes, whereas thepresent development can be applied to eitherdiscrete or continuous outcome distributions.

Fixed-Sample Model

One possible decision strategy is to firstdecide how many observations to buy, andthen choose the uncertain alternative only ifthe average of the sample is greater than thecertain value. This is an inefficient strategyfor the information purchasing task. Unlike,Experiment 1, there is no limit on the numberof observations that can be purchased, andthe number purchased can vary dependingon the informativeness of the sample. Forexample, suppose the subject decides to pur-chase only a small number of observations,and the difference between the sample meanand the certain value is very small. In thiscase, the sample is uninformative, and it maybe worthwhile to continue sampling. Nowsuppose the subject decides to purchase alarge number of observations, but discoversafter the first few observations that the samplemean is much larger than the certain value.In this case, the small sample is very infor-mative, and it may not be worthwhile tocontinue sampling costly observations.

There is a simple way to test whether ornot subjects are using a fixed-sample strategy.If the sample size is selected before anyobservations are purchased, then the actualnumber purchased should be independent ofthe mean and the variance of the uncertainalternative distribution. This is because thedistribution properties of the uncertain alter-native will not be known when the samplesize is selected.

Sequential-Sampling Model

According to the sequential-sampling model,each observation produces an increment inpreference. This increment is denned as thedifference between the utility of the observedvalue sampled from the uncertain alternative

and the utility of the certain value. Positiveincrements increase preference for the uncer-tain alternative, and negative increments in-crease preference for the certain alternative.If the first increment is insufficient to evokea choice response, then another observationis purchased, and a new increment is pro-duced. The new increment is added to theprevious increment to produce a new cu-mulative preference. This accumulation ofincrements continues until a positive criterionis exceeded, evoking the choice of the uncer-tain alternative, or until a negative criterionis exceeded, evoking the choice of the certainalternative.

The criterion are selected by the subjectprior to sampling. Three factors are assumedto influence the selection of criterion bounds.The first factor is an individuals attitudetoward uncertainty. If the subject tends toavoid uncertainty, then the magnitude of thecertain alternative bound will be smaller thanthe uncertain alternative bound. The oppositerelation holds if the subject tends to approachuncertainty. The second factor is informationavailable before sampling begins. For example,knowledge of the certain value may influencethe magnitudes of each bound. The thirdfactor is the observation cost. Under highcost, the bounds must be close to zero sothat very few observations are purchased; butunder the low cost, the boundaries can befarther apart, allowing a stronger preferenceto accumulate before making a final decision.

A mathematical model of this process isillustrated in Figure 7. Sequential samplingproduces a sequence of utilities where w(l) isthe utility of the first sample, u(j) is theutility of the / h sample, and u(N) is theutility of the last sample before making afinal decision. A comparison of the/h samplewith the utility of the certain alternativeproduces an increment, d(j) = [u(j) - u(k)].The cumulative preference after the first sam-ple is d{\\ and the cumulative preferenceafter j + 1 samples equals D(j + 1) = D(j)+ d(j + 1). For example, in Figure 7, d{i) isthe increment and £>(3) - D(2) + d(3) is thecumulative preference following the thirdsample. The solid line in Figure 7 is a samplepath of the cumulative preference for a singlechoice trial. The dotted line in Figure 7represents the mean of the cumulative pref-


Choose Uncertain

.= 0

excess

D(12)

Figure 7. The sample path of the cumulative preferencefor the sequential-sampling model.

erence averaged across trials under identicalstimulus conditions.

The mean of the increments, symbolizedas ii{d), equals [ti(u) - u{k)], where M(W) isthe mean utility of the uncertain alternative.In Figure 7, the mean increment is repre-sented by the negative slope of the dottedline, indicating that for this example themean of the uncertain alternative is less thanthe certain value. As can be seen in thefigure, the mean increment determines thedirection and rate of the sample path. Thevariance of the increments is simply equal tothe variance of the utility of the uncertainalternative, c*{u\ because u(k) is a constant.

The criterion for choosing the uncertainalternative is symbolized as a and the criterionfor choosing the certain alternative is sym-bolized as —/3. The bounds are drawn as theupper- and lower-horizontal lines in Figure7. For example, Figure 7 indicates that thecumulative preference eventually exceededthe criterion for the certain alternative afterobserving iV = 12 samples.

Choice probability. Probability of choosingthe uncertain alternative, P(UA), is deter-mined by the probability that the cumulativepreference exceeds the uncertain criterionbefore the certain criterion. Although thehypothesized decision process is very simple,the behavioral properties that it produces arequite complex. The derivation of /*(UA) isgiven in Appendix A, and only the qualitativeproperties are described here.

Choice probability is a function of threeparameters—a measure of discriminability

(0), a measure of response bias (5), and ameasure of the distance between the bounds{A). More specifically, the distance betweenbounds is the sum of the criterion magnitudes,A = (a + /?). Increasing the distance betweenbounds increases the average strength of pref-erence required to reach a decision.

The discriminability parameter 0 is deter-mined by the distribution of the incrementsd(j). If the increments are normally distrib-uted, then 8 = 2fi(d)/a2(u\ that is, 0 is pro-portional to the mean increment divided bythe variance of the increments.

The parameter 5 is called the relative bias,and it is defined by the ratio 8 = (0 — a)/A.In other words, it reflects the difference be-tween the magnitude of the certain boundand the magnitude of the uncertain bound,relative to the total distance between thebounds. When 8 = +1 , the uncertain alter-native is always chosen; when 6 = 0, then theinitial preference is midway between thebounds; and when b = —1, the certain alter-native is always chosen. It is important tonote that a constant absolute bias, (/? - a),will have a smaller effect on the relative biasif the total distance between the bounds, A,is increased.

