Information Sampling Behavior With Explicit Sampling Costs...time, energy, and money spent acquiring...

Information Sampling Behavior With Explicit Sampling Costs

Mordechai Z. JuniUniversity of California, Santa Barbara

Todd M. Gureckisand Laurence T. Maloney

New York University

The decision to gather information should take into account both the value of infor-mation and its accrual costs in time, energy and money. Here we explore how peoplebalance the monetary costs and benefits of gathering additional information in aperceptual-motor estimation task. Participants were rewarded for touching a hiddencircular target on a touch-screen display. The target’s center coincided with the meanof a circular Gaussian distribution from which participants could sample repeatedly.Each “cue”—sampled one at a time—was plotted as a dot on the display. Participantshad to repeatedly decide, after sampling each cue, whether to stop sampling and attemptto touch the hidden target or continue sampling. Each additional cue increased theparticipants’ probability of successfully touching the hidden target but reduced theirpotential reward. Two experimental conditions differed in the initial reward associatedwith touching the hidden target and the fixed cost per cue. For each condition wecomputed the optimal number of cues that participants should sample, before takingaction, to maximize expected gain. Contrary to recent claims that people gather lessinformation than they objectively should before taking action, we found that partici-pants oversampled in one experimental condition, and did not significantly under-sample or oversample in the other. Additionally, while the ideal observer model ignoresthe current sample dispersion, we found that participants used it to decide whether tostop sampling and take action or continue sampling, a possible consequence ofimperfect learning of the underlying population dispersion across trials.

Keywords: information gathering behavior, optimal stopping, cost-benefit analysis,ideal observer model, sensitivity to sample dispersion

Human decision makers face a challenge:how to balance the advantage obtained fromgathering additional information against thetime, energy, and money spent acquiring it.Stigler (1961) analyzed the economic costs ofprolonging a search for a better price on a

commodity. Since a relatively cheap price isoften obtained after a relatively brief search, thecost of an exhaustive search will often not offsetthe savings gained by finding the cheapest price.Instead, consumers should continue searchingonly so long as the expected savings from find-

This article was published Online First December 14,2015.

Mordechai Z. Juni, Department of Psychological andBrain Sciences, University of California, Santa Barbara;Todd M. Gureckis, Department of Psychology, New YorkUniversity; Laurence T. Maloney, Department of Psychol-ogy and Center for Neural Science, New York University.

The U.S. Government is authorized to reproduce and dis-tribute reprints for Governmental purposes notwithstandingany copyright annotation thereon. The views and conclusionscontained herein are those of the authors and should not beinterpreted as necessarily representing the official policies orendorsements, either expressed or implied, of IARPA, DOI,or the U.S. Government. Parts of this study were presented atthe 33rd Annual Conference of the Cognitive Science Society(Juni, Gureckis, & Maloney, 2011). Laurence T. Maloney

was supported by National Institutes of Health (NIH) GrantEY019889. Mordechai Z. Juni was supported by NIH GrantT32 EY007136, and by the Intelligence Community Postdoc-toral Research Fellowship Program through funding from theOffice of the Director of National Intelligence. Todd M.Gureckis was supported by National Science FoundationGrant BCS-1255538, by a John S. McDonnell FoundationScholar Award, and by the Intelligence Advanced ResearchProjects Activity (IARPA) via Department of the Interior(DOI) Contract D10PC20023. We thank Nathaniel Daw, TimPleskac, and Ed Vul for helpful discussions in the develop-ment of the paper.

Correspondence concerning this article should be ad-dressed to Mordechai Z. Juni, Department of Psychologicaland Brain Sciences, University of California, Santa Bar-bara, CA 93106-9660. E-mail: [email protected]

THIS ARTICLE HAS BEEN CORRECTED. SEE LAST PAGE

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Decision © 2015 American Psychological Association2016, Vol. 3, No. 3, 147–168 2325-9965/16/$12.00 http://dx.doi.org/10.1037/dec0000045

147

mailto:[email protected]

http://dx.doi.org/10.1037/dec0000045

ing a cheaper price are enough to offset theadditional costs in prolonging the search. Sim-ilar ideas are reflected in the mathematical lit-erature on optimal sampling and optimal search(Wald, 1945a, 1945b; Arrow, Blackwell, & Gir-shick, 1949; Stone, 1989), and feature promi-nently in the study of animal foraging (Stephens& Krebs, 1986). The present study exploreshow effective humans are in trading off thecosts and benefits of gathering additional infor-mation in a perceptual-motor estimation taskwith explicit sampling costs.

Prior Work on InformationSampling Behavior

Information gathering (or “sampling”) be-havior was a topic of interest starting in the1960s (Edwards, 1965; Green, Halbert, & Mi-nas, 1964; Rapoport & Tversky, 1970;Wendt,1969). Tversky and Edwards (1966), for exam-ple, had participants perform a probability-learning task. On each of 1,000 trials, partici-pants had to guess which of two mutuallyexclusive alternatives would occur. Althoughparticipants were rewarded at the end of theexperiment for all of their correct guesses, theyreceived no feedback following each guess. Togain information about the relative frequenciesof the mutually exclusive alternatives, partici-pants could forgo guessing on any particulartrial and observe the outcome instead. Theyforfeited any possible reward on trials that theychose to observe which event would occur in-stead of guessing.

The optimal strategy maximizing expectedgain in that study was to observe a certainnumber of trials at the start of the experiment,keep track of the relative frequencies of themutually exclusive alternatives, and thenchoose on the remaining trials the alternativethat had occurred more frequently (Wald,1947). Participants in Tversky and Edwards(1966) greatly oversampled information, ob-serving around 300 trials spread out throughoutthe experiment, compared to the optimal sam-pling strategy of observing fewer than 33 or 38trials (depending on the condition) at the start ofthe experiment. Busemeyer and Rapoport(1988) investigating a similar task also reportedoversampling in some experimental conditions.

Many recent studies, using card-samplingtasks for example, have not reported oversam-

pling however (Hau, Pleskac, Kiefer, &Hertwig, 2008; Hertwig, Barron, Weber, &Erev, 2004; Ungemach, Chater, & Stewart,2009). Participants in these studies were pre-sented with two decks of cards that had differ-ent, unknown payoff distributions. To gain in-formation about the payoff distribution of eachdeck, participants were encouraged to freelysample as many cards as they wished “until theyfelt confident enough to decide from which[deck] to draw for a real payoff” (Hertwig et al.,2004). The single random draw for a real payoffat the end of the task was, in effect, the execu-tion of a lottery where the probabilities of dif-ferent monetary rewards were initially unknownbut could be learned by sampling.

A common conclusion of these recent studieswas that the median number of cards that par-ticipants sampled across both decks (between11 and 19 cards depending on the study) wassurprisingly small. Sampling more cards wouldlead to better estimates of the respective payoffdistributions of the decks (Hau et al., 2008;Hertwig et al., 2004; Ungemach et al., 2009).1

A limitation of these recent studies, however,is that the experimental designs could not beused to test whether implicit costs of sampling(e.g., the participant’s time, boredom, etc.)could explain participants’ sampling behavior.If there were absolutely no costs to sampling,then the optimal strategy maximizing expectedgain would be to continue sampling indefinitely(or until the decks are depleted). No matter howmany cards one has sampled already, additionalsamples lead to better estimates of the respec-tive payoff distributions of the decks. Comparedwih the optimal sampling strategy, participantscould only undersample. On the contrary, if thecosts of sampling were large enough, then theoptimal behavior would correspond to not sam-pling at all, simply picking one of the decks atrandom without learning anything about eachdeck’s payoff distribution and expected reward.

1 The frequent reports of participants taking a small num-ber of samples in recent studies have led a number of papersto argue that taking a small number of samples can beadvantageous in certain situations (Todd, 2007; Vul et al.,2014). For example, Hertwig and Pleskac (2010) present ananalysis showing that taking a small number of samplesmay amplify differences between the payoff distributions ofdifferent options (cf., Kareev, 2000), thus making it easierto choose between them (meeting the criterion of satisficinglaid out by Simon, 1956).

148 JUNI, GURECKIS, AND MALONEY

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In such a case, any sampling at all would countas oversampling.

It is possible that participants in these studiesstopped sampling cards because of implicitsampling costs.2 But because we do not knowwhat those costs are, it is incorrect to concludethat they undersampled.

