39. Decision making and learning while taking sequential risks.txt

Journal of Experimental Psychology:
Learning, Memory, and Cognition
2008, Vol. 34, No. 1, 167–185

Copyright 2008 by the American Psychological Association
0278-7393/08/$12.00 DOI: 10.1037/0278-7393.34.1.167

Decision Making and Learning While Taking Sequential Risks
Timothy J. Pleskac

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Indiana University
A sequential risk-taking paradigm used to identify real-world risk takers invokes both learning and
decision processes. This article expands the paradigm to a larger class of tasks with different stochastic
environments and different learning requirements. Generalizing a Bayesian sequential risk-taking model
to the larger set of tasks clarifies the roles of learning and decision making during sequential risky choice.
Results show that respondents adapt their learning processes and associated mental representations of the
task to the stochastic environment. Furthermore, their Bayesian learning processes are shown to interfere
with the paradigm’s identification of risky drug use, whereas the decision-making process facilitates its
diagnosticity. Theoretical implications of the results in terms of both understanding risk-taking behavior
and improving risk-taking assessment methods are discussed.
Keywords: risk taking, learning, Bayesian, individual differences, cognitive psychometrics

example, create an aversion toward risky alternatives in the gain
domain and an attraction toward risky alternatives in the loss
domain—a pattern typically attributed to how DMs evaluate outcomes (Denrell, 2007; March, 1996). The learning process can
even produce the opposite pattern (Erev & Barron, 2005; Hertwig,
Barron, Weber, & Erev, 2004; Weber, Shafir, & Blais, 2004).
Applying theories of decision making to the Balloon Analogue
Risk Task (BART; Lejuez et al., 2002) or to the Iowa Gambling
Task (Bechara, Damasio, Damasio, & Anderson, 1994) also exposes the necessity of learning. Clinicians use these laboratorybased gambling tasks to study and identify people with specific
clinical or neurological deficits. Cognitive models of these tasks
reveal that decision and learning processes are necessary to account for choices made by both clinical and normal populations
(Busemeyer & Stout, 2002; Wallsten, Pleskac, & Lejuez, 2005).
Besides describing behavior during the tasks, the models also show
how the populations differ on the underlying cognitive dimensions
captured within the models. For example, during the BART, Wallsten et al. (2005) found that people who take unhealthy and unsafe
risks tend to differ from normal populations in how they evaluate
payoffs and the consistency of their responses.
What is unclear, however, is the extent to which real-world risk
takers systematically differ in the learning process used during the
BART. There are two possible roles the learning process could
play. On the one hand, learning may not play a significant role at
all. This prediction comes from studies showing Slovic’s (1966)
devil task discriminates between risk takers without requiring
learning (Hoffrage, Weber, Hertwig, & Chase, 2003). Slovic’s
devil task (henceforth devil task) is of interest because it has an
identical structure to the BART but does not require learning. On
the other hand, the learning process may aid in the BART’s ability
to identify risk takers. In the Iowa Gambling Task, for example,
the learning process is necessary to understand how specific clinical deficits lead to individual differences in risky decision making
(Stout, Rock, Campbell, Busemeyer, & Finn, 2005; Yechiam,
Busemeyer, Stout, & Bechara, 2005). To directly address these
conflicting predictions, this article develops a larger class of sequential risk-taking tasks: the Angling Risk Tasks (ART). This

Learning and decision making are conceptually linked. Typically only after decision makers (DMs) make a decision do they
observe or experience the precise outcome of that decision. For
example, only after commuters select a traffic route do they
determine its effectiveness, and only after athletes use a steroid do
they learn about the precise properties it has on their body. These
observations better inform DMs about the precise properties of
their choice options and shape their next decision among the same
or similar options. Despite this natural association between decision and learning processes, most decision-making theories fail to
incorporate or explicate a learning component (e.g., Busemeyer &
Townsend, 1993; González-Vallejo, 2002; Kahneman & Tversky,
1979). Yet, how DMs learn from experience has proven an important process in understanding risk-taking behavior. It can, for

Timothy J. Pleskac, Cognitive Science Program, Indiana University.
A portion of this article is based on a dissertation submitted in partial
fulfillment of the doctoral requirements at the University of Maryland–
College Park. Parts of this article were also presented as posters at the 2004
annual meeting of the Society for Judgment and Decision Research, Minneapolis, Minnesota, and the 2005 biannual meeting of the European
Association of Decision Making, SPUDM-20, Stockholm. National Institute on Drug Abuse Grant R21-DA14699 (principal investigator: Carl W.
Lejuez) partially funded the study, and National Institute of Mental Health
Research Service Award MH019879, awarded to Indiana University, supported the writing of this article.
I extend my gratitude to Thomas S. Wallsten, Carl W. Lejuez, and Ralph
Hertwig for their continuous support throughout this project. I thank Neil
Bearden, Ana Franco-Watkins, and Eldad Yechiam for constructive comments on an earlier version of this article. I also thank Anthony Bishara,
Sarah Queller, Julie Stout, Ido Erev, members of the Clinical and Cognitive
Neuroscience Laboratory at Indiana University, and members of the ABC
Research Group at the Max Planck Institute for Human Development for
their input on the project. Finally, I am indebted to Laura Wiles for detailed
editing of the article.
Correspondence concerning this article should be addressed to Timothy
J. Pleskac, who is now at the Department of Psychology, 282A Psychology
Building, Michigan State University, East Lansing, MI 48824. E-mail:
tim.pleskac@gmail.com
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PLESKAC

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168

larger class experimentally dissociates the contribution of learning
from the decision processes in a model-free manner. Consequently, we can address whether the learning process gives rise to
systematic individual differences between populations or not.
Equally important to understanding how individual DMs differ
in their learning process is to identify how the learning process
changes across different task environments. One possibility is that
DMs use the same learning process regardless of the task environment. This prediction is derived from the counterintuitive finding that DMs mentally model the BART, a nonstationary probabilistic environment, as a stationary or sampling-with-replacement
environment (Wallsten et al., 2005). The mental representation in
turn affects the specific learning process used. One would then
expect that during a task with a stationary stochastic environment
DMs would adopt the same stationary mental representation. Alternatively, akin to findings in multiattribute decision making,
DMs may use different processes in different environments (see
Gigerenzer, Todd, & the ABC Research Group, 1999; Payne,
Bettman, & Johnson, 1993). To address this issue, DMs completed
the ART with different probabilistic structures (stationary and
nonstationary probabilistic environments). Anticipating the results,
this article will show that DMs do adjust their mental representations to different stochastic environments. Furthermore, during the
ART, in stark contrast to during the BART, they adopt mental
representations that match the task’s actual probabilistic structure.
The article is structured as followed. First, I introduce the
sequential risk-taking paradigm and compare two exemplars from
it: the BART and Slovic’s (1966) devil task. Next, to derive
predictions about how learning may either aid or obstruct in
identifying risk takers, I introduce Wallsten et al.’s (2005) Bayesian sequential risk-taking model (BSR) of the paradigm. I show
how the model can be adapted to account for different mental
representations of different stochastic environments. Then to test
the predictions, I develop the ART and present the results of a
study. During the experiment, participants played the four different
conditions of the ART, reported their past drug-use activity, and
completed the Domain-Specific Risk-Attitude Scale (DOSPERT;
Weber, Blais, & Betz, 2002). The results are analyzed in terms of
the cognitive processes at play during the four tasks and how risky
drug users differ on the underlying cognitive dimensions of the
model. Finally, in the Discussion section I address how the tandem
use of cognitive models and the sequential risk-taking paradigm
can lead to a more precise understanding of the underlying processes used in sequential risky choice.

The Sequential Risk-Taking Paradigm
Two exemplars of the sequential risk-taking paradigm are the
BART and devil task. Both identify real-world risk takers. Performance on the BART, for instance, correlates with a variety of
self-reported risky behaviors such as drinking alcohol, smoking
cigarettes, using illegal drugs, gambling, not wearing a seatbelt,
engaging in unprotected sex, and stealing (Lejuez, Aklin, Jones, et
al., 2003; Lejuez, Aklin, Zvolensky, & Pedulla, 2003; Lejuez et al.,
2002; Lejuez, Simmons, Aklin, Daughters, & Dvir, 2004). The
devil task distinguishes among risk takers at different developmental stages (Montgomery, 1974; Slovic, 1966) and discriminates
between children who cross the street dangerously (Hoffrage et al.,
2003).

Both the BART and the devil task require participants to play
the same structural game for up to a maximum of n choice trials.
On each trial they make a choice between two options, a stop and
a play option. If DMs choose to stop, then the game ends and they
collect the reward in the bank. If they choose the risky play option,
then with some probability one of two possible events occurs. A
successful event can happen in which a constant reward is deposited into the bank, and DMs proceed to the next trial. Alternatively,
a failure event can occur, ending the game, and participants lose
the accumulated reward in the bank for that round.1
In both the BART and the devil task, the same probabilistic
structure determines the probability of a failure event (and by
implication the successful event). For each round, one of the n
choice trials is randomly chosen to produce the failure event. Thus,
all possible trials are a priori equally likely to result in a failure
event. Yet, like drawing balls from an urn and not replacing them,
the probability of a failure increases with each successful play
option, making the probability of a successful event nonstationary.
The games are typically played anywhere between 1 and 90
rounds, and at the beginning of each round the bank is empty.
Despite their underlying similarities, the BART and the devil
task differ in the amount of information participants know about
the task. The devil task is well defined for respondents. During the
game participants— often children—are presented with a board
with n ⫽ 10 switches making them visually aware of the number
of choice trials available for each round.2 True to the structure of
the paradigm, nine of the switches produce a sticker or some fixed
amount of candy when pulled, and the remaining switch yields a
failure event (a devil sticker). Again, players must choose when to
stop the round and collect their cumulated reward; otherwise, if
they pull the devil the round ends and they lose their collected
prizes for that round.
The BART, in comparison, is ill defined for players.3 During the
game, a balloon is shown on a computer monitor and participants
inflate it by pressing a button at the bottom of the screen. With
each successful inflation they receive a fixed amount of money, x¢.
If the balloon explodes (a failure), participants lose the money for
the round. The task is usually constructed to allow a total of n ⫽
128 total possible inflations (choice trials) per round. The BART
is ill defined because participants are unaware of the possible
choice trials for each balloon, nor are they explicitly told that the
chances of a failure (explosion) increase with each successful
choice trial. Participants are told only that (a) they will earn x¢
with each successful pump option; (b) the balloon will explode
somewhere between the first pump and when it fills the screen; (c)
they must decide when to stop pumping and collect their reward
1

This is a different structure from bandit problems (see Berry &
Fristedt, 1985), typically used to study how DMs learn in uncertain
situations. During bandit problems, DMs make repeated choices between
two or more initially unknown options, and unlike in the sequential
risk-taking paradigm, DMs have to learn the possible payoffs as well as
their distribution associated with each option.
2
The children are also shown a graph explaining that the likelihood of
the failure event increases with each pulled switch (see Hoffrage et al.,
2003; Jamieson, 1969; Slovic, 1966).
3
The terms ill-defined task and well-defined task come from experimental economics, where they are frequently used in discussions about deception (see Hertwig & Ortmann, 2001; Hey, 1998).