First consider the predictions when thebias is assumed to equal zero. Under theseconditions, P(UA) is an increasing functionof the product A[n(d)/a2(u)]. Thus, P(\JA)increases as the mean increment increasesfrom negative to positive. The slope of thisfunction decreases as the variance increases.Finally, all slopes increase as the observationcost decreases due to the increase in in thetotal distance, A. In sum, accuracy (see Foot-note 1) always increases as the observationcost decreases. Note that these predictionsare similar to those generated by the fixed-sample model (with no bias) described inExperiment 1.

Now relax the assumption that the bias isfixed at zero. Suppose that subjects adjustthe certain criterion magnitude, & dependingon the certain value, fc, producing a correlatedbias. More specifically, assume that the certainbound equals (/? - bk) where bk is positivewhen k > 0, bk is zero when k = 0, and bk

is negative when k < 0. In other words,subjects have an initial tendency to favor thecertain alternative when the certain value is


positive, and they have an initial tendency todisfavor the certain alternative when the cer-tain value is negative.

In this case, accuracy may decrease as theobservation cost decreases for the followingreason. When the variance is very high, thendiscriminability is near zero, and the cumu-lative preference wanders without any sys-tematic direction. Despite the fact that dis-criminability is low, accuracy may be highdue to the correlated bias. Under the high-cost condition, the correct criterion bound isclose to the starting point, and the cumulativepreference is very likely to hit the correctbound before wandering off in a randomdirection. However, under the low-cost con-dition, the criterion bounds are far apart, andthe correlated bias has very little effect. Thecumulative preference has ample opportunityto wander off randomly before hitting eitherthe correct or the incorrect criterion bound.

Note that these predictions are similar tothe fixed-sample model with correlated biasdescribed in the discussion of Experiment 1.However, the sequential-sampling model pro-vides a simple explanation for the decreasingeffectiveness of the correlated bias as thesample size increases. The decreased effec-tiveness is due to the property that choiceprobability is a function of the relative bias,&k = UP ~ h) -a]/[a + (fi - bk)l It is as-sumed that a and fi increase as the observa-tion cost decreases, but bk remains constant.As a and $ increase, the average sample sizeincreases, and 5* converges towards a singlevalue independent of bk.

If the correlated bias hypothesis is thecorrect explanation for the decreased accuracywith increased sample size observed in Ex-periment 1 (see Table 1), then this resultshould only occur when the certain value ismanipulated. It should not occur if the certainvalue is fixed at zero. This suggests that twodifferent patterns of results should occur inExperiment 2, depending on whether themean of the uncertain alternative or thecertain value is manipulated. If the uncertainalternative mean is fixed at zero as in Exper-iment 1, then the same pattern of resultsshown in Table I should occur in Experiment2, with the time limit variable replaced byobservation cost. However, if the certain valueis fixed at zero, then a different pattern

should occur. Decreasing the observation costshould always increase accuracy under bothvariance conditions.

Number of samples purchased. The pre-dictions for the mean number of samplespurchased, denoted E{N), are derived in Ap-pendix A, and only the qualitative propertiesare discussed here. Intuitively, E(N) is deter-mined by the average value of the finalcumulative preference divided by the meanincrement. (This is analogous to a measureof the distance traveled divided by rate oftravel.)

If the relative bias is zero, and discrimi-nability is held constant, then E(N) is anincreasing function of the total distance, A,When the relative bias is zero, and all factorsother than the mean increment are heldconstant, then E(N) is a decreasing functionof the magnitude of the mean increment.When the mean increment is zero, and allfactors other than the relative bias are heldconstant, then E(N) is a decreasing functionof the magnitude of the relative bias. Finally,a large increase in the variance reduces E(N)if the total distance, relative bias, and meanincrement are held constant. Note that thesepredictions contrast sharply with those of thefixed-sample model.

Method

Subjects, Six psychology students (5 graduate and Isenior, I male and 5 females) from Indiana University-Purdue University at Indianapolis volunteered to partic-ipate for 1 practice and 10 experimental 1-hr dailysessions. Subjects were paid according to their perfor-mance, and they earned an average of $3.75 per session.Five students verbally expressed interest throughout theexperiment. One subject (S4) became a little discouragedmidway due to initially low performance, but she com-pleted the entire experiment.

Apparatus, Same as Experiment 1.Design and procedure. The uncertain alternative gen-

erated monetary payoffs (in units of $.02) according toa normal distribution with a mean of +10, 0, or - 1 0and a standard deviation of 10 or 20 depending on thecondition. The certain alternative was set equal to - 5 , 0,or +5, and the cost of sampling each observation fromthe computer equaled 0 or 2 units depending on thecondition. Factorial combination of the uncertain alter-native mean, uncertain alternative variance, certain value,and observation cost produced 3 X 2 X 3 X 2 = 36conditions, and all 36 conditions were presented in anew random order within each block of 36 trials. Eachsession consisted of three such blocks.


The protocol for a typical trial is illustrated below:

1. Stimulus:

Lottery Batch #2176Observation Cost = 2 Certain Value = 5

Make Your Choice

2. Subject requests a sample observation by typinga " I . "

3. Screen clears.4. Stimulus:

Lottery Ticket #51328 Was WorthValue = 23

5. Delay .5 s6. Screen clears7. Program returns to line 1

(This cycle continues until a final choice is made.)8. Subject requests a terminal choice by typing a "2"

or "3 ."9. Screen clears

10. Subject initiates a new trial by pushing the returnkey.

As the display illustrates, each trial began with therandom selection of an urn or batch of lottery tickets,indicated by a new random number (e.g., #2176). Theobservation cost (e.g., C = 2) and the certain value (e.g.,k = 5) were also indicated at the start of each trial. Ifthe subject requested an observation from the computer(by typing the number 1), then the computer randomlyselected a new lottery ticket (e.g., # 51328) and printedthe value of the ticket (e.g., Value = 23). After a .5-sdelay, the screen cleared, and the subject was again facedwith the original choice problem, but now having moreinformation about the uncertain alternative. This infor-mation request cycle continued until the subject eventuallyselected a terminal choice, the certain alternative (typingthe number 2), or the uncertain alternative (typing thenumber 3). Following a terminal choice, a new trial wasinitiated.4

Subjects were told that their pay would equal theaccumulated wins and losses produced by their terminalchoices minus the cost of sampling observations. Subjectswere never shown the payoffs produced by the terminalchoice on each trial. Instead they were shown the totalaccumulated pay after each block of trials. This procedurewas used to prevent trial by trial sequential effects dueto runs of large wins or losses.

