The importance of learning when making inferences

Judgment and Decision Making, Vol. 3, No. 3, March 2008, pp. 261–277
The importance of learning when making inferences
Jörg Rieskamp ?
Max Planck Institute for Human Development
Abstract
The assumption that people possess a repertoire of strategies to solve the inference problems they face has been made
repeatedly. The experimental ?ndings of two previous studies on strategy selection are reexamined from a learning
perspective, which argues that people learn to select strategies for making probabilistic inferences. This learning process
is modeled with the strategy selection learning (SSL) theory, which assumes that people develop subjective expectancies
for the strategies they have. They select strategies proportional to their expectancies, which are updated on the basis
of experience. For the study by Newell, Weston, and Shanks (2003) it can be shown that people did not anticipate the
success of a strategy from the beginning of the experiment. Instead, the behavior observed at the end of the experiment
was the result of a learning process that can be described by the SSL theory. For the second study, by Bröder and
Schiffer (2006), the SSL theory is able to provide an explanation for why participants only slowly adapted to new
environments in a dynamic inference situation. The reanalysis of the previous studies illustrates the importance of
learning for probabilistic inferences.
Keywords: inferences, strategy selection, heuristics, learning theory, reinforcement learning, cognitive modeling.
1 Introduction
perience and select a strategy based on past success. I
will describe a computational theory that speci?es this
How do people make probabilistic inferences, such as in-
learning process (Rieskamp & Otto, 2006) and will use it
ferring the selling price of a car, the severity of an ill-
to explain people’s probabilistic inferences in two previ-
ness, or the likely winner of a tennis match? Differ-
ous studies. The goal of this article is to demonstrate that
ent strategies can be applied to make these inferences,
probabilistic inferences are strongly in?uenced by learn-
such as integrating all available information to predict
ing and that this learning process can be conceptualized
the criterion. In fact, many researchers have argued that
as learning to select strategies.
people are equipped with a repertoire of different cogni-
tive strategies for making judgments and decisions (e.g.,
1.1 Cost-bene?t approach to strategy selec-
Brown, 1995; Brown, Cui, & Gordon, 2002; Einhorn,
1970; Fishburn, 1980; Gigerenzer, Todd, & the ABC Re-
tion
search Group, 1999; Ginossar & Trope, 1987; Payne,
The contingency model of Beach and Mitchell (1978)
1976; Payne, Bettman, & Johnson, 1988, 1993; Rapoport
conceptualizes the selection of strategies as a cost-bene?t
& Wallsten, 1972; Rieskamp & Hoffrage, 1999, 2008;
analysis. Each strategy is assumed to lead with a spe-
Svenson, 1979).
ci?c probability to a correct solution with a bene?cial
Do people apply different strategies for solving proba-
outcome, and with the remaining probability to an in-
bilistic inferences? And if so, how do they select from
correct solution with a less bene?cial or detrimental out-
their strategy repertoire? The cost-bene?t approach to
come. Each strategy also involves some costs and it is
strategy selection argues that people trade off the strate-
assumed that the probability of a correct solution is posi-
gies’ anticipated costs and bene?ts. In contrast, I will
tively correlated with those costs. Subtracting a strategy’s
argue that selection is achieved via learning—that is, peo-
costs from its bene?ts results in the net bene?t, and the
ple learn the success and failure of strategies through ex-
strategy with the maximum net bene?t is selected by the
decision maker.
?I would like to thank Ben Newell and Arndt Bröder for provid-
ing me with the opportunity to analyze the data of their experiments
Many authors have argued for such a cost-bene?t
and for their helpful comments concerning the design and analysis of
approach to describe how strategies are selected (see
their studies. I also thank Anita Todd for editing a draft of this ar-
Christensen-Szalanski, 1978; Payne et al., 1988, 1993;
ticle. Correspondence concerning this article should be addressed to
Smith & Walker, 1993). Yet despite being conceptually
Jörg Rieskamp. Address: Jörg Rieskamp, Max Planck Institute for
Human Development, Lentzeallee 94, 14195 Berlin, Germany. Email:
straightforward, the approach has not been spelled out as
rieskamp@mpib-berlin.mpg.de.
a computational model that de?nes how the trade-off pro-
261