DECISION MAKING AND LEARNING

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before an explosion ends their round; and (d) they will play the
BART for 30 rounds (Lejuez et al., 2002).
The success of both the BART and the devil task in identifying
real-world risk takers suggests that the ill-defined characteristic of
the BART may not be a necessary element to identify real-world
risk takers. In fact, one aspect of the BSR model suggests that not
fully informing respondents about the BART may even hurt its
clinical diagnosticity. At the same time, though, the model indicates that learning might help its diagnosticity. These conflicting
predictions are not intuitive, so in the next section I introduce the
BSR model and the underlying predictions.

dents with larger values of ␥⫹ will have a target that is greater than
those for people with lower values of ␥⫹ and will typically choose
the play option for more trials in a given round.

Response
DMs use the targeted trial Gh to probabilistically choose between playing or stopping on each trial. The response rule assumes
that the probability of choosing the play option, rh,i, during round
h on trial i strictly decreases with each choice. Formally, the
response rule
rh,i ⫽

The BSR Model
The BSR model was the best performing model of the many
Wallsten et al. (2005) tested, accounting for both choice behavior
during the BART and correlated with self-reported unhealthy and
unsafe risky behaviors.4 The model posits that DMs use three
cognitive processes to complete a sequential risk-taking task:
evaluation, response, and learning. Each component has at least
one free parameter that is used to identify how each individual DM
quantitatively differs on the underlying process. Each process is
briefly described next, beginning with the evaluation process.

Evaluation
To make a choice between playing or stopping, DMs within the
model adopt a choice policy whereby prior to each round they
identify how many trials they should play to maximize expected
gains. That is, at the beginning of each round, DMs can earn x¢,
2x¢, . . ., ix¢, . . ., nx¢ on choice trials 1, 2, . . ., i,. . ., n with
subjective probability ␲h(1), ␲h(2), . . ., ␲h(i), ␲h(n). Thus, the
expected gain on round h for each choice trial i is
⫹

vh,i ⫽ ␲h(i)*(ix)y .

(1)

The exponent ␥⫹ is akin to prospect theory’s diminishing sensitivity parameter where ␥⫹ must be greater than 0 and typical
participants have values less than 1 (Kahneman & Tverksy, 1979;
Tversky & Kahneman, 1992). Lower values of ␥⫹ indicate less
sensitivity to changes in payoffs, and higher values indicate greater
sensitivity.5 The precise subjective probability of a success ␲h(i)
depends on DMs’ mental representations of the stochastic process
(stationary or nonstationary) and their cumulative experience from
the past h – 1 rounds with the task. The best fitting BSR in the
BART assumes participants have a stationary representation,
where the probability of a success on any given trial is q̂h and the
subjective probability of i successes is ␲h(i) ⫽ q̂ih. The Bayesian
learning process (specified later) describes how q̂h changes with
cumulative experience in the task.
The model assumes DMs identify the choice trial that maximizes expected payoffs and use it as a targeted trial, Gh. Optimizing Equation 1—assuming a stationary stochastic process—
produces the following closed form solution:

Equation 2 illustrates how DMs with different values of ␥⫹ will
behave differently during sequential risk-taking tasks. Respon-

(3)

␨h ⫽ exp关z(h ⫺ H/2)兴⫺1,
where H is the total number of rounds that DMs played during the
task. The free parameter z identifies the type of bias a DM has.
Negative and positive values of z characterize an early exploratory
or a late bias, respectively. If z ⫽ 0, then the participant exhibits
no round-dependent bias.

Learning
Reacting to participants’ partial ignorance in ill-defined tasks
like the BART, the BSR allows DMs to have different mental
representations of the stochastic process and assumes they engage
in a Bayesian learning process to discover the parameters of the
task, given their mental representation. During the BART, participants typically represent the task as a stationary one (e.g., drawing
successive balls from an urn and replacing them), despite its true
nonstationary stochastic structure. This mental representation and
Bayesian learning process is described next.
Stationary representation. Before the first round of each
game, DMs have a prior opinion about the probability of a success,
q. The prior beliefs are modeled with beta distributions, f1(q),
The cognitive model is technically number 3 in Wallsten et al. (2005).
Because of the nature of the BART, a probability weighting function
could not be estimated, and so now weighting function was assumed.
However, the larger set of tasks developed here does allow both the value
and weighting function to be estimated for a given respondent and is
therefore addressed later in the article.
5

(2)

1
1 ⫹ exp (␤dh,i⫺␨h)

captures these properties, where dh,i ⫽ i – Gh and ␤ is a free
parameter representing how consistently DMs follow their targeted
evaluation. DMs with lower values of ␤ will be more variable in
their overall gambling behavior during a sequential risk-taking
task.
Departing from the model in Wallsten et al. (2005), the response
function in Equation 3 also contains a response bias module, ␨h,
allowing DMs to have a bias in their response that changes over
rounds. Some DMs, as Wallsten et al. (2005) report, have an
exploratory bias whereby during the first few rounds they continue
choosing the play option past the maximizing trial. Other participants nearing the end of the task tend to choose the play option
much more than would be expected, perhaps exhibiting a house–
money effect (e.g., Thaler & Johnson, 1990). To account for these
two biases, ␨h is set equal to the following expression:

4

⫺␥⫹
Gh ⫽
.
ln(q̂h)

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170

whereby the mean of the beta distribution, q̂1, represents the
estimated subjective probability of a success on the first round for
any given trial and is used in Equation 2. It is a free parameter in
the models.6 DMs with higher levels of q̂1 are more optimistic and
will tend to choose the play option more than would DMs with
lower levels of q̂1.
After each round, DMs observe the total number of successes,
ch, and whether the round ended in a failure (dh ⫽ 1) or not (dh ⫽
0). DMs then update their beliefs about the probability of a success
using Bayes’ rule

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fh⫹1(q兩c1,d1,. . .,cn,dn) ⫽

冕

p(c1,d1,. . .,ch兩q)f1(q)

.

(4)

p(c1,d1,. . .,ch,dh兩q)f1(q)dq

The impact that the observed data have during the updating process depends on the DMs’ uncertainty in their prior beliefs, ␦1. The
free parameter ␦1 is the variance of the prior beta distribution. DMs
with low uncertainty will have a low ␦1 that leads them to discount
the observed data more than would DMs with a higher ␦1. Once the
distributions describing respondents’ beliefs in q are updated, their
new means are used as the estimates of q̂h⫹1 for Round h ⫹ 1.
The mean (q̂1) and the variance (␦1) of the prior distribution
show how the learning process can either improve or impede the
sequential risk-taking paradigm’s clinical diagnosticity. On the one
hand, real-world risk seekers and avoiders may have systematic
differences in their prior beliefs (summarized by q̂1). Specifically,
risk seekers could be more optimistic and believe the initial probability of a success to be higher (high q̂1) than their risk-avoiding
counterparts. As a result they would choose to play for more trials
in a given round, producing a correlation between the adjusted
ART and risky behaviors.
On the other hand, the uncertainty across DMs in their beliefs
(␦1) may only hurt the diagnosticity of the task. In the model, ␦1
moderates the degree to which respondents’ updated beliefs are a
compromise between their observed data from past rounds and
their initial beliefs. Higher levels of uncertainty (␦1) mean that
DMs will update their beliefs faster to reflect the observed data.
And because the observed data are probabilistic, in the short run
the beliefs of DMs will be more variable and thus reduce the
reliability of correlations between performance and real-world
behavior. Of course, this result is further compounded by differences in how far DMs continue to choose the play option for a
given round.
Nonstationary mental representation. The stationary mental
representation is surprising because both the BART and devil task
have a nonstationary probabilistic structure where the a priori
probability of a success, s(i), decreases with each passing Choice
Trial i,
s(i) ⫽

n⫺i
,
n

where n is the total possible number of choice trials. Examining
performance in a larger class of task can address whether DMs use
the same or different mental representation in different tasks.
This question of same or different mental representations can be
examined with an alternative BSR model that allows for a correct
nonstationary mental representation of the task (see also Wallsten

et al., 2005 model number 7). The alternative model assumes that
DMs adopt a correct mental representation but remain uncertain
about the precise properties of the task. That is, during ill-defined
conditions, DMs are uncertain of the maximum number of possible
trials, n. A discretized gamma distribution over n, p1(n), describes
their prior opinion about n’s value for Round 1. The mean, n̂1, and
variance, ␯1, of the gamma distribution are free parameters and
carry the same psychological interpretation as their stationary
counterparts.7 The mean represents the best prior estimate of the
maximum number of trials allowed on Round 1, n̂1, and scales how
optimistic DMs are about the task. The variance, ␯1, indexes
uncertainty and controls the impact that observed successes and
failures have on the DMs’ Bayesian updating process. In this
mental representation, DMs learn and update their opinion of the
likelihood of different values of n using Bayes’ rule
ph⫹1(n兩c1,d1,. . .,ch,dh) ⫽

(c1,d1,. . .,ch兩n)p1(n)
.
兺np(c1,d1,. . .,ch,dh兩n⬘)p1(n⬘)

(5)

.
A derivation of the revision process can be found in Appendix C
of Wallsten et al. (2005). The expected value for the updated
distribution after each round represents the new estimate of the
maximum number of trials, n̂h⫹1.
The remaining evaluation and response components are unchanged. DMs still identify the maximizing trial and probabilistically choose to play or stop based on the distance from that trial.
However, the subjective probability in Equation 1 is set to ␲h(i) ⫽
(n̂h⫺i)/n̂h, and the maximizing trial is
Gh ⫽

n̂h␥⫹
,
␥⫹ ⫹ 1

(6)

where again n̂h is either obtained from the DMs’ estimated beliefs
via the learning process in ill-defined tasks or directly observed in
well-defined games. Comparing Equation 2 with Equation 6 also
shows that ␥⫹ retains the same properties. In both cases, larger
values of ␥⫹ lead to larger target values and consequently riskier
play options taken in a given round.

Summary of Model and Predictions
In summary, the BSR model uses at most five parameters to
predict the decision between playing and stopping during round h
on choice trial i. It assumes DMs evaluate options prior to begin6
The beta distribution was chosen due to its nice mathematical properties as a conjugate distribution to the binomial (see Gelman, Carlin, Stern,
& Rubin, 2003). Typically the beta distribution is modeled with two free
parameters, a0 and b0, where a0 ⱖ 0 and b0 ⬎ 0. The mean and variance
are functions of these two parameters. In fact, the models were estimated
with a0 and b0. For ease of interpretation, a0 and b0 are reparameterized
into the mean and variance in the text.
7
The gamma distribution is a continuous distribution that is sometimes
specified by the parameters ␯ and ␶, where ␮G ⫽ ␯␶ and ␴G2⫽ ␯␶2 (see
Gelman et al., 2003) . It was chosen because of its properties of being
distributed over the positive reals and being flexible enough to characterize
a wide range of different opinions. To obtain the discrete approximation to
the gamma distribution, I integrated the distribution from x ⫽ n – 0.5 to
x ⫽ n ⫹ 0.5 for each n ⫽ 1, 2, . . ., ⬁ and then normalized.