The goal of the present study was to examineinformation sampling behavior in a task whereparticipants incur explicit sampling costs andexplicit rewards for being successful and theoptimal sampling rule is amenable to mathemat-ical analysis. We will show in the Discussionsection that—by comparison of performanceacross experimental conditions—we can testwhether implicit costs or rewards could explainparticipants’ sampling behavior. This approachallows us to characterize participants’ informa-tion sampling behavior: that is, whether theyoversampled, undersampled, or sampled opti-mally relative to a well-defined and realisticideal observer model (Geisler, 2003).

Overview of the Current Experiment

Participants performed a perceptual-motorestimation task similar to those used by Batta-glia and Schrater (2007) and Tassinari, Hudson,and Landy (2006). The goal on each trial was totouch a hidden circular target on a touch-screendisplay to earn points. The target’s radius re-mained constant throughout the experiment.The target’s location changed randomly fromtrial to trial but was cued by dots drawn from acircular Gaussian that was centered on the mid-dle of the hidden target. We refer to these dotsas “cues.” The pointing strategy that maximizesthe probability of touching the hidden target isto touch the screen at the center of gravity of thecues (i.e., the sample mean).

Participants were given the option to samplecues—one at a time—at a fixed cost per cue.The more cues the participant sampled, thegreater the probability that the sample meanwould fall within the target region (because thevariance of the sample means decreases withincreased sample size) and the greater the prob-ability that the participant would successfullytouch the hidden target. Previous studies withsuch tasks have shown that participants cor-rectly interpret the arrival of more cues as re-ducing external uncertainty about the target’slocation, and that they are more successful in

localizing the hidden target with more cues(Battaglia & Schrater, 2007; Tassinari et al.,2006).

In the current experiment, however, the ben-efit of sampling additional cues to reduce exter-nal uncertainty about the target’s location cameat a fixed monetary cost. The critical question iswhether participants sample the optimal numberof cues before taking action (i.e., before at-tempting to touch the hidden target): partici-pants had to balance the benefit of gatheringadditional information (more cues) against thecost incurred to obtain it (Dieckmann &Rieskamp, 2007; Fu & Gray, 2006).

There is a growing interest in exploring dif-ferent forms of decision making where proba-bilities arise from different sources, perceptualjudgments, motor acts, or explicitly given nu-merical values (Maloney & Zhang, 2010; Sum-merfield & Tsetsos, 2012; Trommershäuser,Maloney, & Landy, 2008; Trueblood, Brown,Heathcote, & Busemeyer, 2013 Wu, Delgado,& Maloney, 2009). If undersampling behaviorwere a robust aspect of human judgment anddecision making, we would expect to see itemerge in many types of tasks beyond thosecommonly employed in the cognitive decision-making literature. Thus, our paper represents anattempt to generalize past findings to a widervariety of tasks.

Experimental Task

The participant’s goal was to collect pointsby touching an invisible circular target placed atrandom on a touch-screen display. To gain in-formation about the location of the hidden tar-get, participants could request to observe theoutcome of a theoretical dart thrower who couldsee the hidden target (unlike the participants)and who would throw a dart toward it, aimingfor its center. Participants could use the re-peated outcomes of this theoretical person’s re-peated throws as cues to the location of thehidden target.

2 Another possibility is that participants stopped samplingcards because of working memory limitations (Hertwig etal., 2004). There is little point in sampling additional cardsif the participant cannot use them to update his or herestimate of the deck’s payoff distribution and expectedreward.

149INFORMATION SAMPLING BEHAVIOR

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Each participant alternated between blocks oftwo different conditions that varied in the cost-benefit structure of the task: (a) a low stakescondition and (b) a high stakes condition. In thelow stakes condition, the reward for touchingthe hidden target was initialized at 40 points oneach trial and decreased by two points for everycue that was requested. In the high stakes con-dition, the reward for touching the hidden targetwas initialized at 60 points on each trial anddecreased by six points for every cue that wasrequested.

Figure 1 illustrates two examples of the se-quence of events within a trial. In the schematicexample of a low stakes trial (left-hand panels),

the participant chose to stop sampling and at-tempt to touch the hidden target after requestingnine cues. The bottom panel shows that theparticipant successfully touched the hidden tar-get (the square response dot marking the loca-tion that the participant touched the screen iswithin the target region), and the target is re-vealed for feedback. The reward for this trial is22 points, as indicated numerically at the centerof the target. In the schematic example of a highstakes trial (right-hand panels), the participantchose to stop sampling and attempt to touch thehidden target after requesting four cues. Thebottom panel shows that the participant missedthe hidden target (the square response dot mark-

60 60

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42 42

36 36

0

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36 36

22 22

22

22 22

Figure 1. Schematic of the sequence of events in both experimental conditions. At the leftand right edges of the screen there were identical, vertical reward bars that decreased in heightwith each additional cue sampled. Left-hand panels show an example of a low stakes trial. Inthis case, the participant decided to stop sampling after requesting nine cues. The potentialreward for successfully touching the hidden target was reduced from 40 points to 22 points,as indicated numerically at the top of the reward bars (each cue in this condition reduces thepotential reward by 2 points). The square dot in the bottom left panel marks the participant’sfinal response (which is made by touching the screen and making fine adjustments), and thehidden gray target is revealed for feedback. Since the square response dot is within the targetregion, the trial is coded as a hit and the reward for this trial is 22 points, as indicatednumerically at the center of the target. Right-hand panels show an example of a high stakestrial. In this case, the participant decided to stop sampling after requesting four cues, and thepotential reward was reduced from 60 points to 36 points (each cue in this condition reducesthe potential reward by 6 points). Since the square response dot in the bottom right panel isoutside the target region, the trial is coded as a miss and there is no reward (or penalty) forthis trial, as indicated numerically with a 0 at the center of the target. See the online articlefor the color version of this figure.


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ing the location that the participant touched thescreen is outside the target region), and thetarget is revealed for feedback. Participants didnot have to pay for the cues that they sampledon trials that they failed to touch the hiddentarget. Hence, the reward for this trial is 0 points(i.e., no reward), as indicated numerically at thecenter of the target.

Participants had to request at least one cue pertrial, and they were allowed to continue sampling,one cue at a time, so long as the potentialreward for successfully touching the hidden tar-get remained greater than zero. Hence, theywere limited to sampling nine cues per trial inthe high stakes condition and 19 cues per trial inthe low stakes condition. The two experimentalconditions were designed so that the maximumexpected gain of the ideal participant (see be-low) would be very similar in the two condi-tions: 18.48 points per trial in the high stakescondition and 18.63 points per trial in the lowstakes condition. The ideal participant obtainsthese maximum expected gains by samplingfour cues on every trial in the high stakes con-dition and seven cues on every trial in the lowstakes condition (see derivation of the optimalsampling rule below).

What Is the Optimal Sampling Rule?

In order to analyze behavior in our task, webegin by defining an ideal participant who sam-ples information with the goal of maximizingexpected gain. The radius of the hidden circulartarget remained constant throughout the exper-iment. The underlying bivariate Gaussian distri-bution from which the cues were drawn wasisotropic (“circular”) with equal variance ineach direction: �x

2 � �y2. For convenience, we

refer to this common variance as the populationvariance, denoted �P

2. The population varianceremained constant throughout the experiment.

To maximize expected gain, an optimal par-ticipant must minimize their estimation vari-ance by using the center of gravity of the cues(i.e., the sample mean) as their estimate of thehidden target’s location. The sample mean isthe minimum variance unbiased estimate of theGaussian population mean (Mood, Graybill, &Boes, 1974) which coincides with the center ofthe hidden circular target. Hence, for any num-ber of cues, pointing at the sample mean is thestrategy that maximizes the probability of

touching the hidden target. The sample mean isdistributed as an isotropic Gaussian centered onthe target, but with reduced variance in eachdirection: �P

2/n. The larger the sample size n, themore accurate the participant’s estimate of thecenter of the target.