At the beginning of the experiment, subjects wereinformed (using a graphical display) that the distributionof lottery tickets was normal, but the mean and standarddeviation of the distribution would be randomly selectedfor each choice trial from independent uniform distri-butions. They were told that the mean could range from-100 to 100, and the standard deviation could rangefrom 1 to 50. This instruction was used to introduce agreat deal of uncertainty. During a postexperimentalinterview, subjects indicated that the mean did appear tofollow a uniform distribution. Also during this interview,subjects were asked to construct a relative frequencyhistogram based on their overall experience across all 10sessions, using the same method as described in Experi-ment 1. In general the subjective histograms closelyapproximated the shape of the normal distribution.

Results

After the first practice session, there wereno interactions with training, and the respon-ses were averaged across the 10 experimentalsessions. Although there were individual dif-ferences (noted later), averages across individ-uals provided an accurate representation ofthe qualitative pattern produced by the ma-jority of subjects. Figures 8a and 9a (describedin detail later) provide a graphical illustrationof the pattern of results for the two perfor-mance measures—proportion of trials thatthe uncertain alternative was chosen (denotedZ^UA) in Figure 8a), average number ofobservations purchased on each trial (denotedN in Figure 9a). Each mean is based on 180observations. Figures 8b and 9b illustrate thepredictions generated by the sequential-sam-pling model. A detailed theoretical analysisof the results is presented in the discussionsection.

Choice probability.5 Figure 8a (top) pre-sents the observed P(UA) as a function ofthe programmed mean difference [/*(*) - k],produced by each of nine combinations ofuncertain alternative mean, fi(x)> and certain

4 Subjects were also asked to provide confidence ratingsfollowing each terminal decision, but these results willnot be reported in detail. In general, confidence ratingsdecreased as the observation cost increased. Holding costconstant, confidence ratings decreased as the numberpurchased increased. However, the rate of decrease inconfidence with increased number purchased was smallerfor the high-cost condition.

5 The analysis of variance was computed on the arcsinetransformed proportions. Il should be noted that theideal method for evaluating the sequential-sampling modelwould be to consider the response probability conditionedon the cumulative preference. However, there are twoproblems with this approach. First, this method requiresestimating the sample path of the cumulative preference,D(j), (see the solid line in Figure 7). Although the exactsequence of numeric values which determine the samplepath can be recorded, the sample path underlying thedecision process is unobservable. The second problem isthat it would be impractical to estimate the probabilitiesconditioned on each possible sequence of numeric values.This method would be feasable if the outcomes weregenerated by a discrete (e.g., binary) rather than contin-uous (e.g., normal) distribution because this would reducethe number of outcome sequences to a small number.Because the purpose of Experiment 2 was to replicateand extend Experiment 1, the normal distribution wasused.

1.0

• 9

• 8

.7

.6

P(UA) 5

.4

3

.2

.1

0

DECISION MAKING UNDER UNCERTAINTY

2 a Observed C = 0

553

0=10

0=20

• • ' < • i • | | i p | | i i i | P

+5 0 - 5 +5 0 - 5 + 5 0 - 5 +5 0 - 5 +5 0 -S +5 0 -5

-10 O +10 -10 0+10

C = 2 b Predicted C = o

+5 +5 +10 +15 -13 -10 -7 -3 +3 +7 +10 +13

Figure 8. Probability of choosing the uncertain alternative [P(UA)] plotted as a function of the meandifference, with a different curve for each level of variance (a2) and a different panel for each level ofobservation cost (C). [The lop figure (8a) shows observed results, and the boltom figure (8b) shows thepredictions of the sequential-sampling model.]


N

A

N

18

16

14

12

10

8

6

4

2

0 -

18

16

14

12

10

8

6

C=2

6 - 4 4 4 - 1

bPredicted

c=o

C=2

Figure 9. Mean number of observations purchased (N) plotted as a function of the mean difference, witha different curve for each variance and observation cost condition. [The top figure (Figure 9a) showsobserved results, and the bottom figure (Figure 9b) shows predictions for the sequential sampling model.]


Table 2Probability of Choosing the Uncertain Alternative for Conditions With a ProgrammedMean Difference of+5 or - 5

C = 2

M*) - A]

c = o

Observed

.22

.16

.60

.37

Predicted

.26

.18

.61

.44

Observed

.11

.17

.65

.79

Predicted

.11

.20

.64

.79

- 1 00

0+ 10

c+5

-5+5

—5- 5

+5+5

Note. Each choice proportion was obtained by averaging .F(UA) across the low- and high-variance conditions. Predictionswere derived from the sequential-sampling model.

value, k. The panels on the left and rightdisplay the responses under the high- (C =2) and low- (C = 0) observation cost condi-tions, respectively. The solid and open dotsindicate the choice proportions for the low-and high-variance conditions, respectively.Three major patterns are apparent in Fig-ure 8a:

1. The most obvious feature of Figure 8ais the jagged sawtooth pattern obtained underthe high-cost condition (C = 2), and thesmooth ogival pattern, obtained under thelow-cost condition (C = 0). Under the high-cost condition, a decrease in P(UA) occurredwhenever the certain value changed from k --5 to k = +5 (see the dashed lines for C =2). Under the low-cost condition, an increasein P(UA) occurred whenever the certain valuechanged from k = - 5 to k = +5 (see thedashed lines for C = 0).