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DECISION MAKING AND LEARNING

ning each round and identify a targeted choice trial that would
maximize their expected gains. They then probabilistically choose
to play or stop according to the distance from their target. Finally,
during ill-defined games they use their observed data from past
rounds to update their beliefs about the properties of the task. The
model can account for both stationary and nonstationary mental
representations.
There are two sets of predictions from the BSR that this article
investigates. One set deals with the competing hypotheses regarding learning during ill-defined conditions. On the one hand, individual differences in prior beliefs captured with the parameter p̂h
or n̂h may aid the paradigm’s ability to identify real-world risk
takers. On the other hand, DMs’ general uncertainty seen with
parameter ␦1 or ␯1 may do the opposite and hurt this ability. A
second prediction is that DMs mentally model sequential risktaking tasks as a stationary probabilistic structure even when the
stochastic structure is truly nonstationary. Next, I develop a class
of four sequential risk-taking tasks that I call the Angling Risk
Tasks, which experimentally tests these predictions (see Figure 1).

The Angling Risk Tasks (ART)
The ART uses fishing tournaments akin to decision theory’s prototypical “balls in the urn” laboratory task to aid participants’ understanding of the stochastic environment. The premise of the ART lends
itself well to manipulations of learning and the type of stochastic
process. During a particular game, participants fish in a tournament
for H rounds or trips (typically 30) in a pond that has 1 blue fish and
n – 1 red fish. With each cast of a computerized fishing rod, they hook
a fish (each fish is equally likely to be caught). If it is red, then they

171

earn 5¢ and can cast again. But if it is blue, then the round ends and
the money earned on that round is lost.
Different levels of learning can be manipulated by changing the
weather conditions of the fishing tournament. For example, the
tournament can take place on a sunny day, allowing participants to
see how many fish are swimming in the pond and eliminating the
need to learn their distribution. In contrast, the tournament can take
place on a cloudy day, concealing the fish in the pond, which in
turn forces participants to learn about how many potential fish are
in the pond. If real-world risk takers do systematically differ in
their prior beliefs (q̂1 or n̂1), then the ill-defined cloudy conditions
should correlate equally well or even better than the well-defined
sunny conditions.
Also, DMs’ use of the same or different mental representations
in different stochastic conditions can be tested by changing the
pond’s release law. Participants can catch and keep their fish (the
catch ‘n’ keep tournament), a sampling-without-replacement process that is identical to the BART and devil task. Or participants
can release their fish back into the pond (catch ‘n’ release tournament), a new task with a stationary sample-with-replacement process that matches the assumptions DMs hold about the BART.
In the experiment detailed next, respondents completed all four
tournaments and also reported their past illegal drug use activity.
The activity of drug use was selected because it has been one of the
standard reported behaviors that the BART has been validated with
(see for example Lejuez et al., 2002). In addition, risky drug use is
an activity performed at different levels in and around college
campuses, where this study took place (Johnston, O’Malley, Bachman, & Schulenberg, 2005).

Figure 1. A screenshot of the Angling Risk Tasks’ catch ‘n’ keep tournament on a sunny day. The screen
changes for the three other tournaments. If the weather is cloudy, then the fish in the pond are not shown to the
participants and no information about their number is given in the information panel. During catch ‘n’ release,
the cooler is closed and the fish are returned to the pond rather than the cooler.

PLESKAC

172

A total of 72 participants were recruited from the University of
Maryland community with advertisements placed throughout the
campus. The sample consisted of 38 men and 34 women, ranging
in age from 18 to 34 years (M ⫽ 21.6, SD ⫽ 3.9). Fifty-six percent
described themselves as White, 18% as Black or African American, 17% as Asian or Southeast Asian, and 4% as Hispanic or
Latino; the remaining 6% marked “Other” or chose not to respond
to the question. They were paid $7 for their time. In addition,
participants could earn a bonus based on their winnings in the
games. On average, people received an extra $5, but a few won
nothing and one individual earned an extra $12.

most frequently (i.e., never, one time, monthly or less, 2 to 4 times
a month, 2 to 3 times a week, or 4 or more times a week).
Weighting the categories that participants reported trying with the
frequency rank and summing created the measure identified as
weighted polydrug.
Domain-Specific Risk-Taking (DOSPERT) Scale. Participants
also completed the DOSPERT Scale (Weber et al., 2002), which
contains 40 items that assess the likelihood of engaging in risky
behavior in six domains: ethics, investment, gambling, health/
safety, recreational, and social. Two separate variants of the scale
also assess the perception of the magnitude of the risk for and
expected benefit from each of the 40 risks. The DOSPERT has also
had success in identifying real-world risk takers (Hanoch, Johnson,
& Wilke, 2006).

Materials

Design and Procedure

The ART. In each of the four tournaments participants had
H ⫽ 30 rounds during which they fished in a pond similar to the
one shown in Figure 1. At the beginning of each round the pond
had 1 blue fish and n – 1 red fish. Below the pond were two buttons
and an information panel. One button was labeled “Go Fish.”
Pressing it caused the rod on the left of the screen to cast a line into
the pond and hook a fish. Each fish was equally likely to be caught
on a given cast. If a red fish was caught, then 5¢ was placed into
the “Trip Bank” shown on the information panel. What happened
next depended on the release law. If the law was catch ‘n’ keep,
then the red fish was placed in the cooler on the right of the screen,
reducing the total number of red fish in the pond by one. If the law
was catch ‘n’ release, then the red fish was placed back into the
pond. Either way, the computerized fish swam around the pond
and participants had another opportunity to cast the line into the
pond for that round. However, if a blue fish was caught, then the
round ended and participants lost their accumulated money in the
“Trip Bank.” When participants decided to stop fishing, they
pressed the “Collect” button to transfer the money to the “Tournament Bank” on the information panel and then began a new
round.
In addition to the two release laws, there were two types of
weather conditions. If the weather was sunny—as indicated by the
weather forecast in the bottom right—the pond was clear and
participants could see how many fish were in it at all times.
Additionally, the information panel listed how many red and blue
fish were in the pond before each cast. However, if the weather
was cloudy, then the pond was murky, concealing the number of
fish in the pond and the information panel was left blank. Combining the two release laws with the two weather forecasts produced four different fishing tournaments (conditions). Participants
completed all four tournaments.
Drug and alcohol questionnaire. As a measure of risky illegal/
legal drug use, participants reported with yes/no responses the
number of drug categories ever tried. The 11 different drug categories were cannabis, alcohol, cocaine, MDM (ecstasy), stimulants
(e.g., speed), sedatives/hypnotics, opiates, hallucinogens, PCP,
inhalants, and nicotine. The sum of the number of categories tried
(polydrug) is a validated measure of risky drug use (see Babor et
al., 1992; Grant, Contoreggi, & London, 2000). After each yes/no
question, participants were also asked to report about how often
they used the drug at the time in their life when they used it the

The study used a 2 (release law) ⫻ 2 (weather) within-subjects
design. Participants fished in all four tournaments (conditions) and
completed the drug and alcohol questionnaire along with the
DOSPERT Scale. Participants had 30 rounds per tournament to
cast for as many red fish as they chose, earning 5¢ per red fish
caught. Intertask comparisons were facilitated by setting the ART
parameter settings to be similar to those used in the BART. In the
BART (a nonstationary environment), the total number of possible
choice trials (pumps) is typically set at n ⫽ 128 (Lejuez et al.,
2002). If the goal of participants is to maximize expected value and
they have full knowledge of the structure of the task, then the
strategy that maximizes earnings is for participants to choose the
play option on approximately 64 of the choice trials. This setting
has been found to be the best for distinguishing between real-world
risk seekers and risk avoiders (Lejuez et al., 2002). Consequently,
in the catch ‘n’ keep conditions there were n ⫽ 128 total number
of fish or possible choice trials. Calibrating the catch ‘n’ release
conditions in the ART so that choosing play 64 times per round
was the maximizing strategy (assuming the same conditions as
above) was done by setting the number of fish at n ⫽ 65 fish (see
Equation 2 with ␥⫹ ⫽ 1). Thus, if participants had a precise
understanding of all four tasks and sought to maximize expected
value, they should make an equal number of casts in all four
tournaments.
The order in which participants completed each tournament and
the risk questionnaires was counterbalanced. All eight tasks were
programmed using Sun Microsystem’s Java language and are
available upon request. The entire experiment was administered on
PC computers in separate sound-attenuated laboratory cubicles.
After reading and signing the informed consent form, participants read an introductory set of instructions on the computer.
They were told that they would be playing four different fishing
tournaments with different rules and conditions. The instructions
described the two different release laws and the two different
weather conditions they would experience. Participants were also
informed that between each fishing tournament they would fill out
questionnaires assessing their own risky behavior. No further
instructions were given about the stochastic structure of the tasks.
Next, the participants completed four practice games, one for
each tournament. During the practice games participants were able
to select how many fish they would like in the pond to practice
with (1 to 360). This manipulation was done to demonstrate that

Method

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Participants

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DECISION MAKING AND LEARNING

the ponds in each of the experimental conditions could have any
number of fish so as to minimize knowledge transfer from one
tournament to the next. During the practice round participants
played each condition for two rounds, during which they cast for
red fish as many times as they chose to.
After completing the four practice games, they began the experimental session, alternating between the four questionnaires and
the four tournaments. They started with a questionnaire. Before
each tournament, participants were briefly reminded of the rules
governing the pond they were about to visit. After completing all
of the tournaments, they completed a set of questions regarding the
strategy they used to fish in the tournaments. At the conclusion of
the session, the computer produced four tables showing how much
money participants earned on each round during the four tournaments. A round from each tournament was then chosen randomly
using a bingo basket (four rounds total), and participants were paid
based on their performance during these rounds.

Results
The Results section is organized into two subsections. The first
addresses the cognitive processes used during the four different
conditions of the ART (catch ‘n’ keep, sunny; catch ‘n’ keep,
cloudy; catch ‘n’ release, sunny; and catch ‘n’ release, cloudy).
The BSR model is used in this section to examine whether participants use the same mental representation in all four conditions of
the ART, or whether they change their mental representation
according to the stochastic structure of the task. Additionally, a set
of BSR models with a probability weighting function is compared
to the models without to examine whether DMs overweight rare
events and underweight common events in sequential risk-taking
tasks. The second section examines individual differences in the
cognitive processes and their association to the risky use of drugs.
One participant did not complete the experimental session and is
not included in subsequent analyses.

Cognitive Processes During the ART
Both the average adjusted ART scores and proportion of rounds
ending in a blue fish are listed in Table 1. The adjusted ART score
is the average number of casts taken on rounds when participants
chose to stop fishing. The adjusted score is the standard behavioral
measure of performance in sequential risk-taking tasks (Lejuez et
al., 2002; Pleskac, Wallsten, Wang, & Lejuez, 2007). Hence, it is
used in all subsequent analyses at the behavioral level.