Besides the variability of the sample mean,an optimal participant must take into accountother sources of variability such as their percep-tual-motor error in touching the screen. Werefer to the aggregate of all the other sources ofvariability as additional pointing variability,and we will assume for simplicity throughoutour analyses that it is isotropic Gaussian (Trom-mershäuser, Maloney, & Landy, 2003) and thatits vertical and horizontal variances are bothequal to a value denoted �A

2. Together, these twosources of variance give rise to the participant’seffective estimation variance in each directionas a function of sample size, denoted �E

2(n), asfollows:

�E2(n) � �P

2 ⁄ n � �A2 . (1)

In deriving the optimal sampling rule maxi-mizing expected gain, we will first assume forsimplicity that additional pointing variability iszero: �A

2 � 0. In the Discussion section, we willrevisit this assumption and explore the implica-tions on optimal sampling behavior if �A

2 � 0.We will also assume for now that the idealparticipant has an accurate estimate of the pop-ulation variance. During the practice sessionprior to the experimental trials, the participant isgiven ample opportunity to develop an accurateestimate of �P

2, by observing a very large num-ber of draws from the underlying Gaussian asdescribed below. In the Discussion section,we will revisit this assumption as well andexplore the implications on optimal samplingbehavior if participants do not have an accu-rate estimate of �P

2. Finally, because the targetwas revealed during feedback at the end ofevery trial (including during the practice ses-sion), we will assume for simplicity through-out our analyses that the participant has anaccurate estimate of the radius of the hiddencircular target.

Given �E2(n), we can compute the probability

of touching the hidden circular target as a func-tion of sample size as follows:


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p[hit | n] � ��T

��0¡

, � (n)�dx dy, (2)

where the region of integration T is the invisible

circular target, and ��0¡

, ��n�� denotes theprobability density function of a bivariate

Gaussian distribution centered on 0¡

� �0, 0� withcovariance

� (n) � ��E2(n) 0

0 �E2(n)�. (3)

We can estimate p[hit | n] by computing theprobability that a draw from an isotropic Gauss-ian distribution, whose variance in each direc-tion matches �E

2(n), will fall within the circulartarget region. This depends only on the proba-bility that the distance between the drawn pointand the origin is less than the radius of thetarget. This distance squared is a �2

2 randomvariable multiplied by �E

2(n), and we need onlycompute the cutoff corresponding to the target

radius squared to estimate the probability of ahit.

Given p[hit | n], we can compute the expectedgain as a function of sample size as follows:

EG(n) � p[hit | n]�R � nC�, (4)

where R is the initial point value of the target,and C is the fixed reduction to the target valuethat is incurred for each cue that is sampled.Participants were not penalized for missing thetarget, and they were allowed to sample addi-tional cues only so long as �R � nC� � 0. Allplots will be cut off where EG(n) starts to dipbelow zero.

Figure 2 plots the gain function R � nC, theprobability of hitting the target p[hit | n], andtheir product, which is the expected gain func-tion EG(n), against the number of cues sampledfor the low stakes condition. The expected gainfunction is greatest when n � 7. The idealparticipant who seeks to maximize expectedgain would sample seven cues and then touch

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8

16

24

32

40

Number of cues sampled

Poi

nts

p[hit |n]R−nCEG(n)

p[hi

t |n]

0.2

0.4

0.6

0.8

1

0 0

Figure 2. Expected gain function. The expected gain function EG(n) is the product of theprobability p[hit | n] of touching the hidden target with n cues, and the gain R � nC that isearned for successfully touching the target with n cues (which takes into account the cost ofsampling n cues). Probability of touching the hidden target is plotted on the right verticalscale, gain and expected gain on the left. This figure shows the construction of the expectedgain function for the low stakes condition. See the middle panel of Figure 7A and top panelof Figure 7B for a similar construction of the expected gain function for the high stakescondition. See the online article for the color version of this figure.


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the mean of the sampled cluster. A similar con-struction (see middle panel of Figure 7A and toppanel of Figure 7B) shows the optimal numberof cues that maximizes expected gain for thehigh stakes condition: n � 4.

Dispersion

The ideal participant derived above makes itsdecision whether to stop or continue samplingwithout reference to how spread out a sampledcluster is. Since the ideal model has an accurateestimate of the population variance �P

2, there isno need for the ideal participant to make use ofany aspect of a sampled cluster beyond its mean(just as, in computing a z test with known pop-ulation variance, we make no use of samplevariance). We will revisit this assumption in theDiscussion section and it will prove convenientto define a measure for how spread out a sam-pled cluster is. While the underlying bivariateGaussian was isotropic, the vertical and hori-zontal variances of a sampled cluster are usuallydifferent from each other and from one clusterto the next because of normal sampling vari-ability. We define sample dispersion as S �

��Sx2�Sy

2� ⁄ 2, where Sx2 and Sy

2 are the respec-tive horizontal and vertical variances of a sam-pled cluster. The corresponding population dis-persion is simply �P, the square root of thepopulation variance �P

2.

Method

Participants and Apparatus

Thirteen women and 15 men participated oneat a time in the experiment, for a total of 28participants. Participants were graduate stu-dents, postdocs, or lab managers between theages of 23 and 43, with a median age of 26 anda mean age of 27.6 (SD � 4.71). None wereaware of the purpose of the experiment. Partic-ipants were paid $10 plus any monetary bonusthat they earned, which depended on their per-formance.

Eight participants were run at New York Uni-versity, and 20 participants were run at theUniversity of California, Santa Barbara. Allconditions were identical except that the touchscreen used at UCSB (19-in. diagonal) waslarger than the one used at NYU (17-in. diago-nal). Both LCD touch screens had the same

resolution of 1,280 pixels � 1,024 pixels, andwe report all measurements in pixels. The con-version factor from pixels to millimeters is0.293 for the UCSB touch screen (375 mm �300 mm) and 0.2637 for the NYU touch screen(337.5 mm � 270 mm).

Participants were asked to seat themselves ata comfortable distance and to adjust the heightof the chair. The experiment was programmedand run using MATLAB and the PsychophysicsToolbox libraries (Brainard, 1997; Pelli, 1997).

Stimuli

The hidden target, which was revealed duringfeedback at the end of every trial, was a graycircle with radius � 48 px. The cues, whichremained on the screen throughout the trial,were small white circular dots with radius � 4px. The cues were drawn from an isotropic,bivariate Gaussian distribution (�x � �y ��P � 80 px) that was centered on the middle ofthe hidden target. At the left and right edges ofthe screen there were identical, vertical rewardbars that decreased in height with each addi-tional cue sampled (see schematic in Figure 1).Additionally, there was a number at the top ofeach bar indicating how many points would beawarded for successfully touching the target ifno additional cues are sampled. To help partic-ipants remember which condition they were in,the reward bars were colored green during lowstakes trials and red during high stakes trials.

Design

Each participant ran in a practice sessionfollowed by the experimental session. The prac-tice session consisted of 30 trials of each con-dition in alternating blocks of 10 trials, for atotal of six practice blocks and 60 practice trials.The actual experiment consisted of 100 trials ofeach condition in alternating blocks of 25 trials,for a total of eight experimental blocks and 200experimental trials. The ordering of the con-ditions was randomly assigned and counter-balanced between participants. However, theordering was kept constant between the prac-tice session and the experimental session.Hence, a participant assigned to start with thehigh stakes condition would start both thepractice and the experimental sessions with ahigh stakes block. Each participant ran alone


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and the task was self-paced. Participants’ par-ticipation time (including practice) rangedfrom 34 min to 74 min, with an average of 53min.

Monetary Bonus

Points earned were converted into bonusmoney at a rate of five cents per point. Thismeans that the maximum possible reward fortouching the target was 38 points � $0.05 �$1.90 per low stakes trial, and 54 points �$0.05 � $2.70 per high stakes trial. Participantswere informed that they would receive a mon-etary bonus on 5% of the trials, randomly cho-sen from the trials they completed in each con-dition at the end of the experiment (5 randomtrials from each condition were selected forpayment). The total expected monetary bonusof the ideal participant who maximizes ex-pected gain was: 10 trials � 18.55 points �$0.05 � $9.28.

Procedure

Participants were told that the theoretical dartthrower was standing behind the participant atsome unknown distance away from the touch-screen display. They were told that this theoret-ical person could see the hidden target (unlikethe participants) and aimed each dart throw atthe center of the target, but that when peoplethrow darts, they rarely hit exactly where theyaim. The experimenter emphasized that the the-oretical dart thrower was always going to throwthe dart in the same way, always aiming for thecenter of the target, and always standing in thesame exact location and at the same exact dis-tance away from the target. Participants wereexplicitly told that this theoretical person’sthrowing ability never improved or deterioratedthroughout the experiment. Finally, participantswere explicitly told that the size of the circulartarget would remain the same throughout theexperiment.