Four of the 6 subjects produced the saw-tooth pattern under the high cost and theogival pattern under the low cost. Of the 2that failed to show this pattern, 1 subject (S2)produced sawtooth patterns, and the othersubject (S4) produced ogival patterns underboth cost conditions. A repeated measuresANOVA indicated a statistically significant Un-certain Alternative Mean X Certain Value XObservation Cost interaction, F(4, 20) =4.05, p < .05.

Table 2 analyzes this interaction effect inmore detail. It presents P(UA) for four dif-ferent combinations of v(x) and k, averagedacross variance conditions. The first twocombinations result in a constant mean dif-ference equal to [n(x) - k] = -5, and thelast two combinations result in a constant

mean difference equal to [tt(x) — k] = +5.The first pair of columns indicate the exper-imental conditions, the next pair of columnsindicate the probabilities obtained under thehigh-cost condition, and the last pair of col-umns indicate the probabilities obtained un-der the low-cost condition. Note that changingfrom k = —5 to k = +5 decreased P(UA)under the high cost, but this same manipu-lation increased P(UA) under the low cost.The results in Table 2 suggest that the certainvalue is more effective under the high cost,and the uncertain alternative mean is moreeffective under the low-cost condition.

2. A Variance X Mean Difference interac-tion occurred under both cost conditions.This can be seen by comparing the curveunder the low-variance condition (solid dots)with the curve under the high-variance con-dition (open dots) within each panel of Figure8a. Note that for both cost conditions, thehigh-variance curve ties above when the meandifference is negative, but the low-variancecurve lies above when the mean difference ispositive. This interaction is similar to the a Xk crossover interaction observed in Experi-ment 1. Theoretically, this interaction wasdue to a reduction in discriminability underthe high-variance condition. All 6 subjectsproduced this crossover pattern, and the effectwas statistically significant.

3. Table 3 provides a detailed analysis ofthe effects that observation cost had on ac-curacy. First consider the top part of Table3, which only involves conditions with theuncertain alternative mean equal to zero, asin Experiment 1. Each of the four Variance XObservation Cost cells contain the proportion


of correct responses calculated by averagingP(UA) when k = - 5 , with 1 - P(UA) whenk = +5.

The results presented in the top half ofTable 3 replicate the pattern of results fromExperiment 1. Accuracy increased as theobservation cost decreased under the low-variance condition (.75 vs. .82). Accuracydecreased as the observation cost decreasedunder the high-variance condition (.69 vs. .66).

Next consider the bottom half of Table 3,which only involves conditions with the cer-tain value equal to zero. The proportioncorrect within each cell was obtained byaveraging the ^(UA) when n(x) = +10, with1 - P(UA) when n(x) = - 10 . As can be seenin the bottom half of Table 3, accuracyincreased as the observation cost decreasedfor both the low- and the high-variance con-ditions.

The results shown in Table 3 agree withthe predictions of the correlated bias hypoth-esis proposed in the discussion of Experiment1 and the introduction to Experiment 2. Amore detailed analysis is given in the discus-sion.

Table 3Proportion of Correct Responses for EachCombination of Variance and ObservationCost Condition

Variancecondition

a = 10observedpredicted

a - 2 0observedpredicted



C = 2

n{x) = 0

.75

.75

.69

.67

k = 0

.80

.79

.71

.71

C = 0

.82

.80

.66

.63

.98

.98

.85

.84

Note. The proportion within each cell of the top table wascalculated by averaging ^UA) given n(x) - 0 and k = - 3with [1 - .F(UA)] given /i(*) = 0 and fc = +3. The pro-portion with each cell of the bottom table was calculatedby averaging P(\JA) given k = 0 and p(x) - +10 with[1 - f(UA)] given k - 0 and p(x) = -10 . Predictionswere derived from the sequential-sampling model.

Four subjects produced the same patternof results as shown in top half of Table 3,and all 6 produced the pattern shown in thebottom half of Table 3. Two subjects (S2 andS4 again) failed to produce the pattern in tophalf of Table 3. For both of these subjects,accuracy consistently increased as observationcost decreased under both variance condi-tions.

Number of samples purchased. Figure 9a(top) presents the observed TV as a functionof the programmed mean difference, \jx(x) —k]. The pair of curves located on the top halfof the figure represent the results for the low-cost condition, and the pair of curves locatedon the bottom half of the figure represent theresults for the high-cost condition. The linesconnected by the solid and open dots repre-sent the low- and high-variance conditions,respectively. Two major patterns are apparentin Figure 9a:

1. The most salient feature of Figure 9a isthe difference in the functions relating N tothe mean difference for the low- and high-cost conditions. When the cost was high, Nwas primarily determined by the certainvalue, and N increased as certain value de-creased from +5 to —5. Five of the 6 subjectsproduced this pattern of results for the high-cost condition (S4 produced an inverted U).A repeated measures ANOVA performed onthe high cost data produced a significantmain effect due to the certain value, F(2,10) = 4.6, MSe= 1.951, p< . 01.

The effect of the certain value on N underthe high-cost condition is similar to the meanchoice time results from Experiment 1 underthe short (L = 1) deadline. This result suggeststhat the criterion bounds were more extendedwhen the certain value was negative.

2. Under the low-cost condition, N gener-ally followed an inverted U-shape function ofthe mean difference as predicted by the se-quential-sampling model. The inverted Ushape was much flatter under the high vari-ance as compared to low-variance condition.All 6 subjects produced the inverted U-shapepattern shown in Figure 9a. A repeated mea-sures ANOVA performed on the low cost dataproduced a significant Uncertain AlternativeMean X Uncertain Alternative Variance XCertain Value interaction effect, F(4, 20) =5.26, MSe - 1.4125, p < 01.