Table 1
Mean (SD) Adjusted ART Scores and Percentage of Rounds
Ending With a Participant Catching a Blue Fish
Adjusted ART scores
Release law

Sunny day

Cloudy day

Catch ‘n’ keep
38.96 (18.77) 31.59 (15.35)
Catch ‘n’ release 32.19 (15.95) 28.06 (15.22)

Percentage of rounds
ending with a blue fish

173

To analyze the effect of the weather conditions and release
conditions, I conducted an ANOVA on the adjusted ART scores
where each experimental condition was a within-groups variable.
To test for order effects, two other between-groups variables were
entered: (a) the order of completing sunny and cloudy tournaments; and (b) the order of completing the release conditions. The
ANOVA utilized the full model, allowing interactions between the
individual participant and the variables of interest. The appropriate
error terms are therefore the interactions between the participant
and the variable of interest, MSPar ⫻ Variable (see Howell, 1997).
The left panel in Figure 2 shows the average adjusted ART score
in the four experimental conditions.8 Participants had a tendency to
cast more frequently in the catch ‘n’ keep condition than during
catch ‘n’ release, F(1, 67) ⫽ 17.70, p ⬍ .01, MSPar ⫻ Release
Rule ⫽ 103.84. They also made more casts in the sunny condition
than in the cloudy condition, F(1, 67) ⫽ 18.18, p ⬍ .01, MSPar ⫻
Weather Condition ⫽ 121.55. Finally, there was a significant
interaction between release law and weather, F(1, 67) ⫽ 6.01, p ⬍
.05, MSPar ⫻ Release Rule ⫻ Weather Condition ⫽ 35.76. As
Figure 2 indicates, the change from sunny to cloudy had a larger
effect in the catch ‘n’ keep condition than in catch ‘n’ release,
t(67) ⫽ 2.46, p ⬍ .05.
There were no main effects related to the order in which participants completed the four fishing conditions. Participants who
completed sunny tournaments before cloudy tournaments did not
have significantly different adjusted ART scores. Nor did the order
in which they completed the different release laws affect their
casting behavior. Despite all efforts to minimize information transformation from the sunny to cloudy condition, there were two
significant interactions with the order of completing the weather
conditions. The order of completing the different release laws
interacted with the weather variable, F(1, 67) ⫽ 6.00, p ⬍ .05,
MSPar ⫻ Weather Condition ⫽ 121.55. Paired comparisons revealed participants did cast significantly more in sunny conditions
(M ⫽ 36.7) than in cloudy (M ⫽ 27.9) when completing catch ‘n’
release conditions first, t(67) ⫽ 4.8, p ⬍ .05. But when the catch
‘n’ keep conditions were completed before catch ‘n’ release conditions, there was not a significant difference between casting
behavior between sunny (M ⫽ 34.2) and cloudy (M ⫽ 31.8)
conditions, t(67) ⫽ 1.3. The order of completing the weather
conditions also interacted with the weather manipulation, F(1,
67) ⫽ 7.55, p ⬍ .05, MSPar ⫻ Weather Condition ⫽ 121.55. The
adjusted ART scores for the cloudy conditions were significantly
greater when participants completed sunny conditions first (M ⫽
33.4) as compared with cloudy conditions first (M ⫽ 26.4), t(67) ⫽
3.8, p ⬍ .05. But there was not a significant difference between
adjusted ART scores on the sunny conditions based on the order in
which they completed the weather conditions (M ⫽ 35.4 and 35.5,
respectively). The only other significant order effect was the
four-way interaction, but it was due to the two significant two-way
interactions. Neither significant interaction pertaining to the order
in which participants completed the four conditions influenced any

Sunny day Cloudy day
.34 (.16)
.38 (.14)

.25 (.15)
.35 (.14)

Note. Calculations are based on a total of 30 rounds. ART ⫽ Angling
Risk Tasks.

8
The mean and median number of casts made on rounds that participants stopped on is approximately equal. All conclusions drawn from
analyses based on the mean, including all the ANOVAs and all the
correlations, are identical to those done with the median. Consequently,
only the means will be reported in the text.

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174

PLESKAC

Figure 2. Average adjusted Angling Risk Tasks (ART) scores across participants for the four different fishing
tournaments. In the left panel, the bars represent the average adjusted ART scores and the vertical lines denote
standard errors of the mean, estimated from the MSPar ⫻ Release Rule ⫻ Weather Condition interaction. The
right panel plots the predicted average adjusted ART score calculated from the 71 sets of individual parameter
estimates using the Bayesian sequential risk-taking model with correct mental representations (see Appendix B
for the analytic solutions for predicting the adjusted ART scores from the cognitive models). Stars indicate the
average number of casts that the decision makers intended to take on all rounds (see Appendix B).

of the conclusions drawn from all subsequent analyses and will not
be considered further.
A final observation concerning the adjusted ART score is that
choice behavior in the ART conditions appears relatively consistent with that of the BART. The average adjusted ART scores for
all four conditions were near or slightly greater than the adjusted
score typically observed with the BART (M ⫽ 30.3; N ⫽ 448;
SE ⫽ 2.2; 95% confidence interval ⫽ 26.0 ⬍ score ⬍ 34.6; see
Aklin, Lejuez, Zvolensky, Kahler, & Gwadz, 2005; Crowley,
Raymond, Mikulich-Gilbertson, Thompson, & Lejuez, 2006;
Jones & Lejuez, 2005; Lejuez, Aklin, Bornovalova, & Moolchan,
2005; Lejuez, Aklin, Jones, et al., 2003; Lejuez, Aklin, Zvolensky,
& Pedulla, 2003; Lejuez et al., 2002). Next I use the two versions
of the BSR model to test whether respondents use the same mental
representation in all four conditions or whether they adapt their
mental representation to the different conditions.

BSR Analyses
Two BSR models were fit to the data from all four ART
conditions at the individual level. Both predict the probability of
casting, rh,i, during round h on trial i. The first model—the best
fitting model for the BART—tests the hypothesis that participants
believe a stationary stochastic process governs the task. For cloudy
conditions it has one evaluation parameter, two response parameters, and two learning parameters. In sunny conditions there is no
learning and q̂h in Equation 2 is set to the observed parameters of
the task (e.g., 64/65 in catch ‘n’ release). The second model tests
whether participants mentally represent the structure as nonstationary. It has the same allocation of free parameters, and in sunny
conditions n̂h is set to the observed number of fish (e.g., 128 in
catch ‘n’ keep). In addition, a baseline statistical model was
estimated from the data (see Wallsten et al., 2005, for details). It
has one parameter, the probability of casting on trial i. It assumes
no cognitive processes and serves as an initial test for the BSR
models.

Model estimation and comparison. The models were fit to
each individual’s choice data for all rounds from each fishing
tournament using maximum likelihood estimation methods described in Appendix A. The data have a substantial number of
observations per participant, but the precise number of independent observations depends on the intercorrelation between trials
(captured within the model) and differs for each individual. As a
rough estimate, the number of independent observations for each
tournament ranged between 30 rounds and the total number of
trials on which participants actually chose between playing and
stopping, which was on average 840.4 trials per person per tournament.
Standard maximum likelihood ratio tests are not available to
evaluate and compare model fits due to the nonnested nature of the
models. Instead descriptive-level comparisons were made with the
Bayesian information criterion (BIC; Schwarz, 1978; Wasserman,
2000). The BIC is one method to compare the fit of nonnested
models with different numbers of parameters, where
BIC ⫽ 关⫺2 ⫻ ML兴 ⫹ 关k ⫻ log(N)兴.

(7)

In the expression, ML is the maximum log-likelihood of the data
given the model, k is the number of parameters in the model, and
N is the number of independent observations that contributes to the
likelihood. The term on the right-hand side of Equation 7 accounts
for model complexity (i.e., the more parameters the more complex
the model). The model with the lowest BIC is chosen as the best
fitting model. As a very liberal estimate of N, I used the total
number of choices a participant made in a given tournament. This
estimate puts the greatest handicap on models with more parameters.
Table 2 shows how many participants were best fit by each
model in the two sunny tournaments using BIC. Recall no learning
process parameters for the BSR models were fit in these conditions, and the number of free parameters reflects this. The model
fits in the sunny tournaments also reveal that the models with

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DECISION MAKING AND LEARNING

congruent representations fit the data better than does the statistical
baseline model.
However, as is apparent in Table 2, in the sunny tournaments the
BSR model that assumes participants have an incorrect mental representation of the task could not fit the data. Here is why. Recall that
in the well-defined sunny tournaments participants knew how many
fish were in the pond. Consequently, in the catch ‘n’ release tournaments the model assuming a nonstationary representation would set
the value of n̂h to 65 in Equation 6 for all 30 rounds (the number of
fish in the pond at the beginning of the round). With this setting, the
model does not predict that DMs would make more than 65 casts on
a given round. That is because in their sample-without-replacement
model they would have caught all 65 fish in the pond, leaving none in
the pond. Yet, a third of the participants in the sunny catch ‘n’ release
tournaments cast more than 65 times for at least one round. For the
remaining participants the evaluation parameter would have to reflect
extreme sensitivity to payoffs (␥⫹ ⬎ 1.5 in Equation 6) to capture
their data—an estimate that is highly unlikely and inconsistent with
decision theory literature (e.g., Tversky & Kahneman, 1992).
The same issue arises when fitting the model that posits a
stationary representation to the data from the sunny catch ‘n’ keep
condition. Here q̂h would be set to 127/128 in Equation 2. Using
this value, all participants would appear incredibly insensitive to
outcomes (␥⫹ ⬍ 0.3)—an estimate that is quite incongruent with
the literature. Thus, these results paired with the superior performance of the BSR model over the baseline model offer preliminary
support that in nonlearning conditions DMs can correctly adapt to
and represent the probabilistic structure of the sequential task they
face. This result is in contrast to the hypothesis derived from the
BART that DMs use the same representation in both conditions.
The results from the cloudy tournaments support the same
conclusion of correct mental representations. Table 3 shows that
both BSR models fit the data better than do the baseline statistical
models in the catch ‘n’ keep and catch ‘n’ release conditions.
Moreover, the models that assume participants correctly represent
and learn about the task structure best fit a majority of participants
in both conditions. That is, the stationary model fits best in catch
‘n’ release, and the nonstationary fits best in catch ‘n’ keep. These
results are in contrast with the BART results, where the model

Table 2
Model Comparisons of the Sunny Tournaments
Catch ‘n’ keep

Model

df

Mean
BIC

Number of
DMs best
fit with
BIC

Baseline
Stationary mental
representation
Nonstationary mental
representation

1

192.16

4

3
3

156.86

Catch ‘n’ release

Mean
BIC

Number of
DMs best
fit with
BIC

176.15

1

145.53

70

67

Note. During sunny conditions, only the Bayesian sequential risk-taking
models that assumed the correct stochastic process could be fit to the
respective conditions. BIC ⫽ Bayesian information criterion; DM ⫽ decision maker.

175

Table 3
Model Comparisons of the Cloudy Tournaments
Catch ‘n’ keep

Catch ‘n’ release

Model

df

Mean
BIC

Number of
DMs best
fit with
BIC

Baseline
Stationary mental
representation
Nonstationary mental
representation

1

206.00

4

176.34

3

5

179.35

3

144.83

49

5

171.51

65

146.34

19

Note.