Participants were asked to use their dominanthand throughout the experiment. The experi-menter stayed in the room during the practiceblocks to explain the task. The experimenterfirst demonstrated how one could sample as fewas one cue per trial, and as many as nine and 19cues per trial (depending on the condition). Theexperimenter then told the participant that he orshe should use the practice trials to explore and

observe the outcome of different decision strat-egies. The experimenter remained in the roomduring the practice session to answer any ques-tions that came up. Once the practice trials werecomplete, the experimenter left the room, clos-ing the door behind him.

At the start of each block, participants wereshown an instruction screen that specified theinitial point value of the target on each trial (40points or 60 points) and the cost per cue (2points or 6 points). Participants were required toconfirm the appropriate cost per cue by pressing“2” or “6” before the block would begin.

At the start of each trial, the screen wascompletely black except for the reward bars,which indicated the current reward for touchingthe target. To sample a cue, participants pressedthe spacebar causing a small white circular dotto appear on the screen (the schematic in Figure1 shows the cues as small black circular dots ona white background). Concurrent with the ap-pearance of the cue, the reward bars and num-bers would decrease to indicate the decreasedreward available for touching the target. If par-ticipants wanted more cues, they simply pressedthe spacebar repeatedly until they were ready toreach for the hidden target. This easy and rapidmechanism of requesting cues and recomputingthe value of hitting the target ensured that theinformation-access costs associated with sam-pling information were minimized.

To attempt to touch the invisible target, par-ticipants simply touched the screen with theirfinger causing a small purple response dot toappear (the schematic in Figure 1 shows theresponse dot as a black square dot). Participantswere allowed to adjust their response by movingthe response dot with their finger or by pressingthe arrow keys. Once satisfied with their re-sponse, participants pressed the spacebar to re-ceive feedback. During the feedback phase ofeach trial, the hidden target would appear alongwith the response dot and all the cues sampledon that trial. If the response dot was within thetarget, the trial was counted as a hit and thenumber of points awarded for that trial wouldappear at the center of the target (see the left-hand side of Figure 1). If, however, the responsedot was not within the target, then the trial wascoded as a miss and a 0 appeared at the centerof the target (see the right-hand side of Figure1). A second spacebar press began the next trial.


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Finally, at the end of both the practice sessionand the experimental session, participants weregiven feedback as to how they performed oneach of 10 randomly selected experimental tri-als and how much bonus money they receivedas a result. The feedback at the end of thepractice session was not actually paid out, butserved to give the participant a rough sense ofthe range of actual bonuses possible.

Results

Number of Cues Sampled

The solid curves in Figure 3 show the respec-tive high stakes and low stakes expected gainfunctions for the ideal participant with addi-tional pointing variability equal to zero (�A

2 �0). The expected gain for this ideal participant ismaximized when sampling four cues in the highstakes condition and seven cues in the lowstakes condition. The horizontal displacementof data points in Figure 3 shows each partici-pant’s mean and standard deviation of the num-ber of cues sampled for each of the two exper-imental conditions. In addition, the individualdata points appear at heights that correspond totheir average gain per trial. For similar sampling

behavior (i.e., similar location along the x-axis),some participants where better able to success-fully touch the hidden target and collect morepoints (i.e., their data points are located higherup along the y-axis). The average number ofpoints per trial that each participant earnedacross both experimental conditions rangedfrom 9.04 to 18.75 with a mean of 14.03 (SD �2.08), which is 76% of the maximum expectedgain of 18.55 points per trial.

Participants adapted to changes in thecost-benefit structure of the task. Overall,participants sampled significantly more cues pertrial in the low stakes condition (M � 6.94,SD � 1.65) compared to the high stakes condi-tion (M � 5.65, SD � 1.05), t(27) � 6.22, p �.001. This indicates that participants adapted tochanges in the cost-benefit structure of the task.Individually, 24 participants sampled signifi-cantly more cues per trial in the low stakescondition than in the high stakes condition (p �.05), one sampled significantly more cues pertrial in the high stakes condition than in the lowstakes condition (p � .05), and three showed nosignificant difference in the number of cuessampled per trial between conditions (p � .05);all individual significance tests were t tests with

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Figure 3. Results. The horizontal displacement of data points shows each participant’s meanand standard deviation of the number of cues sampled for the high stakes condition (datapoints shown in red, otherwise dark gray circles) and the low stakes condition (data pointsshown in green, otherwise light gray squares). The data points appear at heights thatcorrespond to each participant’s average gain per trial. The expected gain curve for eachcondition is plotted in the corresponding color, with corresponding vertical lines marking thenumber of cues that the ideal participant should sample to maximize expected gain. For thisfigure, we plot the corresponding expected gain curves assuming that participants’ additionalpointing variability �A

2 � 0. See Figure 6 and the accompanying discussion where we addressthe consequence that additional pointing variability (i.e., �A

2 � 0) has on the optimal samplingstrategy. See the online article for the color version of this figure.


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198 degrees of freedom. This result rules out thepossibility that participants weren’t doing anycost-benefit analysis at all, and that they wereonly concerned with the challenge of touchingthe hidden target, a challenge that is identical inboth conditions (up until nine cues, which is thelimit in the high stakes condition).

Participants did not exhibit systematicundersampling. Overall, participants did notsignificantly differ from sampling the idealseven cues per trial in the low stakes condition(M � 6.94, SD � 1.65, t(27) � 0.18, p � .05).Individually, 11 participants sampled signifi-cantly more than seven cues per trial (p � .05),12 sampled significantly less than seven cuesper trial (p � .05), and five did not significantlydiffer from sampling seven cues per trial (p �.05); all individual significance tests were t testswith 99 degrees of freedom.

On the contrary, participants sampled signif-icantly more than the ideal four cues per trial inthe high stakes condition (M � 5.65, SD �1.05, t(27) � 8.35, p � .001). Individually, 26participants sampled significantly more thanfour cues per trial (p � .05), one sampled sig-nificantly less than four cues per trial (p � .05),and one did not significantly differ from sam-pling four cues per trial (p � .05); all individualsignificance tests were t tests with 99 degrees offreedom.

Taken together, participants did not exhibitany systematic tendency to undersample infor-mation, and actually oversampled informationrelative to ideal in the high stakes condition.

Variability in Sampling BehaviorAcross Trials

Because of normal sampling variability, thecurrently visible cluster is sometimes morespread out and sometimes less spread out. Theideal observer model described above has anaccurate estimate of the underlying populationdispersion �P and should pay no attention to thesample dispersion S of the currently visiblecluster (defined above). Specifically, becausethe cues are independent and identically distrib-uted draws from a Gaussian distribution, thevariance of the sample means in each direction,which is equal to �P

2/n, is the same for clustersof size n that happen to have low sample dis-persion as for clusters of the same size n thathappen to have high sample dispersion. Conse-

quently, the probability of touching the hiddentarget p[hit | n], which is maximized by pointingat the mean of the sampled cluster, is the samefor high-dispersion clusters of size n as forlow-dispersion clusters of the same size n. Weverified this fact through computer simulation.The only factor that affects the variance of thesample means (besides for the sample size n) isthe underlying population dispersion �P, whichwas constant throughout the experiment.

In the high stakes condition, an optimal de-cision maker will always request four cues ir-respective of how dispersed the cues are on thescreen, while in the low stakes condition theywill always request seven cues. Nonetheless,almost all participants in our experiment exhib-ited variability in the number of cues that theyrequested from one trial to the next (see Figure4 for each participant’s trial by trial samplingbehavior). Combined with our finding of over-sampling in the high stakes condition (seeabove), this result reveals a second way inwhich the participants’ sampling behavior dif-fered from optimal.