Recall that the fixed-sample model pre-dicted no effect of the uncertain alternativemean and variance on the number sampled.Contrary to this prediction, the mean andvariance had an effect on the number pur-chased for all 6 subjects.

Discussion

Summary of major results. Probability ofchoosing the uncertain alternative was anincreasing function of the difference betweenthe uncertain alternative mean and the certainvalue (see Figure 8). Under the high-obser-vation cost condition the shape of this func-tion was sawtooth, suggesting that the certainvalue had a much stronger effect on choicethan the uncertain alternative mean whensampling was severely restricted. Under thelow-cost condition, the shape of this functionwas smooth and ogival, suggesting that theuncertain alternative mean had a muchstronger eifect on choice than the certainvalue when sampling was unrestricted.

The slope of the function relating the meandifference to choice probability was muchsteeper under the low- as compared with thehigh-variance condition. This produced aVariance X Mean Difference interaction, ex-tending the a X k interaction reported inExperiment 1. Theoretically, discriminabilitywas reduced by increasing the variance.

On the one hand, when the uncertainalternative mean was fixed at zero, then de-creasing the observation cost increased ac-curacy under the low variance, but decreasedaccuracy under the high variance, replicatingthe pattern obtained in Experiment 1 (com-pare Table 1 and the top half of Table 3). Onthe other hand, when the certain value wasfixed at zero, then decreasing the observationcost always increased accuracy under bothlow- and high-variance conditions (see thebottom half of Table 3). Both patterns (Table3) were exactly in accord with the predictionsof the correlated bias hypothesis.

Under the high-cost condition, the meannumber of observations purchased increasedas the certain value decreased, similar to thechoice, response time results of Experiment1. Under the low-cost condition, the numberpurchased was an inverted U-shape functionof the mean difference. The inverted U-shape

function was flatter under the high- as com-pared with the low-variance condition (seeFigure 9). The strong influence of the uncer-tain alternative mean and variance on thenumber purchased rules out the fixed-samplemodel.

Relation to prior research. Irwin andSmith (1956, 1957) investigated the effects ofmean and variance on number of observationsrequired to determine whether the mean ofa deck of cards was positive or negative.Similar to the present results, they found thatthe number of observations requested was aninverted U-shape function of the card deckmean. Unlike the present results, they foundthat increasing the variance always increasedthe number of observations required to reacha decision independent of the mean of thedeck.

There are several differences between thepresent study and those of Irwin and Smith(1956, 1957). This includes the form of thestimulus distribution, selection of mean andvariance parameters, amount of training,payoff procedure, task instructions, and sub-ject populations. Further research is neededto isolate the particular cause for the differentresults.

Theoretical analysis. One may questionwhether or not the sequential-sampling modelcan explain the complicated pattern of resultsobserved in Experiment 2. To answer thisquestion, it was necessary to obtain quanti-tative fits. The major assumptions used to fitthe model are described in the next fewparagraphs, and the details are given in Ap-pendix A. In brief, eight parameters wererequired to fit the 36 observed choice pro-portions in Figure 8. These same eight pa-rameters were then used to generate predic-tions for the number of observations pur-chased shown in Figure 9. The predictionsfor number purchased were adjusted by fittinga slope and intercept parameter separatelyfor each observation cost condition. The fittedresults are shown in Tables 4 and 5 andFigures 8b and 9b. The first four columns ofTables 4 and 5 indicate the experimentalcondition, the next two columns indicate thederived parameters,6 and the last two pairs

6 Only eight parameters were fit to the data. Thederived parameters were calculated from these eight. The


Table 4Observed P(UA) and Predicted P(UA) Probabilities of Choosing the Uncertain Alternative,Observed N and Predicted N Number of Observations Purchased

c000

000

000

000

000

000

Experimental condition

•W

101010

101010

101010

202020

202020

202020

*W

- 1 0- 1 0- 1 0

000

+ 10+ 10+ 10

- 1 0- 1 0- 1 0

000

+ 10+ 10+ 10

1

k

+50

- 5

+50

- 5

+50

- 5

+50

—5

+50«

+50

- 5

Derivedparameters

AB

-9 .6-7.2-4.8

-2.402.4

4.87.29.6

-3.02-2.26-1.52

-.760

.76

1.522.263.02

b

-.3-.3-.3

-.3-.3-.3

-.3-.3-.3

.3-.3-.3

-.3-.3-.3

.3-.3-.3

f(UA)

.02

.01

.05

.11

.39

.74

.87

.96

.98

.06

.09

.17

.23

.40

.56

.71

.79

.86

Observed and

f\UA)

.001

.004

.02

.11

.38

.72

.90

.97

.99

.06

.11

.19

.29

.42

.56

.69

.79

.86

predicted values

N

8.310.012.5

12.914.316.3

14.611.611.5

9.210.812.1

12.913.812.8

13.512.413.1

N

9.910.611.9

14.116.115.7

13.812.311.2

10.811.311.8

12.212.512.5

12.311.911.5

Note. Predictions were derived from the sequential-sampling model. Global fit index R2 = .996 for P0JA) and .708for N; Global fit index RMS = .022 for P(VA) and 1.02 for N.

of columns present the observed and predictedvalues. Included in the note for Table 4 arethe two global fit indices for each performancemeasure—R2 is the percentage of variancepredicted by the model, and RMS is the rootmean square error.