Mean
BIC

Number of
DMs best
fit with
BIC

BIC ⫽ Bayesian information criterion; DM ⫽ decision maker.

assuming a stationary representation fit the nonstationary task best
(Bishara et al., 2007; Wallsten et al., 2005).9
Finally, the best-fitting BSR models also reproduce the adjusted
ART scores well. The bar graph in the right panel of Figure 2 plots the
predicted adjusted ART scores for the four conditions averaging
across all participants and all rounds. The predicted adjusted ART
scores were calculated for each individual and for each round using
their respective parameter estimates from each condition. The analytical solutions for the predictions are developed in Appendix B.
Comparing the predicted and empirical scores does show a
slight miscalibration in the models. For all four conditions, the
predicted adjusted ART scores are about five casts higher than in
the empirical data. This is likely due to the fact that the empirically
estimated adjusted ART scores are calculated from only the rounds
that the participant ended with a stop choice. The models, in
contrast, are estimated from all the rounds. To understand why this
is important, recall first that in the ART the more that participants
cast the more likely the round is to end in a blue fish. Thus, the
rounds that end with a blue fish are also the rounds in which
participants are more likely to exhibit the greatest level of risk
seeking. The models— being fit to all the rounds—reveal this
aspect of increased risk taking in their parameters, and as a result
the predicted adjusted ART scores (even when conditionalized on
the stop rounds) will reflect a slightly more risk-seeking nature
than in the empirical scores. The models in fact even permit us to
estimate the intended casting behavior of participants for all
rounds (see Appendix B for a solution). These predictions are
plotted as stars in the right panel of Figure 2. They show that if
casting behavior were observable on all rounds, then the scores
would be even slightly higher.10 Further implications of these results
are addressed in the discussion. Next, I examine whether the models
should be expanded to include a probability weighting function.
Probability weighting function. One final untested hypothesis
is whether during the evaluation process rare events are overweighted and common events underweighted. A probability
9
Note Wallsten et al. (2005) used Akaike’s information criterion (AIC)
as a goodness of fit measure. BIC and AIC give rise to the same conclusions in this dataset.
10
Appendix B also uses the analytic solutions for the adjusted ART
scores to show an error theory is not sufficient alone to account for the
observed data.

PLESKAC

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176

Figure 3. Plots and properties of the weighting function. The left panel shows a weighting function exhibiting
overweighting of low probabilities and underweighting of high probabilities. The right panel varies ␣ between
0 and 1; as ␣ decreases, the weighting function approaches a step function.

weighting function with these properties is displayed in the left
panel in Figure 3. These properties have been found to be a
necessary component in explaining preferences over static gambles (see Fox & Tversky, 1998; Gonzalez & Wu, 1999; Kahneman
& Tversky, 1979; Luce, 2000; Prelec, 1998; Tversky & Kahneman, 1992; Wakker & Tversky, 1996; Wu & Gonzalez, 1996,
1999). A probability-weighting function may be necessary in sequential risk-taking tasks as well. This study presents a unique
opportunity to investigate this hypothesis because participants
used two different mental representations in two different tasks.
This occurrence makes it possible to simultaneously estimate a
probability weighting function and a value function (like the one
used in Equation 1) due to the fact that participants completed
more than one sequential risk-taking task and used different mental
representations (see Pleskac, 2004). If these conditions are not
met—as in Wallsten et al. (2005)—then the two functions are not
simultaneously identifiable.
To expand the BSR model to include a weighting function, I
used Prelec’s (1998) one-parameter function 11 : w(t) ⫽
exp共–关–log共t)]␣), where 0 ⱕ ␣ ⱕ 1 and t is the probability of an
event. For ␣ ⫽ 1 the weighting function is an identity function,
illustrating Wallsten et al.’s (2005) assumptions. The parameter ␣
measures the degree of over/underweighting of probabilities. As ␣
decreases there is more over/underweighting (see the right panel of
Figure 3). In Equation 1, ␲h(i) is now w[␲h(i)]. Consequently, the
maximizing trial for either mental representation is a function of ␣
as well. If we revise Equation 2, the closed form solution for the
maximizing trial assuming a stationary mental representation is

Gh ⫽

冉 冊

␥⫹ 1/␣
␣
.
ln(qh)

⫺

(8)

There is not a closed form solution for the maximizing trial in the
nonstationary mental representation (see Equation 6 for the expression
without a weighting function), but it can be estimated numerically.
The weighting function has similar implications for both mental
representations. With ␥⫹ held constant, as the degree of over/
underweighting of probabilities increases (␣ decreases), the maximizing trial increases and DMs are predicted to make more and more
casts. In comparison, if the weighting function is held constant, as
individuals exhibit less and less diminishing sensitivity to changes in
payoffs (␥⫹ decreases), then the maximizing trial decreases.
Equation 8 reveals that for a given condition of ART, ␣ and ␥⫹
cannot be simultaneously estimated. The same holds for the nonstationary model. However, the parameters can be estimated if
they are set equal across tasks in which participants use different
mental representations (see Pleskac, 2004). In this experiment we
have two conditions in which participants used a stationary representation (the catch ‘n’ release conditions) and two with a nonstationary representation (the catch ‘n’ keep conditions; see Tables 2
and 3). The most reasonable constraint for model estimation is to
set ␥⫹ and ␣ equal across catch ‘n’ keep and catch ‘n’ release for
a given weather condition. Using this constraint the BSR models
with a weighting function were fit at the individual level for all
four conditions using maximum likelihood procedures. The reader
is referred to Appendix C for a detailed explanation of the estimation and model comparison.
Briefly, the results of the model comparison showed that a
weighting function made a substantial contribution in describing
the data only when DMs were constrained to show only diminishing sensitivity (0 ⬍ ␥⫹ ⱕ 1). If, however, ␥⫹was allowed to take
11

The two-parameter version is not identifiable in the present models.

DECISION MAKING AND LEARNING

177

Table 4
Summary Statistics of and Correlation With MLE Parameters for the Cloudy Tournaments, Assuming Mental Representations Were
Congruent With the Structure of the Task
Catch ‘n’ keep
Evaluation
Statistics

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Mean
SD
1st quartile
Median
3rd quartile
Adjusted ART score (KC)
Adjusted ART score (RC)
Health Attitude score
Social Attitude score
Weighted polydrug

␥

⫹

␣

0.59
0.25
0.43
0.59
0.84

0.75
0.20
0.61
0.79
0.98

.26**
.24**
.01
⫺.07
.24*

⫺.07
⫺.10
⫺.18
⫺.22
.01

Catch ‘n’ release

Learning
n̂1
145.3
141.5
50.78
96.79
183.65
.30**
.13
⫺.01
.07
⫺.09

Response

Learning

Response

v1

z

␤

q̂1

␦1

z

␤

9.74 ⫻ 106
8.20 ⫻ 107
83.08
645.22
3,867.18

0.01
0.05
⫺0.01
0.01
0.05

0.3
0.58
0.09
0.15
0.28

0.92
0.21
0.96
0.98
0.99

1.73 ⫻ 10⫺3
8.11 ⫻ 10⫺3
1.41 ⫻ 10⫺5
6.50 ⫻ 10⫺5
3.31 ⫻ 10⫺4

⫺0.01
0.08
⫺0.05
0.00
0.04

1.26
8.18
0.14
0.17
0.31

.22
.11
.01
.04
.43x

.08
⫺.02
.05
.01
⫺.10

⫺.34**
⫺.16
⫺.01
.05
⫺.15

.20
.14
⫺.17
⫺.01
⫺.37*

⫺.09
⫺.11
⫺.01
⫺.18
.01

.10
.05
.16
.08
⫺.04

⫺.37**
⫺.52**
⫺.19
.03
⫺.10

Note. The evaluation parameters (␥⫹ and ␣) are set constant across the release conditions for a given weather condition. The Pearson r correlations
between the MLE parameters and the adjusted ART scores, the DOSPERT scores from the Health Attitude and Social Attitude domains, and the selfreported weighted polydrug use are in the bottom five rows. MLE ⫽ maximum-likelihood estimates; ART ⫽ Angling Risk Tasks; KC ⫽ catch ‘n’ keep
(cloudy day); RC ⫽ catch ‘n’ release (cloudy day); DOSPERT ⫽ Domain-Specific Risk-Taking Scale.
*
p ⬍ .05. **p ⬍ .01. x ⫽ not a meaningful correlation (see Footnotes 11 and 12).

any value between 0 and 3—meaning DMs could have diminishing, equal, or increasing sensitivity—then the original set of models taking probabilities at face value without a weighting function
fit a majority of participants best.
Next, the models with the weighting function and the diminishing
sensitivity constraint are used to assess individual differences in the
cognitive processes. Though perhaps not the best fitting models— but
still adequate (see Appendix C)—the models with a weighting function make it possible to test whether risk takers differ in how they
evaluate payoffs, weight probabilities, or both. Behavioral decision
theory would predict that risk takers might differ on both dimensions
(Tversky & Fox, 1995; Tversky & Kahneman, 1992). Tables 4 and 5
summarize the maximum likelihood estimates (MLE) of the parameter values for models with the weighting function in cloudy and
sunny tournaments, respectively.12,13

Individual Differences in Cognitive Processes
The relationship between performance during sequential risktaking tasks and real-world risky behavior makes it possible to examine individual differences in the cognitive processes used during
the tasks. In particular, the relationship between the four adjusted
ART scores and risky behaviors can reveal whether learning helps or
hurts the clinical diagnosticity of sequential risk-taking tasks. On
average, participants reported using 2 of the 11 drug categories. The
average score on weighted polydrug was 10.0 (SD ⫽ 10.5; Mdn ⫽
7.0). The scale had good internal reliability with a coefficient alpha
value of .89. Three participants did not complete the drug questionnaire, so they are not included in this set of analyses.
Table 6 lists the Pearson correlations between the adjusted ART
scores and weighted polydrug.14 The table shows that in the sunny
conditions (when participants saw and knew how many fish were
in the pond) there was a significant correlation between the adjusted ART score and weighted polydrug. However, in support of
the prediction that the learning process hinders the diagnosticity of

the ART, the adjusted ART scores in the cloudy tournaments were
not significantly correlated with the self-reported risky use of
drugs. An a priori planned comparison showed that the correlations
in the sunny conditions were significantly greater than in the
cloudy, z ⫽ 3.10, p ⬍ .05 (see Meng, Rosenthal, & Rubin, 1992).
That is, only in well-defined versions of the ART that do not
require a learning process do the adjusted ART scores identify the
risky use of drugs. In fact, Table 7 shows that behavior was
consistent enough across the sunny conditions that the adjusted
ART scores calculated on the first, second, and last set of 10
rounds were also significantly correlated with weighted polydrug.