One possible explanation is that variation insampling behavior from trial to trial is the resultof unsystematic noise in each individual’s deci-sion-making process. A second possibility,which we evaluate below, is that some randomcharacteristics of the visible cues during eachtrial, specifically their sample dispersion S(which is an index of how spread out the currentcluster is), influenced the participants’ decisionwhether to stop or continue sampling, inducingvariability in sampling behavior across trials.3

Stop clusters versus continue clusters.During a trial, participants sampled cues one ata time and had to decide, after sampling eachcue, whether to stop sampling and attempt totouch the hidden target (given the currentlyvisible cluster) or to continue sampling. Thecumulative number of cues sampled so far dur-

3 We considered several different hypotheses for thisanalysis, including (a) whether participants used the area ofthe convex hull of the currently visible cluster to guide theirsampling behavior, (b) whether participants used the outli-erness of the current cue relative to the previous cues of thecurrently visible cluster to guide their sampling behavior,and (c) whether participants used the sample dispersion S ofthe currently visible cluster to guide their sampling behav-ior. We give a full account of the analysis of this lasthypothesis because it was the best predictor of participants’sampling behavior among those we considered.


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ing the trial (i.e., the current sample size) can berepresented by the symbol N. In this section, wecomputed the sample dispersion S of the cur-rently visible cluster for all values of N, andcompared said sample dispersion when partici-pants stopped sampling after the Nth cue towhen they continued sampling after the Nth cue.

Table 1 shows the total number of “stop”clusters and “continue” clusters across all par-ticipants as a function of N in each experimentalcondition. For example, participants almost al-ways continued sampling after seeing only onecue (the exception being just one case in thehigh stakes condition). At the other extreme(N � 9 in the high stakes condition, and N � 19in the low stakes condition), participants wereobligated to stop sampling and take action (i.e.,attempt to touch the hidden target). As a result,these two extremes were excluded from thisanalysis. For intermediate values of N, some-times there were enough stop clusters and con-tinue clusters to conduct this analysis meaning-fully, but other times there were not. Thus, wehave emboldened the cells in Table 1 for whichwe decided not to conduct a comparison be-cause of the limited number of stop clustersand/or continue clusters. Overall, we carried out

this analysis 16 times across both conditions(see the cells in Table 1 that are not in bold),and readers should keep in mind a Bonferronicorrection of .05/16 � .0031 when consideringwhether any particular result is statistically sig-nificant.

As shown in Table 1, there were 2,641 trialsin which participants sampled at least four cuesin the high stakes condition. In 464 of thosetrials they decided to stop sampling and takeaction, while in the other 2,177 they decided torequest another cue. In Table 2 (high stakescondition; N � 4) we show that the currentlyvisible cluster when participants chose to con-tinue sampling had a mean sample dispersion of78.05 px (SD � 21.86), compared to a meansample dispersion of 71.65 px (SD � 23.43)when they chose to stop sampling and attemptto touch the hidden target. A statistical testindicates that this difference in mean sampledispersion was significant: t(2639) � 5.66, p �.001. Likewise, there were 2,177 trials in whichparticipants sampled a fifth cue in the highstakes condition. In 704 of those trials theydecided to stop sampling and take action, whilein the other 1,473 they decided to continuesampling. The mean sample dispersion for con-

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Figure 4. Trial by trial sampling behavior. Some participants (most notably UCSB 2 andUCSB 12) showed little or no variability in the number of cues that they requested acrosstrials. Most participants however did exhibit said variability. In the main text we explore whatdrives this trial-to-trial variation in sampling behavior. Note that there did not seem to be anysystematic learning effects as the experiment progressed toward the optimal sampling rule,which is four cues in the high stakes condition (data points shown in red, otherwise dark graycircles) and seven cues in the low stakes condition (data points shown in green, otherwise lightgray squares). See the online article for the color version of this figure.


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tinue clusters (M � 80.72 px, SD � 19.03) wassignificantly larger than the mean sample dis-persion for stop clusters (M � 74.72 px, SD �19.03), t(2175) � 6.86, p � .001.

Table 2 shows the respective mean sampledispersion for “stop” clusters and “continue”clusters as a function of N in each experimentalcondition. Statistical tests are shown in boldwhen the mean sample dispersion was signifi-cantly larger for continue clusters than for stopclusters at the Bonferroni corrected p of .0031.The table shows that for almost all values of N,and irrespective of the experimental condition,

continue clusters tend to have higher sampledispersion than stop clusters.

These observations suggest that participantswere influenced by how spread out the currentlyvisible cluster was in deciding whether to stopsampling and attempt to touch the hidden targetor to continue sampling. The larger the sampledispersion S, the more likely it is that partici-pants would continue to sample.

High dispersion clusters versus low disper-sion clusters. For each value of N, and sepa-rately for each experimental condition, wesorted all trials across all participants based onthe sample dispersion S of the currently visiblecluster and grouped them into three equal-sizedgroups: high dispersion clusters, medium dis-persion clusters, and low dispersion clusters.When the total number of clusters for a partic-ular value of N was not divisible by three, weassigned an equal number of clusters to thehigh- and low-dispersion groups, with the sur-plus assigned to the medium-dispersion group.

The analysis of most interest in this section isthe proportion of clusters that participants choseto continue sampling, denoted P[continue]. Thecircle data points in Figure 5 show P[continue]across all clusters, irrespective of their sampledispersion, as a function of N in each experi-mental condition. Notice that P[continue] tendsto decrease as the number of cues sampledincreases. This pattern is expected given thecosts of requesting additional cues. However, asalso shown in the figure, P[continue] in bothconditions is consistently greater for clusterswith high sample dispersion than for clusterswith low sample dispersion, with P[continue]for clusters with medium sample dispersionnestled in between.

This analysis supports our conclusion fromthe previous section that increased sample dis-persion S seems to prompt a greater tendency tocontinue sampling. For example, in the highstakes condition the proportion of clusters thatparticipants chose to continue sampling for N �6 was 39% greater for clusters with high sampledispersion (P[continue] � .57) than for clusterswith low sample dispersion (P[continue] �.41).

Discussion

Overall, our results indicate that participantscorrectly sampled more cues in the low stakes

Table 1Total Respective Number of Clusters ThatParticipants Chose to Stop Sampling and ContinueSampling as a Function of the Number of CuesSampled so far in the Trial andExperimental Condition

N Stop Continue

High stakes condition1 1 2,7992 1 2,7983 157 2,6414 464 2,1775 704 1,4736 749 7247 431 2938 174 1199 119 N/A

Low stakes condition1 0 2,8002 0 2,8003 120 2,6804 221 2,4595 488 1,9716 518 1,4537 494 9598 339 6209 230 390

10 194 19611 69 12712 55 7213 28 4414 17 2715 8 1916 8 1117 3 818 3 519 5 N/A

Note. We have emboldened the cells for which we did notconduct our comparison analyses (cf., Table 2 and Figure 5)because of the limited number of “stop” clusters and/or“continue” clusters.


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condition compared to the high stakes condi-tion. However, the results also show two dis-tinct ways in which participants diverged fromoptimal sampling behavior. First, in the highstakes condition—but not the low—participantsrequested more cues relative to the optimal sam-

pling rule that maximizes expected gain. Sec-ond, participants varied the number of cues thatthey requested from one trial to the next. Wedemonstrated that this trial-to-trial variationwas correlated with sample dispersion: whenthe currently visible cluster was more widely

Table 2Mean Sample Dispersion (in Pixels) of the Current Cluster for “Stop” Clusters and “Continue” Clustersas a Function of the Number of Cues Sampled so far in the Trial and Experimental Condition

N Stop Continue t-test

High stakes condition3 M � 73.50 (SD � 28.32) M � 75.04 (SD � 27.26) t(2796) �.69, p � .054 M � 71.65 (SD � 23.43) M � 78.05 (SD � 21.86) t(2639) � 5.66, p < .0015 M � 74.74 (SD � 19.03) M � 80.72 (SD � 19.03) t(2175) � 6.86, p < .0016 M � 77.76 (SD � 16.50) M � 83.36 (SD � 16.73) t(1471) � 6.47, p < .0017 M � 82.48 (SD � 15.77) M � 84.31 (SD � 15.33) t(722) � 1.55, p � .058 M � 83.96 (SD � 15.18) M � 82.70 (SD � 11.53) t(291) � �.77, p � .05

Low stakes condition3 M � 74.05 (SD � 33.59) M � 74.42 (SD � 27.03) t(2798) � .14, p � .054 M � 68.99 (SD � 21.99) M � 76.86 (SD � 22.42) t(2678) � 5.01, p < .0015 M � 73.33 (SD � 19.31) M � 78.23 (SD � 19.52) t(2457) � 4.99, p < .0016 M � 74.53 (SD � 17.47) M � 80.62 (SD � 17.62) t(1969) � 6.78, p < .0017 M � 78.03 (SD � 17.09) M � 81.49 (SD � 15.75) t(1451) � 3.88, p < .0018 M � 77.89 (SD � 14.16) M � 82.86 (SD � 15.27) t(957) � 4.94, p < .0019 M � 82.19 (SD � 13.79) M � 82.83 (SD � 14.83) t(618) � .53, p � .05

10 M � 79.94 (SD � 13.74) M � 84.79 (SD � 13.44) t(388) � 3.52, p < .00111 M � 81.15 (SD � 10.47) M � 85.75 (SD � 14.19) t(194) � 2.37, p � .0212 M � 84.26 (SD � 12.78) M � 86.99 (SD � 14.27) t(125) � 1.17, p � .05

Note. Statistical tests are shown in bold when mean sample dispersion is significantly larger for continue clusters than forstop clusters at the Bonferroni corrected p of .0031. See the dispersion section in the introduction for how we define andmeasure sample dispersion.