The first question is how does the modelexplain the interaction shown in Table 2?According to the correlated bias hypothesis,subjects select a larger or smaller certaincriterion magnitude depending on the certainvalue. This bias has a very strong effect ohchoice probability under the high-cost con-ditions, because the criterion bounds areclose to the starting point of the cumulativepreference. Thus the results shown under the

derived parameter AS represents the discriminabilitytimes the total distance between the bounds, and 5 is therelative bias. The measures A and 8 are not presentedseparately because they are not uniquely identified, butthe product is uniquely identified. See Appendix A formore details.

C = 2 column of Table 2 were due to thestrong influence of the correlated bias.

Under the low-cost condition, subjectsmaintain a correlated bias, but they alsoincrease the total distance between the boundsto allow stronger preferences to accumulatebefore reaching a decision. As noted earlier,a large increase in the total distance practicallyeliminates the effect of the correlated bias.This analysis implies that the cell differencesunder column C = 0 in Table 2 should havebeen in the same direction as those foundunder column C = 2, but reduced nearly tozero in magnitude. On the contrary, theobserved differences in each cell under thecolumn C = 0 are in the opposite direction.

The results in Table 2 suggest that thecertain value was less effective than the un-certain alternative mean for determiningchoice under the low observation cost con-dition. One reason may be that subjects oc-casionally forget to subtract the utility ofcertain value during preference accumulation,


particularly after a large number of observa-tions. Define p as the probability that thesubject forgets to subtract the utility of thecertain value when computing the incrementd(j). Then the mean increment equals ft(d) =[fi(u) — (1 — p)u(k)]. For simplicity, assumethat the utility function is linear, so that themean increment can be expressed as fi(d) =Mx)-(\~p)k], where pt(x) is the pro-grammed mean for the uncertain alternative,and k is the certain value. Setting p = 0 forthe high observation cost and setting p = (1/3) for the low observation cost produces themean increments shown on the abcissa ofFigure 8b. Note that the mean increments,fi(d), are in perfect rank order agreementwith the observed choice probabilities for thelow observation cost condition in Figure 8.

The second question is how does the modelexplain the crossover interaction between themean difference and the variance shown inFigure 8? This is easily explained as the resultof a reduction in the discriminability param-

eter, 6, which is the ratio of the mean incre-ment divided by the variance of the incre-ment. In other words, preferences bounce upand down haphazardly under large variances,but they cumulate systematically in the cor-rect direction under small variances.

The third question is how does the modelexplain the interaction shown in Table 3.First consider the top half of Table 3 withthe uncertain alternative mean equal to zero.In this case, the correct decision depends onthe sign of the certain value. By adjusting thecertain criterion magnitude according to thecertain value, accuracy can be improved.However, this improvement is largely elimi-nated when the total distance between boundsis large. Thus, decreasing the observation costhas two opposing effects. On the one hand,it reduces the effect of the correlated biaswhich reduces accuracy and, on the otherhand, it increases the average sample sizewhich increases accuracy. When the varianceis large and discrimination is very poor, the

Table 5Observed P(UA) and Predicted f*(UA) Probabilities of Choosing the Uncertain Alternative,Observed N and Predicted N Number of Observations Purchased

c222

222

222

222

222

222

Experimental condition

•w101010

101010

101010

202020

202020

202020

MW

-10-10-10

000

+ 10+ 10+ 10

-10-10-10

000

+ 10+ 10+ 10

k

+50

-5

+50

—5

+50

- 5

+50

-5

+50

-5

+50

-5

Derivedparameters

Ad

-2.66-2.0-1.32

-.8801.32

.882.04.0

-1.42-1.06

-.70

-.460

.70

.461.062.14

5

.94

.72

.3

.94

.72

.3

.94

.72

.3

.94

.72

.3

.94

.72

.3

.94

.72

.3

AUA

.02

.05

.16

.13

.32

.63

.42

.64

.89

.10

.16

.27

.19

.37

.57

.32

.57

.72

Observed and predicted values

) W A )

.03

.07

.21

.14

.31

.64

.44

.66

.91

.09

.16

.30

.22

.35

.57

.43

.58

.80

N

1.11.52.2 :

1.31.82.4 ;

1.41.92.3 ;

1.01.52.0

1.21.62.2

1.21.62.1

N

.1

.4

.5

.81.6

.7

.91.0

.2

.3

.9

.3

.5

.9

.2

.3

.9

Note. Predictions were derived from the sequential-sampling model. Global fit index R2 = .977 for P(UA) and .808for N\ Global fit index RMS = .0375 for P(UA) and .188 for N,


correlated bias has a larger effect on accuracythan does the increased sample size. Theopposite is true when the variance is small.This explains the crossover interaction in thetop half of Table 3.

Next consider the bottom half of Table 3,with the certain value fixed at zero. In thiscase the bias does not vary, and only thedistance between bounds changes as the ob-servation cost decreases. Increasing the totaldistance increases the average number ofsamples purchased, and accuracy improvesfor both variance conditions.

The last question is how does the modelexplain the different pattern of results ob-tained for the mean number purchased underthe low- and high-observation cost conditions(see Figure 9)? Under the high cost, thecriterion magnitudes are very small and thenumber purchased is largely determined bythe adjustment of the certain criterion mag-nitude correlated with the certain value. Un-der the low-cost conditions, the criterionmagnitudes are very large, and the meannumber purchased is largely determined bythe mean increment or average rate of move-ment towards each criterion bound.

In sum, by comparing the predicted withthe observed patterns in Figures 8 and 9, andTables 2 and 3, it is clear that the modelprovides a fairly complete description of thequalitative pattern of results. Although thereare some minor quantitative deviations, thesedo not appear severe enough to warrantrejecting the model.