12

The bias component significantly improved the fit of the models. The
bias-free nonstationary representation models provided a significantly worse
fit to the data in the catch ‘n’ keep conditions, G2(71) ⫽ 915.87, p ⬍ .01 and
G2(71) ⫽ 105.54, p ⬍ .01, respectively. A similar result was found with the
stationary representation in catch ‘n’ release, G2(71) ⫽ 436.85, p ⬍ .01 and
G2(71)⫽ 168.34, p ⬍ .01. All conclusions regarding the mental representation
of respondents remain if the data are analyzed without the bias component.
13
The mean MLE of ␣ in Tables 4 and 5 indicate that on average individuals were less sensitive to changes in probability in the sunny tournaments than
in the cloudy, t(70) ⫽ 3.80, p ⬍ .01. The remaining possible hypothesis tests
regarding changes between conditions of cognitive processes with both leastsquares methods and nested model comparisons did not identify any significant or consistent trends across the tournaments.
14
All correlations were run with both Goodman and Kruskal’s nonparametric measure of association (K) and Pearson’s r. All conclusions drawn
from the association with weighted polydrug remain, whether one uses a
Pearson’s r correlation or K. One minor inconsistency is that response
sensitivity in the sunny catch ‘n’ keep condition was negatively associated
only with the nonparametric measure, K ⫽ –.29, p ⬍ .01. Given that none
of the other conditions showed this association, it will not be interpreted.
Pearson’s r correlations will be reported in the text to support the interpretation of the linear regression analyses.

PLESKAC

178

Table 5
Summary Statistics of and Correlation With MLE Parameters for the Sunny Tournaments, Assuming Mental Representations Were
Congruent With the Structure of the Task
Evaluation
Statistics

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Mean
SD
1st quartile
Median
3rd quartile
Adjusted ART score (KS)
Adjusted ART score (RS)
Health Attitude score
Social Attitude score
Weighted Polydrug score

␥⫹
0.68
0.33
0.41
0.87
0.99
.74**
.54**
.24*
.00
.34**

Catch ‘n’ keep response

Catch ‘n’ release response

␣

Z

␤

z

␤

0.59
0.28
0.45
0.55
0.89

⫺0.01
0.05
⫺0.06
⫺0.01
0.06

0.73
3.31
0.1
0.14
0.19

⫺0.01
0.06
⫺0.04
0.00
0.02

0.15
0.21
0.03
0.09
0.18

⫺.07
⫺.16
⫺.13
⫺.10
⫺.08

⫺.16
⫺.08
.17
.18
.02

.00
⫺.07
.19
⫺.04
⫺.15

⫺.15
⫺.07
.19
⫺.04
.16

⫺.46**
⫺.40**
⫺.30*
⫺.09
⫺.22

Note. The evaluation parameters (␥⫹ and ␣) are set constant across the release conditions for a given weather condition. The Pearson r correlations with
the DOSPERT risk attitude factor and self-reported drug use are in the bottom five rows. MLE ⫽ maximum-likelihood estimates; ART ⫽ Angling Risk
Tasks; KS ⫽ catch ‘n’ keep (sunny day); RS ⫽ catch ‘n’ release (sunny day); DOSPERT ⫽ Domain-Specific Risk-Taking Scale.
*
p ⬍ .05. **p ⬍ .01.

In comparison, for the cloudy conditions, none of the adjusted
ART scores shown in Table 7 were statistically significant.
To identify the cognitive processes responsible for the tasks’ reliable identification of the risky use of drugs, the MLE parameters for
each individual from the four conditions were correlated to weighted
polydrug. The correlations with each individual parameter value are
listed in the bottom rows of Tables 4 and 5. In both cloudy and sunny
tournaments the payoff sensitivity parameter, ␥⫹, was significantly
correlated to past drug use, not the probability weighting parameter, ␣.
The significant correlations with the learning process parameters, q̂1
and ␷1, are due to a few extreme parameter values and do not remain
with nonparametric associations.15
The significant correlation between weighted polydrug and ␥⫹ in
the cloudy conditions (see Table 4) is particularly noteworthy. Recall
that the adjusted ART scores were not correlated with weighted
polydrug in these cloudy conditions (see Table 6). This highlights one
merit of the models; more specifically, once the effect of different
experiences on learning was controlled for within the BSR models,
systematic differences in real-world risk takers were identified. Further discussion of the implications of this finding in terms of both the
learning and evaluation processes is left for the discussion.
To establish whether the differences in how people evaluate
payoffs (␥⫹) during the ART were responsible for the correlation
between the adjusted ART scores and risky drug use, I performed
a series of linear regressions. This was done by first forming a
Sunny Tournament factor for each participant from the two sunny
adjusted ART scores by normalizing the two scores and averaging
them together. The Sunny Tournament factor is correlated both
with the MLE ␥⫹ from the sunny tournaments (r ⫽ .70, p ⬍ .05)
and with weighted polydrug (r ⫽ .37, p ⬍ .05). Next the estimates
for ␥⫹ in the sunny tournament and the Sunny Tournament factor
were simultaneously entered in a linear regression predicting
weighted polydrug. The linear regression model with both ␥⫹ and
the Sunny Tournament factor was significant, adjusted R2 ⫽ .13,
F(2, 65) ⫽ 5.77, p ⬍ .01. But the standardized regression coefficient for ␥⫹ was not significant, ␤ ⫽ 0.17, sr2 ⫽ .01, t(65) ⫽ 1.07,
nor was the Sunny Tournament factor, ␤ ⫽ 0.26, sr2 ⫽ .03,

t(65) ⫽ 1.64. This pattern implies that the sensitivity that people
have to payoffs is responsible for the individual differences of
real-world risk takers observed in the ART.
In a final set of analyses, correlations between weighted polydrug and Weber et al.’s (2002) DOSPERT scale were computed.
Recall the DOSPERT scale identifies a person’s risk attitude
toward six different domains of risk-taking behavior. The
DOSPERT scores reveal that participants who reported greater
drug use also tended to report that they were more likely to engage
in health and social risks in the future (see Table 6 for the health
and social domains). None of the other four subscales were significantly correlated with the risky use of drugs.16 The DOSPERT
score for each of the six risky domains were also not significantly
correlated with the adjusted ART scores in the four fishing conditions (again, see Table 6 for the health and social domains).
This pattern of correlations suggests that both the DOSPERT
and the sunny ART conditions account for unique variance in the
risky use of drugs. As a test of this prediction, responses to the social
and health subscales were normalized and averaged to form a Risk
Attitudes factor. The Risk Attitudes factor is significantly correlated
with weighted polydrug (r ⫽ .34, p ⬍ .05) but not with the Sunny
Tournament factor (r ⫽ .07). Next a simultaneous linear regression
was run using both the Risk Attitudes factor and the Sunny Tournament factor to predict weighted polydrug. Together they significantly
accounted for 24% of the variance in weighted polydrug, F(2, 65) ⫽
10.03, p ⬍ .01, MSE ⫽ 86.54. Individually both the Risk Attitudes
factor, ␤ ⫽ .31, sr2 ⫽ .10, t(65) ⫽ 2.88, p ⬍ .05, and the Sunny
15
This was revealed by a nonsignificant rank-ordinal K correlation (see
footnote 11) for both of these parameters.
16
Besides assessing the likelihood of engaging in risky behavior in each
of the six domains, I asked respondents to rate the perceived benefits and
perceived risks associated with each activity in the six domains. In all six
domains the perceived benefits were significantly positively correlated
with the likelihood (r៮ ⫽ .60) and perceived risks were significantly
negatively correlated with the likelihood (r៮ ⫽ –.52).

DECISION MAKING AND LEARNING

179

Table 6
Intercorrelations Between ART Scores for Each of the Four Tournaments, Responses to the DOSPERT Health Attitude and Social
Attitude Subscales, and Self-Reported Drug Use
Variable

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1.
2.
3.
4.
5.
6.

Adjusted ART score (KS)
Adjusted ART score (KC)
Adjusted ART score (RS)
Adjusted ART score (RC)
Health Attitude score
Social Attitude score

Mean (SD)

Weighted polydrug

38.96 (18.77)
31.59 (15.35)
32.19 (15.95)
28.06 (15.22)
2.58 (0.64)
3.76 (0.60)

.36**
.14
.32**
.11
.29*
.27*

1
—
.69**
.75**
.51**
.04
⫺.09

2

3

4

5

—
.57**
.72**
.08
⫺.01

—
.64**
.21
.10

—
.12
.02

—
.28*

Note. A total of 71 participants completed the four ART tournaments and the Health Attitude and Social Attitude subscales, and 68 participants reported
their past drug use. ART ⫽ Angling Risk Tasks; DOSPERT ⫽ Domain-Specific Risk-Taking Scale; KS ⫽ catch ‘n’ keep (sunny day); KC ⫽ catch ‘n’
keep (cloudy day); RS ⫽ catch ‘n’ release (sunny day); RC ⫽ catch ‘n’ release (cloudy day).
*
p ⬍ .05. **p ⬍ .01.

Tournament factor, ␤ ⫽ .33, sr2 ⫽ .11, t(65) ⫽ 3.12, p ⬍ .05,
accounted for unique variance in weighted polydrug.
To identify whether the DOSPERT accounted for significant
incremental variance in weighted polydrug above participants’
payoff sensitivity (␥⫹) and the Sunny Tournament factor, I conducted a hierarchical linear regression. Table 8 shows that, consistent with the previous analysis, entering the Sunny Tournament
factor after ␥⫹ does not produce a significant change in R2.
However, Step 3 of the regression reveals that Risk Attitudes
accounts for unique variance in weighted polydrug over and above
the Sunny Tournament factor and the cognitive parameter ␥⫹. This
implies that the DOSPERT and sequential risk-taking tasks account for different psychological aspects in real-world risk taking.
In the discussion, I will address the implication of this result as
well as the more specific results dealing with the cognitive processes used during sequential risk taking.

Discussion
During sequential risk-taking tasks, DMs make consecutive
decisions to either stop the current round and collect their earnings
or continue to play and take a gamble. The gamble offers them the
chance to win more money, but if they lose the gamble they lose
their earned money for that round and the round ends. People differ
in the risks they take during the laboratory paradigm, and these
systematic differences can be attributed to differences in their
cognitive processes. The purpose of this article was to examine
whether the learning process contributed to the paradigm’s clinical

Table 7
Correlations Between Adjusted ART Scores Calculated From
Three Sets of Rounds for the Two Sunny Tournaments
ART condition
Catch ‘n’ keep
Sunny day
Cloudy day
Catch ‘n’ release
Sunny day
Cloudy day

Rounds 1–10

Rounds 11–20

Rounds 21–30

.37**
.13

.42**
.17

.29*
.11

.29*
.14

.24*
.06

.36**
.13

Note. ART ⫽ Angling Risk Tasks.
*
p ⬍ .05. **p ⬍ .01.

diagnosticity and whether the cognitive processes people use during the tasks depend on the stochastic environment. In the discussion that follows, I will summarize the results of the study. Then
I will discuss their implications in terms of the specific cognitive
dimensions of learning, mental representations, and decision making. Finally, I will address the broader issue of developing a
multitheory framework for studying risk-taking behavior.