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Figure 5. P[continue]. Proportion of clusters that participants chose to continue sampling ineach condition is plotted as a function of (a) the sample dispersion S of the currently visiblecluster, and (b) the numbers of cues sampled so far in the trial. The circle data points showP[continue] across all clusters irrespective of their sample dispersion. Triangles mark P[con-tinue] for clusters with high sample dispersion, squares mark P[continue] for clusters withmedium sample dispersion, and inverted triangles mark P[continue] for clusters with lowsample dispersion. See the online article for the color version of this figure.


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dispersed participants were more likely to re-quest another cue. But for the ideal participantmaximizing expected gain, the sample disper-sion of the current cluster—as opposed to thefixed dispersion of the underlying Gaussian—should play no role in deciding whether to stopsampling and attempt to touch the hidden targetor to continue sampling. In the following sec-tions we consider a range of hypotheses thatmight explain the observed patterns of samplingbehavior.

Possible Explanations of Participants’Oversampling Behavior

Asymmetry around the peak of the ex-pected gain function. The expected gainfunctions in Figure 3 are asymmetric around themaximum. One hypothesis is that participantsselected a cautious sampling strategy that isbiased toward the shallower side of the expectedgain function to avoid the risk of ending up onthe steeper side. Such a strategy of “erring onthe shallower side” would lead to oversamplingin our case.

We doubt, however, that the observed over-sampling can be explained in this way. First, thekeyboard presses to request cues were discrete,readily distinguishable, and completely underthe participant’s control. There is no need tocautiously plan to press the sampling key fivetimes, instead of the optimal four, to avoid thenegligible risk of pressing it only three times.Second, the difference in expected gain betweenthe shallower side (oversampling) and thesteeper side (undersampling) is smaller than thedifference in expected gain between both sidesand the peak of the expected gain function. Ifparticipants in the high stakes condition, forexample, were sensitive to the difference inexpected gain between sampling five cues(17.80 points) and three cues (17.52 points) andthus oversampled to err on the shallower side ofthe expected gain function, then they shouldalso be sensitive to the larger difference in ex-pected gain between the optimal four cues(18.48 points) and five cues (17.80 points).

Optimal compensation for additionalpointing variability. In deriving the optimalsampling rule, we assumed for simplicity thatadditional pointing variability (due to motorerror, perceptual error, etc.) was negligible(�A

2 � 0): that is, that the participant is perfectly

accurate in touching the mean of the sampledcluster. One hypothesis is that participants sam-pled additional cues (relative to an ideal ob-server who has negligible pointing variability)to reduce external uncertainty about the target’slocation as an attempt to compensate for addi-tional variability in their perceptual-motorpointing response.

Sampling additional cues when �A2 � 0, how-

ever, actually reduces the participants’ expectedgain relative to the maximum expected gain thatthey could achieve given any additional point-ing variability. Figure 6 shows that as additionalpointing variability increases, the ideal observerwould sample fewer cues—not more—in orderto maximize expected gain. To see why, con-sider an extreme case where an individual’sperceptual-motor pointing error is so high thatthe chances of even touching the screen are verysmall. In such a case, the reduced external un-certainty in localizing the hidden target that isobtained by sampling additional cues is negli-gible compared to the individual’s extremepointing variability. Instead, it would be best tojust try to touch the screen (aiming for themiddle of the screen) without sampling any cuesat all, so that on the lucky occasions that thehidden target is touched the reward will begreat.

In summary, taking into account additionalpointing variability (i.e., �A

2 � 0) would lead toa decrease in the number of cues one shouldoptimally sample in order to maximize expectedgain. However, as shown in Figure 3, mostparticipants oversampled. Hence, participants’oversampling behavior cannot be explained asoptimal compensation for any additional vari-ability in their perceptual-motor pointing re-sponse.

Risk aversion. Oversampling cues in thistask is reminiscent of common descriptions ofrisk aversion in the literature. We explored theimplications of having a nonlinear utility func-tion, specifically a power function of the formU�x� � x� (Kreps, 1990, p. 82), rather than thelinear gain function used in our ideal observermodel. The gain function R � nC in Equation 4is replaced by a power utility function: U�R �nC� � �R � nC��.

When � � 1 the power utility function isstrictly convex (“convex utility function”) andwhen 0 � 1 the power utility function is


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strictly concave (“concave utility function”) asshown in Figure 7A. As shown in the top panelof the figure, a convex utility function (� �1.5) shifts the optimal number of cues necessaryto maximize expected utility in each conditionto the left (i.e., fewer cues) relative to an idealobserver with a linear utility function (middlepanel). Conversely, and as shown in the bottompanel of the figure, a concave utility function(� � 0.5) shifts the optimal number of cuesnecessary to maximize expected utility in eachcondition to the right (i.e., more cues).

The hypothesis, then, is that the observedoversampling (relative to an ideal observer witha linear utility function) is a consequence ofhaving a concave utility function. However,while oversampling in any one experimentalcondition can be fit with a concave utility func-tion, no single concave utility function can ac-count for participants’ pattern of sampling be-havior across both experimental conditions. Wefound (by simulation) that any single concaveutility function for our task will generate agreater oversampling of cues for the low stakescondition than for the high stakes condition.That is, the rightward shift of the peak of theexpected utility function, relative to the peak ofan ideal observer with a linear utility function,will be greater for the low stakes condition thanfor the high stakes condition.

For example, and as shown in the bottompanel of Figure 7A, a concave utility functionwith � � 0.5 results in a rightward shift of two

cues to the peak of the expected utility functionfor the low stakes condition (from 7 to 9) versusa rightward shift of one cue for the high stakescondition (from 4 to 5). However, only twoparticipants oversampled more in the low stakescondition than in the high stakes condition. Therest of the 24 participants who oversampled inthe experiment (2 participants did not over-sample in either condition) either oversampledonly in the high stakes condition (15 partici-pants), or oversampled more in the high stakescondition than in the low (9 participants).4

Hence, participants’ oversampling behaviorcannot be explained as the consequence of hav-ing any single utility function that is a powerfunction.

We also investigated (by simulation) the im-plications of having a nonlinear subjectiveprobability function, using the model of Tver-sky and Kahneman (1992) as a reference. Whilenonlinear transformations of p[hit | n] alter theexpected gain function EG(n) and could ac-count for participants’ sampling behavior whenlooking at any one experimental condition inisolation, we found no single subjective proba-

4 For example, one participant (NYU 1 in Figure 4)sampled an average of 6.83 cues per trial in the high stakescondition (optimal is 4) and 8.15 cues per trial in the lowstakes condition (optimal is 7). Hence, this participant over-sampled an average of 2.83 cues per trial in the high stakescondition compared to 1.15 cues per trial in the low stakescondition.

Figure 6. Effect of having additional pointing variability. The expected gain function EG(n)changes with increased pointing variability (due to motor error, perceptual error, etc.). Noticethat as additional pointing variability increases, one would need to sample fewer and fewercues to maximize expected gain (indicated by the solid and dashed vertical lines for differentamounts of �A). See the online article for the color version of this figure.