Individual differences. Four subjects pro-duced the pattern of results illustrated inFigure 8a. However, 2 subjects produced in-consistent patterns—subject S2 produced sawtooth patterns, whereas subject S4 producedogival patterns in both observation cost con-ditions. The behavior of subject S4 can beexplained by assuming that this subject wasunbiased across all conditions. The behaviorof subject S2 is more difficult to explain.Previously, it was assumed that the adjustmentof the certain criterion bound was constant,whereas the relative bias, 6, varied acrossobservation cost conditions. One explanationfor the pattern produced by subject S2 is thatthis subject maintained a constant relativebias, 5, by allowing the adjustment of thecertain criterion magnitude to vary across

observation cost conditions. If the relativebias remains constant across observation costconditions (as assumed for both S2 and S4),then the model asserts that accuracy alwaysincreases as the observation cost increasesbecause of the increased total distance. Thiswas exactly the pattern of results obtainedfrom subjects S2 and S4.

Conclusion. When individuals are giventhe opportunity to learn the outcomes pro-duced by an uncertain action, and the numberof observations are limited only by the costof sampling, then they tend to use a sequen-tial- rather than a fixed-sample decision pro-cess. According to the sequential-samplingmodel, individuals continue sampling obser-vations until their cumulative preference ex-ceeds a criterion bound for one of the alter-natives. The criterion boundaries are deter-mined by individual attitudes towardsuncertainty, by prior information such as thevalue of the certain alternative, and by theobservation cost. This model provided a fairlycomplete explanation for the complex patternof results observed in the second experiment.

Several assumptions were required to ex-plain the results. Most of the assumptionsare not new, but are similar to those com-monly made by all sequential-sampling the-ories of decision making (e.g., the effects ofmean and variance on discriminability, andthe effects of observation costs on the criterionmagnitudes). Two assumptions were new—the correlated bias hypothesis was needed toexplain the fact that the certain value had astronger effect than the uncertain alternativemean when sampling was severely limited,and the forgetting hypothesis was needed toexplain the fact that the uncertain alternativemean had a stronger effect than the certainvalue when sampling was unlimited. A sim-pler set of assumptions may be able to explainpart of the results, but it appears that boththe correlated bias and the forgetting hypoth-esis are needed to explain the entire patternfor both choice probability and number ofobservations purchased.

General Discussion

This article began by making a distinctionbetween decision making under risk (perfectknowledge of outcome probabilities) and de-


cision making under uncertainty (outcomesprobabilities initially unknown and learnedthrough experience). Although the latterproblem seems more realistic, the majorityof past research has been concerned withdecision making under risk. Two experimentswere reported which investigated the cognitiveprocesses involved in decision making underuncertainty—the first used a probabilitylearning task, and the second used an infor-mation purchasing task. What has beenlearned from these experiments? Three con-clusions are discussed as follows:

First, two new empirical findings werediscovered. One finding was the crossoverinteraction shown in Tables 1 and 3. Intu-itively, one would expect that if a decisionmaker has more time to decide, or observesmore sample outcomes, then the likelihoodof choosing the correct alternative (see Foot-note 1) should increase. However, if the choiceis between a nonzero sure thing alternative,and an alternative that produces a wide dis-persion of outcomes centered at zero, thenthe opposite tends to occur—the likelihoodof choosing the correct alternative decreases.

Another finding was the crossover interac-tion shown in Table 2. If a decision must bebased on a small number of observationssampled from an uncertain alternative, thenthe value of a certain alternative is moreeffective than the mean of the uncertainoutcomes for determining choice. The oppo-site tends to be true when the decision isbased on a large number of observations.

The second conclusion is that many ofthe algebraic-deterministic models developedwithin the risky decision-making paradigmcannot be applied to decision making underuncertainty in any simple manner.7 Thesetheories typically assume that each alternativeis assigned a single scale value, such as asubjective expected utility. Simple scalabilitymodels are probabilistic versions of thesealgebraic-deterministic models. In both of thereported experiments, a fundamental propertyof simple scalability, the independence be-tween alternatives property, was violated byall subjects.

The third conclusion is that decision mak-ing under uncertainty involves learning andmemory processes when the outcome prob-abilities are learned through experience. There

are two ways that an individual can decideunder uncertainty: one is to recall outcomesof previous decisions under similar circum-stances, and another is to observe new out-comes sampled from the unknown distribu-tion before making a final decision. In eithercase, a fixed- or sequential-sampling strategycould be used. If a fixed-sample strategy isused, the number of past outcomes recalledor the number of new outcomes observed isdetermined first. Then a decision is made byselecting the alternative producing the largeraverage utility. If a sequential-sampling strat-egy is used, the decision maker continuesrecalling past outcomes or continues observingnew sample outcomes until a cumulativepreference exceeds a critical bound for oneof the alternatives. The cumulative preferenceis formed by comparing the utilities of theoutcomes produced by each alternative, andsumming these individual comparisons.

What determines whether a fixed- or se-quential-sampling strategy will be selected?In the first experiment, the use of a deadlinetime limit was likely to encourage a fixed-sample strategy for the following reason. Pre-sumably, subjects believe that the probabilityof making the correct decision is increasedby recalling more past outcomes. Althoughthe number retrieved is limited by the dead-line, no loss is produced by recalling as manypast outcomes as possible within this limit.Thus the maximum number possible is re-called. In the second experiment, the numberof observations that could be purchased wasunlimited, but each one cost a fixed amount.This procedure would tend to encourage asequential-sampling strategy because the sub-ject must simultaneously maximize accuracyand minimize the number purchased. Thussubjects will continue sampling if the cumu-lative preference is weak in order to increaseaccuracy, but they will stop sampling if thecumulative preference is strong in order tominimize the observation cost.

The results of Experiment 2 clearly indicatethat all subjects adopted a sequential-samplingstrategy. The results of Experiment 1 do notprovide a clear cut way to determine whether

7 One exception is the additive difference model (Tver-sky, 1969) which is similar to the sequential-samplingmodel.


a fixed- or sequential-strategy was used. Bothmodels are capable of accounting for theobserved pattern of results if a correlated biasis assumed.