Summary
The BSR model describes how people make decisions during
the gambling paradigm and reveals how they differ on the underlying cognitive dimensions of learning, evaluation, and response.
The model hypothesizes that prior to each round DMs evaluate
how many trials they should play that would maximize their
subjective expected payoff for each round. Then they probabilistically choose to play or stop on each trial as a function of the
distance from the maximizing trial. In ill-defined conditions, like
the cloudy conditions of the ART, DMs use the data they observed
on previous rounds to learn about the task and employ Bayes’ rule
to update their beliefs about the parameters of the task.
How people evaluate expected payoffs and how they learn about
the task depends on their mental representation. The results from this
article show that DMs react to changes in their environment by
changing their mental representation of the task. In contrast to their
Table 8
Hierarchical Linear Regression Analysis Testing the Incremental
Validity Accounted For in Risky Drug Use
Variable

R2 Adjusted R2 ⌬R2 df error

Step 1
Payoff sensitivity (␥⫹)a .12
Step 2
Adjusted ART scoresb .15
Step 3
Risk Attitudes factorc
.24

F

p

.10

.12

66

8.6 ⬍.01

.13

.04

65

2.7

.21

.09

64

7.7 ⬍.01

ns

Note. ART ⫽ Angling Risk Tasks.
Refers to participants’ payoff sensitivity during sunny tournaments.
b
Data refer to scores during sunny tournaments.
c
Refers to the Health Attitude and Social Attitude domains from the
Domain-Specific Risk-Taking Scale (DOSPERT). DOSPERT scores do
account for significant unique variance in Weighted Polydrug scores.
a

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PLESKAC

approach on the BART (see Bishara et al., 2007; Wallsten et al.,
2005), here DMs adopted mental representations that were congruent
with their stochastic environment. They used stationary representations during catch ‘n’ release and nonstationary representations during
catch ‘n’ keep. Finally, it was not the learning process but rather the
decision-making process, specifically how sensitive people were to
changes in payoffs, that accounted for differences between selfreported drug users and non-users. In fact, the Bayesian learning
processes were shown to interfere with this association, reducing the
paradigm’s ability to identify risky drug use.

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Learning
How people learn about choice options influences their preferences and consequently their risk-taking behavior (see Barron &
Erev, 2003; Denrell, 2007; Erev & Barron, 2005; Hertwig et al.,
2004; March, 1996; Weber et al., 2004). During the Iowa Gambling Task (Bechara et al., 1994), the learning process even dissociates between different clinical populations and the risks they
take. For example, during the Iowa Gambling Task both individuals with bilateral damage to the ventral-medial prefrontal cortex
and patients with Huntington’s disease choose the riskier option
more often than normals do, and this difference is due to how these
populations learn about the task (Busemeyer & Stout, 2002;
Yechiam et al., 2005). Stout et al. (2005) even found that when the
payoff structure of the Iowa Gambling Task was made explicit to
participants, thereby removing the learning requirement, its clinical diagnosticity disappeared.
In the sequential risk-taking paradigm, the results presented here
suggest the opposite conclusion. Learning obstructs the paradigm’s
clinical diagnosticity, and removing the learning process improves
its ability to identify real-world risk takers. Only in the welldefined sunny versions were the adjusted ART scores correlated
with self-reported past drug use. The BSR model, in turn, reveals
that the general uncertainty of DMs reduces the sequential risktaking paradigm’s reliability. Once uncertainty and different experiences are accounted for with the BSR models, there is a
significant association between the evaluation process (␥⫹) and
real-world risky behaviors.
The next logical question is, why does learning not appear to
affect BART’s validity? The BSR model can be used to answer
this question as well. During the BART, Lejuez and colleagues
(Lejuez, Aklin, Jones, et al., 2003; Lejuez, Aklin, Zvolensky, &
Pedulla, 2003; Lejuez et al., 2002) try to control the betweenparticipants variance by fixing experience between participants so
that all 30 balloons explode at the same trial number for all
participants. Besides restricting conclusions drawn from the BART
to one particular sequence of failures and successes, this can only
partially account for experience-based learning differences. DMs
still have different experiences based on their own stopping decisions in previous rounds. For example, Lejuez and colleagues
often fix the first balloon to explode on pump 65. Some DMs stop
before this point, whereas others reach the point and find that it
explodes, leading to different beliefs for the next round. This
article indicates that a better alternative—if one is only interested
in identifying risk takers—is to eliminate the learning process
entirely and use well-defined variants of the sequential gambling
paradigm like the sunny tournaments. Only when theoretically
necessary (e.g., if a clinical population is thought to have different

levels of optimism in their prior beliefs) should the learning
process be implicated within the paradigm.

Mental Representations
Although across individuals the learning process does not show
systematic differences, there are differences in the mental representations across different stochastic environments. In this experiment, DMs were remarkably adaptive in their learning process
and their more general mental representations. During the nonstationary tasks (catch ‘n’ keep), the best fitting models were those
that assumed respondents adopted a nonstationary representation;
and during stationary tasks (catch ‘n’ release), the best fitting
models assumed a stationary representation.
Certainly this congruence between representation and environment does not occur with every task. In the BART (a nonstationary
environment), the best fitting BSR model in two different datasets
assumes a stationary representation (Bishara et al., 2007; Wallsten
et al., 2005). The congruence does not even occur with every
model positing different mental representations. When models
assuming DMs sequentially evaluate their options after each play
option (as opposed to the BSR model used in this article that
assumes DMs evaluate options prior to each round) were fit to the
four conditions of this study, a stationary representation fit all four
conditions better than did the same model assuming a nonstationary representation. Importantly, the quantitative fits for the sequential models were dismal compared with those for the BSR models
presented in this article, giving more support to the prior evaluation assumption and the change in mental representation results.
For a more detailed assessment of the sequential evaluation models, see Wallsten et al. (2005) and Pleskac (2004).
The more informative fishing task, as well as the within-subjects
design, may have contributed to the ability of participants to
discriminate between the different stochastic environments. The
study’s design cannot distinguish between these two explanations.
Regardless, the experiment shows that DMs can be remarkably
adaptive to their environment and that the often-used implicit or
explicit assumption that DMs use the same decision strategy in
these or similar experience-based tasks (e.g., Barron & Erev, 2003;
Busemeyer & Stout, 2002; Wallsten et al., 2005; Yechiam et al.,
2005) may be an overly strong assumption (see Erev & Barron,
2005 for a similar conclusion). As in multiattribute decisions, the
cognitive strategies likely depend on other variables related to the
structure of the task, the cognitive ability of the DMs, and the
social context (see Gigerenzer et al., 1999; Payne et al., 1993).

Decision Making
Decision making, more specifically the evaluation process,
plays a crucial role in the sequential risk-taking paradigm. For one
thing, it can explain the relationship seen between the adjusted
ART scores in the catch ‘n’ release and catch ‘n’ keep conditions
(see left panel in Figure 2). Take for instance a risk-averse person
with a constant level of payoff sensitivity across environments
(␥⫹ ⫽ 0.8 in Equations 2 and 6) and no probability weighting
function. This risk-averse person will cast less in the catch ‘n’
release than in the catch ‘n’ keep sunny tournaments. In fact, she
will set a target of making 48 casts in the sunny catch ‘n’ keep
tournament and 39 casts in the sunny catch ‘n’ release tournament.

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DECISION MAKING AND LEARNING

The difference between the conditions converges as the person
grows more sensitive to changes in payoffs and reverses when
there is increasing sensitivity to larger payoffs (␥⫹ ⬎ 1). A similar
pattern holds in cloudy conditions if one assumes participants
adopt a similar level of relative optimism in both catch ‘n’ keep
and catch ‘n’ release conditions.17 This prediction and empirical
result speaks to the strength of formal cognitive models to bring
clarity to seemingly peculiar and counterintuitive results.
Furthermore, the evaluation process and how sensitive people were
to changes in payoffs showed systematic differences across respondents. Participants who reported lower levels of risky drug use were
also less sensitive to changes in payoff gains in all four conditions of
the tasks (for a similar conclusion see Stout, Busemeyer, Lin, Grant,
& Bonson, 2004; Wallsten et al., 2005). More importantly, the regression analyses show that payoff sensitivity is responsible for the
ability of the adjusted ART score to identify real-world risk takers. No
stable association was found between how respondents weighted
probabilities and drug use. Taking this result in combination with the
fact that the BSR model without the weighting function adequately
accounted for the data suggests that the added weighting function
component is unnecessary at this point.
The evaluation process has several implications for the sequential risk-taking paradigm itself. First, it suggests that the tedious
button pressing needed in the BART (to pump the balloon) and the
ART (to cast the rod) may be unnecessary. If participants already
have a target number of play options in mind, then an alternative
response mode would be for them to enter the number into the
computer before each round and watch the computer play for them,
an automatic type of game. Such adaptations have been developed,
and in accordance with the model prediction, there is little difference in performance between the automatic version and the standard button-pressing version (see Pleskac et al., 2007). One advantage, besides decreasing the total amount of time required to
complete the BART, is that the automatic version also makes it
possible to empirically observe intended risky behavior in all
rounds (see stars in the right panel of Figure 2 for the expected
targeted casts for all rounds).
A second implication of the evaluation process is the role that
losses play in the sequential gambling paradigm. The prior evaluation policy indicates that respondents frame the task as a gain
domain, not a mixed outcome domain (see Equation 1 for an
example). As a result, losses are not currently considered in the
already clinically diagnostic ART. Behavioral decision theory,
though, usually predicts that losses and aversion to losses play a
critical role in risk-taking behavior (see Kahneman & Tversky,
1979). Consequently, one might predict that adjustments to the
payoff structure to include losses would only improve the paradigm’s already notable ability to identify real-world risk takers.
This prediction is beyond the scope of this article, but certainly it
opens an intriguing avenue of research— one that would not have
been revealed without the cognitive model.

A Multitheory Framework for Understanding Risk Taking
Risk-taking behavior does not arise from one single psychological
process. The challenge then, for social and cognitive scientists with a
goal of understanding risky behavior, is to build a theoretical framework that draws on these different processes and reaches beyond the
traditional boundaries in psychology. The results of such a framework

181

have great potential. For example, in this article the combined approach of a sequential risk-taking paradigm, with its roots in the
clinical sciences, and the DOSPERT, developed based on theories of
risk perception, together explained 24% of the variance in the reported
risky use of drugs. Separately they accounted for 12%–14% of the
variance. Cognitive models, in turn, led to a better understanding of
how learning and decision processes co-occur in the gambling paradigm and identified the processes critical for its assessment ability: In
identifying risky drug users, evaluation processes are essential,
whereas learning reduces the paradigm’s reliability. Only by integrating these and other approaches will the social and cognitive sciences
come to not only appreciate the complexity of risky behavior at the
theoretical level, but also be able to make substantial contributions in
understanding the potential public-health problems of unsafe and
unhealthy risk taking.

17

This equal optimism assumption is much like setting the catch ‘n’ release
ponds to have 65 fish and the catch ‘n’ keep ponds to have 128 fish, so that if
a DM was an expected value maximizer and had perfect knowledge of the
ponds, she would cast approximately 64 times to maximize her earnings.