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bility function that could account for partici-pants’ pattern of sampling behavior across bothexperimental conditions.

Implicit costs and rewards. Our experi-mental paradigm allows us to test whether im-plicit costs or rewards could explain partici-pants’ sampling behavior. If participants haveimplicit rewards associated with success andalso implicit costs associated with sampling,then only the sum of the two can be estimatedfrom performance: that is, the net implicit cost/reward. Had participants undersampled (relativeto an ideal observer who does not have anyimplicit costs or rewards) we would attempt toestimate the participants’ net implicit costs totest whether they could explain participants’

sampling behavior. Given that they did not un-dersample, the participants’ net implicit cost/reward must be positive or 0.

The hypothesis, then, is that participants ex-perience an implicit reward for successfullytouching the hidden target—independent of theactual points earned—and that this implicit re-ward leads them to oversample (cf., Juni, Gur-eckis, & Maloney, 2012). We evaluated thishypothesis by putting a fixed point value on theimplicit reward for successfully touching thehidden target (a free parameter). This free pa-rameter shifts the diagonal reward function R �nC in Figure 2 upward, which then causes arightward shift to the peak of the expected gainfunction EG(n), with the magnitude of the right-

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Figure 7. Effect of having a power utility function or implicit rewards. (A) Power utilityfunction. Middle panel shows the construction of the expected gain functions for bothconditions when the utility function is the identity line with � � 1. Top panel shows the effectof having a convex utility function with � � 1.5. Such participants seeking to maximize theirexpected utility will sample fewer cues compared with a participant who has a linear utilityfunction. Bottom panel shows the effect of having a concave utility function with � � 0.5.Such participants seeking to maximize their expected utility will sample more cues comparedwith a participant who has a linear utility function. (B) Implicit rewards. Top panel shows theconstruction of the expected gain functions for both conditions when there is no implicitreward for successfully touching the hidden target. Bottom panel shows the effect of havingan implicit reward for successfully touching the hidden target that is equivalent to 20 points.Such participants seeking to maximize their total expected explicit and implicit reward willsample more cues compared with a participant maximizing explicit rewards only. See theonline article for the color version of this figure.


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ward shift to the peak of the expected gainfunction being controlled by the magnitude ofthe upward shift to the diagonal reward func-tion.

However, no single value of implicit rewardcan account for participants’ pattern of sam-pling behavior across both experimental condi-tions. We found (by simulation) that any fixedimplicit reward for successfully touching thehidden target will generate a greater oversam-pling of cues (relative to an ideal observer whodoes not have any implicit reward and shown inthe top panel of Figure 7B) for the low stakescondition than for the high stakes conditions.

For example, and as shown in the bottompanel of Figure 7B, a fixed implicit reward of 20points results in a rightward shift of two cues tothe peak of the expected reward function for thelow stakes condition (from 7 to 9) versus arightward shift of one cue for the high stakescondition (from 4 to 5). However, as mentionedin the previous section, 24 of the 26 participantswho oversampled in the experiment over-sampled more in the high stakes condition thanin the low. Hence, participants’ oversamplingbehavior cannot be explained as the conse-quence of having any fixed implicit reward forsuccessfully touching the hidden target.

Sensitivity to Sample Dispersion

Our results indicate that participants wereinfluenced by how spread out the currently vis-ible cluster was in deciding whether to stopsampling and attempt to touch the hidden targetor to continue sampling. We next consider whyparticipants were more likely to continue sam-pling when the currently visible cluster wasmore dispersed and less likely when it was lessdispersed.

The ideal observer model has an accurateestimate of population dispersion. In deriv-ing the optimal sampling rule, we assumed thatthe ideal participant has full knowledge of theunderlying distribution from which the cues aredrawn except for its mean (which coincideswith the center of the hidden target). In partic-ular, we assume that the ideal participant knowsthe population dispersion �P of the underlyingGaussian. As explained in the Results section,the ideal participant is insensitive to the inci-dental sample dispersion S of the current clus-ter, and will request the same number of cues on

every trial regardless of how spread out thecurrent cluster is. It is true that the dispersion ofthe current cluster provides information aboutthe underlying population dispersion; but theideal observer already knows the populationdispersion. As a consequence, the ideal partici-pant always samples the same number of cueson every trial (either 4 or 7, depending on thecondition).

Participants could, in principle, get an accu-rate estimate of the population dispersion viathe practice session prior to the experimentaltrials (the actual population dispersion �P of theunderlying Gaussian was 80 px). The lowestcumulative number of cues sampled by anyparticipant over the course of the 60 practicetrials was 188 cues (the average across all 28participants was 410 cues, and the maximumwas 774 cues). A participant with perfect mem-ory and computational power would use 188random draws to estimate the population dis-persion of the underlying Gaussian at an aver-age of 79.95 px (SD � 2.92). However, arrivingat such an accurate estimate of this task param-eter requires accumulation of informationacross many trials.

Hypothetical memory-less model. Suppose,in an extreme case, that a hypothetical “memory-less” participant carries over no informationabout the underlying population dispersionacross trials (though it is afforded knowledgethat the cues are drawn from a circular Gauss-ian). In such a case, the participant could onlyuse the information about cue dispersion that isavailable within the current trial to decide whento stop sampling (remember that the cues re-mained on the screen throughout the trial, so nomemory is required to compute the dispersionof the current cluster). Every trial for such ahypothetical memory-less participant is, in ef-fect, the first and only trial in terms of estimat-ing the dispersion of the underlying distribution,which is, in turn, reestimated with each addi-tional cue that is sampled within a trial.

For such a memory-less model, a good esti-mate of population dispersion is the sampledispersion S of the current cluster. This esti-mated dispersion squared replaces the popula-tion variance �P

2 in Equation 1 in estimating theeffective estimation variance �E

2(n) as a functionof sample size n, and the consequent estimatedprobability of successfully touching the hiddentarget p[hit | n] that is given by Equation 2.


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Using this estimated p[hit | n], the memory-lessmodel uses Equation 4 to determine whether theexpected gain EG(n) is higher or lower withmore cues. If expected gain is estimated todecrease with more cues, then the memory-lessmodel stops sampling and takes action (bypointing at the mean of the current cluster). Ifexpected gain is estimated to increase with morecues, the memory-less model samples anothercue and then repeats all these steps, startingwith a reestimation of population dispersiongiven the new cue that was added to the cluster.

We simulated a version of this hypothetical“memory-less” participant5 and summarize itshypothetical sampling behavior in Figure 8 andTable 3. The first thing to note is that while theideal participant (with full knowledge of theunderlying population dispersion) always sam-ples the same number of cues on every trial(either 4 or 7, depending on the condition), thishypothetical memory-less participant samples avariable number of cues across trials.

Table 3 shows that this memory-less partici-pant’s variable sampling behavior across trialsqualitatively matches human behavior (see Ta-ble 2) as follows: for all values of N, and irre-spective of the experimental condition, continueclusters for this memory-less participant have a(much) higher sample dispersion than stop clus-ters. Furthermore, the mean sample dispersionof the clusters increases (greatly) for this mem-ory-less participant as the number of cues sam-pled increases.

In summary, the memory-less model stopssampling when the sample dispersion S of thecurrent cluster is relatively low and continuessampling when it is relatively high. This patternqualitatively resembles human behavior, thoughthe difference in mean sample dispersion be-tween “stop” and “continue” clusters is muchmore extreme for the hypothetical memory-lessparticipant that carries over no informationacross trials about the underlying populationdispersion.

We conjecture that a compromise modelbased on the assumption that the participantcarries over partial information across trialsabout population dispersion, perhaps in theform of an updating prior distribution on popu-lation dispersion, might better match humanvariability in sampling behavior across trials. Ifthis conjecture is correct, then the observed linkbetween the number of cues sampled and the

sample dispersion S of the current cluster wouldbe due to imperfect learning of the underlyingpopulation dispersion across trials.

If, in future work, we were to abruptly changethe dispersion of the underlying Gaussian afterseveral hundred trials, we could likely observethe effect of the prior. The memory-less partic-ipant’s sampling behavior would be affected bythe new population dispersion immediately. Theparticipant with a prior would lag behind untiltheir prior is updated to agree with the newpopulation dispersion, and the relative impor-tance of prior and present might determine therate of adaptation.