In summary, the present research suggeststhat more attention should be given to theoriesof decision making that emphasize learningand memory retrieval (e.g., the fixed- andsequential-sampling models) rather than con-centrating exclusively on deterministic-alge-braic theories (e.g., subjective expected utilitymodels). There are a number of questionsabout the nature of the decision processesthat remain to be answered. First, how arepreferences accumulated when a sequential-sampling strategy is assumed? The presentanalysis proposed one idea, but others arepossible such as the "accumulator" modeldescribed by Vickers (1979). Second, whatfactors influence the selection of criterionbounds for the sequential-sampling strategy?For example, suppose the observation costsrapidly increased across samples rather thanremaining constant as in Experiment 2. Ac-cording to the optimal model (Rapoport &Burkheimer, 1971), the criterion bounds de-crease in magnitude after each observation ispurchased as a consequence of rising obser-vation costs. Finally, how could the model beextended to more than two alternatives? Rat-cliff (1978) has proposed one approach, butthere are several other possibilities. The an-swer to these questions must await furtherresearch on decision making under uncer-tainty.

References

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Appendix A

The predictions generated by the sequential-sampling model are based on the theory of randomwalk stochastic processes (Cox & Miller, 1965,chap. 2 & 3). There are two methods for derivingthese predictions, and both methods provide onlyapproximations. The first method requires one toassume that the excess over the boundaries (seethe difference between D(12) and the lower boundin Figure 7) is small relative to the criterionmagnitude, so that it can be ignored. This is nota reasonable assumption for the present study(because of the small mean number of samplespurchased in Experiment 2 under the high-costcondition), so an alternative approximation methodwas used. Stein & Rapoport (1978) recommendedusing a finite state Markov chain to approximatethe continuous state Markov process of the randomwalk D(j). This approach was realized by discre-tizing the continuous interval [—fiy a] into mequally spaced intervals, and constructing a tran-sient state matrix that approximated the transitiondensities f[D(j + l)\D(j)] = / K / ) ] for the normaldistribution. The theory for finite state Markovchains with absorbing states (—oo, —0) and (a,+oo) was then used to calculate P(UA) and E(N).

The width of each discrete state was defined asw = (a + (3)/m. A set of m discrete preferencestates was constructed from the original continuouspreference states by denning the discrete states asD; = a — (i — .5)H>. The probability of a transitionfrom state D{ to state D} was set equal to Tu =F[tt ~ j)w + -5w] — F[(i — j)w — ,5w], where Fis the cumulative normal distribution functionwith mean n(d) and variance o2(u). Let T = [Ty]represent the m X m transient state probabilitymatrix.

The probability of exceeding a from state Z),equals rn = 1 - F[{i ~ .S)w\, and the probabilityof exceeding p from state D{ equals ra = F[(i -m — .5)w]. Let R = [/•» ra\ represent the m X 2response probability matrix.

Define ir as the initial probability row vector. Itwas assumed that the cumulative preference alwaysstarts at zero so that all elements of -K were set tozero except for the element corresponding to the

zero preference state which was set to unity. P(\JA)can be obtained from the matrix equation

[P(UA) 1 - P(UA)] = *(I - T)' lR, C

where I is the identity matrix. The mean numberof samples purchased can be calculated from thematrix equation

E(N) = T)"2RJ,

where J is a two-element column vector with eachelement set to unity.

For both cost conditions, the mean incrementwas set to ti{d) - ix{x) - (1 - p)K where fi(x) isthe programmed mean of the uncertain alternative.For the high cost p was set to zero, and for thelow cost p was set to 1/3. The variance of theutility of the uncertain alternative was set equalto the programmed variance.

The criterion bounds were estimated from thechoice data using the following assumptions. Forboth cost conditions, two uncertain criterionbounds were estimated—one for each variancecondition. Note that changes in the criterion mag-nitude with changes in variance may simply reflectthe possibility that the variance of the utilities wasnot equal to the programmed variance. This isdue to the fact that the variance of the incrementsand the criterion magnitudes trade off, so changesin one parameter can be offset by changes inanother.

The treatment of the certain criterion bounddepended on the observation cost condition. Forthe high-cost condition, the certain criterion mag-nitude was set equal to the uncertain criterionbound plus an adjustment factor, bk, which de-pended on the certain value k. Under the low-costcondition, it was assumed that for all practicalpurposes the effect of the correlated bias wasreduced to zero. Thus the relative bias was assumedto be constant, and this was achieved by forcingthe certain criterion magnitude to be proportionalto the uncertain criterion magnitude.

In sum, a total of three parameters were esti-mated under the low-cost condition: one relative


bias, and two uncertain criterion bounds. A totalof five parameters was estimated from the choiceproportions obtained under the high cost: twouncertain criterion bounds, and three adjustmentsfor the certain criterion bound.

The parameters were estimated by minimizingthe sum of squared error between the modelpredictions and the sample means shown in Figure8a. A hill climbing grid search algorithm was usedwith a fairly course grid. The fits could probablybe improved with a more efficient search algorithm.

The parameter estimates obtained from thechoice proportions were then used to generatepredictions for the mean number of samples pur-chased. However, the predicted values were relatedto the observed values by an arbitrary multiplicativeconstant because the mean increment is only

denned up to an arbitrary constant on the basisof the choice data. Thus a linear regression equationwas used to generate the predictions reported inTable 4. A separate linear regression equation wasused for the low- and high-observation cost con-ditions, because the mean increment changedacross these two conditions.

The discriminability parameter 0 can be multi-plied by an arbitrary constant, and the criterionbounds can be divided by this same constant toproduce identical predictions for choice probability.Thus only the product A6 and the relative bias 5can be unambiguously interpreted. These param-eter values are reported in Table 4.

Received June 24, 1984Revision received November 30, 1984 •

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