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DECISION MAKING AND LEARNING

183

Appendix A

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This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

The Likelihood Function for the Bayesian Sequential Risk-Taking (BSR) Model and Details on
Fitting the Model
Each model was fit to each individual’s data from each tournament using maximum likelihood methods. Let the vector Yw,l ⫽
(cw,l,1, dw,l,1, 䡠䡠䡠, cw,l,h, dw,l,h, 䡠䡠䡠, cw,l,30, dw,l,30) represent the observed data from tournament w,l, where w ⫽ s or c for sunny or
cloudy conditions, respectively; l ⫽ k or r for keep or release,
respectively; cw,l,h is the number of casts on Round h; and dw,l,h is
whether the DM stopped (1) or not (0). The likelihood of the
observed data, Yw,l, for each of the models is defined as
L(cw,l,1, dw,l,1, . . ., cw,l,h, dw,l,h, . . ., cw,l,30, dw,l,30)

写写

30 c w,l,h

⫽

r̂h,i(1⫺r̂ h,c w,l,h⫹1 ) d w,l,h,

(A1)

h⫽1 i⫽1

where each model predicts rh,i. Taking the natural logarithm gives
the log-likelihood of the data.
None of the BSR models have a closed form solution to find the
maximum likelihood estimates (MLE) of the parameters. Consequently, the solutions were estimated with numerical optimization
techniques, of which there are many. For computational purposes,
I constrained the valuation parameter to 0 ⱕ ␥–, ␥⫹ ⱕ 3 and the
mean of the ␥ distribution to 0 ⬍ ␮o ⱕ 1,000.
Both past experience and simulations have shown that a
Nelder–Mead downhill simplex routine (available in Mathwork’s Matlab) combined with a grid-search technique is the

most successful at both reaching a solution and guarding against
local maxima. The Nelder–Mead downhill simplex routine (see
Nelder & Mead, 1965) in conjunction with a grid-search technique uses a two-step approach to arrive at a solution. During
the first step, I divided the parameter space into three plausible
sectors. For example, the plausible space for ␥⫹ was set between 0 and 3, but the divisions were weighted toward the lower
spectrum of the space: (0, 0.5), (0.5, 1.5), and (1.5, 3). A
starting value for each parameter was then randomly selected
from one of its divisions. The set of starting values was then
tested to ensure the starting values would lead to a solution
below a prespecified criterion (e.g., log-likelihood ⬎ –2,000).
If not, then the set was iteratively perturbed with random noise,
and tested, until the criterion was met or a cutoff was reached.
Parameter sets that did meet the criterion were input into the
Nelder–Mead method, beginning the second step. The full
two-step process was then repeated for 50 to 100 iterations. The
maximum log-likelihood from the full set was taken as the
estimate.
Examples of other procedures used include nonlinear programming, genetic algorithm, or an iterative annealing Nelder–Mead
method (see Van Zandt, 2000; Van Zandt, Colonius, & Proctor,
2000). However, simulations and past experience showed that the
aforementioned procedure performed the best.

Appendix B
Derivation of the Predicted Adjusted Scores
To derive the predicted adjusted scores from the Bayesian
sequential risk-taking (BSR) model, we must find the probability
of observing score y given the round ended in a success, P(y | s).
First, note that the BSR model predicts the target number of trials
for which DMs should select the play option for each round (see
for example Equations 2 and 6). The intended casting behavior for
each round, called score y, is a function of the response rule in
Equation 3. This score is empirically unobservable, and the adjusted score is an estimate of it. Using the model, the probability
of score y for all rounds is the probability of choosing the play
option up to and stopping on Trial y,

写

E(y) ⫽ 兺yPh(y).

The adjusted score is an estimate of the expected score given the
round ended in a success, Eh(y | S).B1 Bayes’ rule identifies the
probability of observing score y given a success, S, on Round h,
Ph(y兩S) ⫽

ri⬘,h关1 ⫺ ry,h兴,

P(S兩y) ⫽

(B1)

i⬘⫽1

where ri,h is the probability of choosing the play option and is
found with Equation 3. In catch ‘n’ release conditions the score y
can take the values [0, ⬁]. In catch ‘n’ keep conditions y can only
take values [0, n]. The expected score, or intended casting behavior, for each round is then

P(S兩y)Ph(y)
.
兺P(S兩y)Ph(y)

(B3)

The values of P(S | y) depend on the stochastic environment. In the
catch ‘n’ keep conditions, the a priori probability of the failure
event, F, on any trial is 1/n; the probability of no failure (i.e., of a
success S, given y sequential successful trials) is

y⫺1

Ph(y) ⫽

(B2)

n⫺y
.
n

(B4)

In catch ‘n’ release, the a priori and the conditional probability of
any one trial leading to a failure is probability p. The probability,
therefore, that y observations will all be successful is
B1

My thanks to Thomas Wallsten for showing me the analytic solution
for distribution of scores given a success.

(Appendixes continue)

PLESKAC

184

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This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

P(S兩y) ⫽ (1 ⫺ p)y.

(B5)

Calculating Eh(y | S) with Equation B5 is straightforward. In
cloudy tournaments, both n and p are replaced with estimated
values from the Bayesian learning models (see Equations 5 and 6).
The bar graph in the right panel of Figure 2 plots the expected
adjusted ART scores averaged across all rounds and across all
participants using the parameter estimates from the models estimated without a weighting function. The models with the weighting function produce a similar graph. The stars in the panel are the
predicted scores from Equation B2. Pleskac et al. (2007) proved
that in sequential risk-taking tasks, the adjusted scores will always
be less than the predicted unconditional latent scores, Eh(y | S) ⬍
Eh(y). This is because the more trials the DM chooses the risky
play option on a given round, the more likely it is to end in a failure
(i.e., catching a blue fish). Therefore, the adjusted score tends to
filter out the longer response sequences.
This tendency of the longer response sequences to result in a
failure and be excluded from the adjusted score suggests that an
error theory might be sufficient to explain both the observed risk
aversion with the adjusted scores and the systematic individual
differences. Indeed, for bias-free DMs maximizing expected value
(␥⫹ ⫽ 1 in Equations 2 and 6), the predicted adjusted ART scores
will fall as response consistency (␤) decreases (or error increases;
see Equation 3). To show this prediction, Table B1 lists the
predicted adjusted ART scores for three bias-free expected value

Table B1
Expected Value Maximizers for Three Levels of Response
Variability (␤) Playing Catch ‘n’ Keep on Sunny Days
␤

Predicted adjusted
ART score

Predicted SD of casts
taken on stop rounds

0.3
0.5
0.1

58.0
52.7
34.8

4.7
6.9
11.9

Note.

ART ⫽ Angling Risk Tasks.

maximizers (␥⫹ ⫽ 1) in the catch ‘n’ keep, sunny condition. The
calculations were done using Equations 3, 6, B1, and B3. As ␤
decreases, the predicted adjusted ART score also falls. However,
the predicted standard deviation of the number of casts made on
the stop rounds increases. Thus, the models predict that if an error
theory is sufficient to explain the data, then the adjusted ART
scores and the standard deviation of casts taken on stop trials
should be negatively correlated. This is not the case for the dataset
presented here. There was a significant positive correlation between the adjusted ART scores and standard deviation of casts
taken on rounds ending in a stop for all four conditions, r(keep,sunny)
⫽ .44; r(keep,cloudy) ⫽ .57; r(release, sunny) ⫽ .74; r(release,cloudy) ⫽
.75, thus indicating that response variability is not sufficient and
other cognitive processes are necessary to explain the data.

Appendix C
Simultaneous Estimation of the Value and Weighting Function for the Bayesian Sequential
Risk-Taking Model
The data set from the entire experimental session is needed to
simultaneously estimate both the value and weighting function,
because the weighting and the value function for each individual can be estimated for each tournament only if their respective
parameters are constrained across different stochastic representations (i.e., across catch ‘n’ keep and catch ‘n’ release; for a
proof see Pleskac, 2004). The most reasonable constraint is
setting ␥⫹ and ␣ equal across the nonstationary and stationary
models for the two release laws within a weather condition.
None of the other parameters are constrained across tournaments. This constraint is summarized in Table C1, under Model
Framework 2. Model Framework 1 is the set of models without
the weighting function that has already been estimated for the
four tournaments. Each framework has 16 total free parameters
that account for all 120 rounds in the entire experimental
session, with an average of 3,360 choice trials.
The frameworks are not nested and have an equal number of
parameters, so comparisons of the maximum log-likelihood
(ML) of the data for the experimental session given each
framework can only serve descriptive purposes. The results in
Table C2 show that Model Framework 1 provided a better fit
than did Model Framework 2 for a majority of the participants
when the value function allows both diminishing sensitivity to
outcomes and increasing sensitivity to outcomes (␥⫹ ⬎ 0).

Table C1
Constraints Imposed on the BSR Model to Estimate the
Weighting Function
Model Framework 1

Model Framework 2

Variable

Cloudy

Sunny

Cloudy

Sunny

Learning
Evaluation
Value function
Weighting function
Response
Bias
Payoff sensitivity (␥⫹)
Total parameters

Yes (4)

No

Yes (4)

No

Yes (2)
No

Yes (2)
No

Yes (1)
Yes (1)

Yes (1)
Yes (1)

Yes (2)
Yes (2)

Yes (2)
Yes (2)

Yes (2)
Yes (2)

Yes (2)
Yes (2)

16

16

Note. Model Framework 1 is the set of models without the weighting
function. Model Framework 2 is the set constrained to estimate the weighting function. On cloudy and sunny days, both catch ‘n’ keep and catch ‘n’
release tournaments were held. For example, Model Framework 1 has a
value function and has 2 ␥⫹ parameters in the cloudy conditions and 2 in
the sunny conditions. BSR ⫽ Bayesian sequential risk-taking model.

An alternative constraint is to allow only diminishing sensitivity
(0 ⬍ ␥⫹ ⱕ 1). With this constraint, Model Framework 1 was a better
fit for only a few more participants than Framework 2 (see Table C2).

DECISION MAKING AND LEARNING

Table C2
Comparison of Models With Different Constraints to Estimate a
Weighting Function, With Varying Constraints on ␥⫹

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This article is intended solely for the personal use of the individual user and is not to be disseminated broadly.

Increasing and
decreasing sensitivity
to outcomes (0 ⬍ ␥⫹)

Decreasing sensitivity
to outcomes
(0 ⬍ ␥⫹ ⱕ1)

Model

Mean ML

Number of
DMs best
fit

Mean ML

Number of
DMs best
fit

Baseline
Model Framework 1
(without
weighting
function)
Model Framework 2
(with weighting
function)

⫺362.01

0

⫺362.01

0

⫺270.32

69

⫺343.98

38

⫺280.76

2

⫺282.27

33

185

However, the mean ML for Model Framework 2 is greater than the mean
ML for Framework 1. This indicates that for some individuals, Model
Framework 1’s fit is quite poor as compared with Framework 2. A better
sense of this for each participant was ascertained by calculating the
absolute difference between ML for Model Framework 1 and Model
Framework 2. For the 38 individuals for whom Model Framework 1 (no
weighting) was a better fit, the average absolute difference was 6.01 (in
log-likelihood space). But for the 33 individuals for whom Model Framework 2 (weighting) was a better fit, the average deviation was 200.65. In
other words, when Model Framework 2 (with the weighting function)
was a poor fit, it was much less a poor fit than was Model Framework 1
(without the weighting function). These results indicate that, based on fit
alone, the weighting function can contribute to the explanatory power of
the model, but only when ␥⫹ is constrained to show only diminishing
sensitivity.

Note. ML ⫽ maximum log likelihood; DM ⫽ decision maker.

Received July 10, 2006
Revision received June 25, 2007
Accepted September 29, 2007 䡲

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