Finally, our analysis and simulation of thehypothetical “memory-less” participant showsthat the observed human sensitivity to sampledispersion (cf., Table 2 and Figure 5) likelycontributed to their trial-to-trial variation in thenumber of cues that they sampled. However, thehypothetical memory-less participant’s slighttendency to undersample relative to ideal (seeFigure 8 and compare to the ideal 4 and 7 cues,depending on the condition) is a constant error(bias) that can be readily corrected. A modifiedhypothetical memory-less model that correctsthis bias would no longer on average under-sample (or oversample) relative to ideal, butwould still sample a variable number of cuesacross trials. That is, a sensitivity to sampledispersion does not, on its own, lead to over-sampling or undersampling, but it likely doescontribute to variability in the number of cuessampled across trials.

Conclusion

We reported an experiment assessing howeffective humans are in trading off the costs andbenefits of sampling additional information in aperceptual-motor estimation task with explicitsampling costs. In our analyses we focused onthe intermediate decisions that participantsmake after sampling each cue: whether to stopsampling and take action or continue samplingto reduce uncertainty. These intermediate deci-sions are similar to those in other cognitivedecision-making tasks, such as the card-

5 Since our participants almost never stopped samplingafter seeing only one or two cues (cf., Table 1), we forcedthis simulated version of the memory-less model to sampleat least three cues per trial.


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sampling tasks used in many recent studies(Hau et al., 2008; Hertwig et al., 2004; Hertwig& Pleskac, 2010; Ungemach et al., 2009).

Our results have relevance to a broad range ofinformation-sampling tasks. For example, whenlearning new concepts or categories, people canoften choose to actively sample additionalmembers of a group (Markant & Gureckis,2014). In such cases, issues like stopping rulesfor terminating active sampling, the number ofsamples actively requested, and the need to bal-

ance information-gain against sampling-costsplay central roles.

Our experimental design has several advan-tages. First, given the explicit costs of samplingand explicit rewards for being successful, wecan compute the optimal number of cues that anideal participant would sample before takingaction to maximize expected gain. Second, wetried to minimize any implicit costs of sampling(e.g., the participant’s time, boredom, etc.). Thetask was engaging; sampling a cue entailed

Table 3Hypothetical Mean Sample Dispersion (in Pixels) of the Current Cluster for aSimulated Memory-Less Participant for “Stop” Clusters and “Continue”Clusters as a Function of the Number of Cues Sampled so far in the Trial andExperimental Condition

N Stop Continue

High stakes condition (simulated memory-less model)3 M � 43.93 (SD � 10.63) M � 88.37 (SD � 21.22)4 M � 81.93 (SD � 15.44) M � 126.15 (SD � 10.30)5 M � 116.38 (SD � 10.70) N/A

Low stakes condition (simulated memory-less model)3 M � 30.71 (SD � 6.72) M � 79.42 (SD � 24.66)4 M � 43.94 (SD � 3.89) M � 81.83 (SD � 19.76)5 M � 55.06 (SD � 4.12) M � 84.92 (SD � 15.84)6 M � 67.84 (SD � 4.95) M � 91.15 (SD � 11.99)7 M � 83.43 (SD � 6.61) M � 104.68 (SD � 8.00)8 M � 101.31 (SD � 8.04) M � 135.01 (SD � 4.64)9 M � 129.78 (SD � 5.13) N/A

Note. We ran 100,000 simulated trials per experimental condition. See the dispersion sectionin the introduction for how we define and measure sample dispersion.

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Figure 8. Hypothetical sampling behavior for a simulated memory-less participant (100,000simulated trials per experimental condition). The memory-less model does not accumulateinformation across trials about the dispersion of the underlying distribution from which thecues are drawn. Instead, the memory-less model estimates population dispersion using thesample dispersion S of the current cluster. As a result, this hypothetical memory-lessparticipant samples a variable number of cues across trials. In Table 3 we show that thishypothetical memory-less participant will sample more cues when the current cluster is moredispersed and fewer cues when it is less dispersed, a qualitative match to human samplingbehavior (see Table 2). See the online article for the color version of this figure.


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pressing a key, which is a very simple and rapidaction; and we limited the number of cues thatparticipants could sample on every trial (either 9or 19 depending on the condition). Third, ourdesign allows us to test whether any observeddeviations from optimal sampling behavior canbe explained by implicit costs or rewards. Thisallows us to draw meaningful conclusions abouthow to characterize participants’ sampling be-havior relative to a well-defined and realisticideal observer model (Geisler, 2003).

We found that in one condition (high stakes)participants sampled significantly more cuesthan that predicted by the ideal observer model,while in the other condition (low stakes) thenumber of cues that participants sampled wasnot significantly different from that predicted bythe ideal observer model. In contrast to thepredictions of recent theories intended to ac-count for why people undersample in cognitivedecision-making tasks (Hertwig & Pleskac,2010; Vul, Goodman, Griffiths, & Tenenbaum,2014), we found no evidence of undersamplingin our task.

We explored a range of hypotheses to ac-count for participants’ oversampling behaviorin our experiment including (a) optimal com-pensation for any additional pointing variability(due to motor error, perceptual error, etc.), (b)risk aversion induced by any single concaveutility function, and (c) any fixed implicit re-ward for successfully touching the hidden targetthat is independent of the explicit rewardearned. However, none of these hypothesescould account for the observed patterns of over-sampling behavior in our experiment.

The existing literature on information sam-pling behavior consists of experiments with de-signs that differ in many respects and it is dif-ficult to isolate factors that might lead tooversampling or undersampling. Further studiesare needed to determine the conditions underwhich people oversample, undersample, orsample optimally relative to ideal. We empha-size, however, that it is difficult, if not impos-sible, to assess human sampling behavior rela-tive to a realistic ideal without taking intoaccount any implicit costs and rewards. In thisstudy we demonstrated that this could be ac-complished by comparing performance acrossdifferent conditions with different explicit costsand rewards.

An additional novel finding from our study isthat participants were more likely to continuesampling when the currently visible cluster wasmore dispersed and less likely when it was lessdispersed. Since variance estimation is impor-tant in many decision-making problems (Kar-eev, Arnon, & Horwitz-Zeliger, 2002), this islikely an important factor to consider in studiesof information-gathering behavior.

Participants in our study were given ampleopportunity during the practice session to learnthe dispersion of the underlying Gaussian fromwhich the cues were drawn. Given an accurateestimate of the population dispersion, the idealsampling strategy maximizing expected gain ig-nores the current sample dispersion. Hence, sen-sitivity to sample dispersion in our experimentcan only reduce the participant’s expected earn-ings relative to the ideal observer model. How-ever, if the participant for whatever reason doesnot have an accurate estimate of the populationdispersion, then the optimal sampling strategymaximizing expected gain should take into ac-count the current sample dispersion. We con-jecture that, for whatever reason, participantsdid not accumulate an accurate estimate of theunderlying population dispersion across trials,and instead relied in part on the current sampledispersion to decide whether to stop samplingand take action or continue sampling.

As an everyday analogy, imagine trying toestimate the overall typicality of a genre ofmusic by paying to listen to random examplesof the genre online. Instead of deciding in ad-vance how many examples to pay for, it wouldmake sense to assess after each examplewhether or not to pay for another sample. Thelistener would pay for more samples if the earlyexamples sound very different from one another(heightened early variance suggests greater un-certainty), and would stop sampling if the earlyexamples sound quite alike. As Edwards (1965)notes, “if a small set of observations happens tobe sufficiently conclusive, there is no need tobuy more” (p. 312).

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Received October 2, 2014Revision received August 28, 2015

Accepted September 7, 2015 �


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Correction to Juni, Gureckis, and Maloney (2015)

In the article “Information Sampling Behavior With Explicit Sampling Costs”by Mordechai Z. Juni, Todd M. Gureckis, and Laurence T. Maloney (Decision,Advance online publication. December 14, 2015. http://dx.doi.org/10.1037/dec0000045), there was a typo in the degrees of freedom reported for one of thesignificance tests. In the Results section, the third sentence in the Participantsadapted to changes in the cost-benefit structure of the task. subsection, thedegrees of freedom reported should be 198, not 99.


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Information Sampling Behavior With Explicit Sampling Costs...time, energy, and money spent acquiring...

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