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
262
cess is cognitively determined. The ?rst obvious barrier
trade-off process assumed by the cost-bene?t approach
to a computational description consists in the fact that
could in?uence the initial preferences for speci?c strate-
several bene?ts (e.g., monetary gains, accuracy) and costs
gies assumed by the SSL theory. However, the SSL the-
(e.g., cognitive effort, time) can be distinguished, yet they
ory argues that individuals’ initial preferences for spe-
have to be mapped onto one single scale. Second, it is
ci?c strategies are wiped out by the strategies’ success
not clear which strategy is applied to evaluate the costs
or failure when repeatedly applied, so that over time the
and bene?ts, nor how this strategy is selected, potentially
selected strategy for a decision problem is a function of
leading to an in?nite regress problem of strategy selec-
the strategies’ success. Proponents of the cost-bene?t
tion.
approach have also suggested that the selection process
could be triggered by learning (Payne et al., 1993).
The learning approach predicts that the probability
1.2 Learning approach to strategy selection
with which a speci?c strategy is selected for a decision
Rieskamp and Otto (2006, see also Rieskamp, 2006a)
problem will change over time when outcome feedback
took an alternative, bottom-up approach to strategy selec-
is provided. In particular, when an individual initially
tion that is based on learning. They proposed the strategy
selects an unsuccessful strategy, the individual should
selection learning (SSL) theory, which assumes that peo-
switch to another more successful strategy during the
ple most likely select the strategy that they expect to be
course of learning. I will call this the learning prediction,
most successful in solving an inference problem. How-
and it implies that in a repeated decision-making situa-
ever, instead of assuming that people deliberately trade
tion, the strategy that most successfully predicts the ma-
off strategies’ costs and bene?ts, the theory states that the
jority of a person’s decisions might not be the only strat-
strategies’ expectancies are the result of a learning pro-
egy this person has applied during the course of learning.
cess. When facing a decision situation for the ?rst time,
Instead, it may be that only after some time did a prefer-
people have initial expectancies for each strategy they
ence for a speci?c strategy develop. Therefore according
might use, based on past experience with similar deci-
to the learning approach, which strategy will predict the
sion situations. These expectancies are updated depend-
majority of inferences strongly depends on the speed of
ing on whether the selected strategy succeeds or fails to
learning and the provided learning opportunities. When
solve the decision problem. On the basis of this learn-
people change the strategies they select on the basis of
ing process the theory predicts that after suf?cient expe-
learning, then these learning processes need to be taken
rience an individual is most likely to select the strategy
into account in any accurate description of the cognitive
that performs best in a speci?c environment. Consistent
processes underlying inferences.
with this basic assumption of the SSL theory, previous
research has provided substantial empirical evidence that
1.3 The SSL theory
depending on the statistical properties of environments,
different cognitive models are best at predicting people’s
In the following, the SSL theory is described in more de-
behavior, in particular when substantial outcome feed-
tail. Individuals have a set S of N cognitive strategies. An
back is provided (e.g., Bröder, 2003; Garcia-Retamero,
individual’s preference for a particular cognitive strategy
& Rieskamp, in press; von Helversen & Rieskamp, 2008;
i is expressed by positive expectancies qt(i), so that the
Rieskamp, 2006a). It should be emphasized, however,
probability of selecting strategy i at trial t is de?ned by
that the cognitive strategies people are “selecting” are un-
qt(i)
observable and can only be inferred from the observed
pt(i) =
(1)
N
behavior, such as information search or the ?nal choices.
q
j=1 t(j)
Thus, when a strategy is capable of predicting a person’s
with j as an index for the cognitive strategies. The strate-
information search and ?nal choices, this can be inter-
gies’ expectancies in the ?rst period of the task can differ
preted as indicating the person had selected the strategy.
and are de?ned by
However, this is always just an interpretation and alterna-
tively cognitive models that do not assume any cognitive
q1(i) = r
· w · ?i
(2)
strategies could provide a better account of the inference
correct
process. Hence, when herein participants’ cognitive pro-
where rcorrect is the payoff received for a correct decision,
cesses are described, for simplicity, as selecting speci?c
w is the initial association parameter, and ? is the initial
strategies, this is always just an interpretation of a strat-
preference parameter. The payoff rcorrect received for a
egy’s good ?t in describing the observed behavior.
correct decision in a particular task is a scaling constant
Although the SSL theory follows a bottom-up ap-
that allows comparisons across tasks with different pay-
proach to strategy selection, it is not necessarily to be
offs. The initial association parameter w is restricted to
seen in opposition to the cost-bene?t approach. The
w > 0 and expresses an individual’s initial association

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
263
with the available strategies relative to later reinforce-
2 Study 1: Learning in Stable Envi-
ment, and thus it essentially describes the learning rate.
ronments
The theory assumes that individuals have initial prefer-
ences for selecting particular strategies at the beginning
The ?rst study that I have reanalyzed was conducted by
of a task. The initial preference parameter ?i for each
Newell, Weston, and Shanks (2003). The authors stud-
strategy i is restricted to 0 < ? < 1 and
N
?
i=1
i = 1.
ied the problem of inferring which of two objects had a
After a decision is made, the strategies’ expectancies are
higher criterion value on the basis of several cues. For
updated by
this inference problem Gigerenzer and Goldstein (1996)
q
proposed a simple lexicographic heuristic called Take
t(i) = qt?1(i) + It?1(i)rt?1(i)
(3)
The Best (TTB), which only considers the most valid
where It?1(i) is an indicator function and rt?1(i) is the
cue for making an inference; if this cue does not dis-
reinforcement. The reinforcement of a strategy is de?ned
criminate the second most valid cue is considered, and
as the payoff rt?1(i) that the strategy produced. The indi-
so on. Gigerenzer and Goldstein illustrated through com-
cator function It?1(i) equals 1 if strategy i was selected
puter simulation that the heuristic performs as well as
and equals 0 if the strategy was not selected. For the fol-
and sometimes even better than more complex alterna-
lowing two studies it is assumed that a strategy was se-
tive strategies, among them a weighted additive strategy
lected if the choice coincides with the strategy’s predic-
(WADD). The WADD strategy computes a score for each
tion. When two or more strategies make the same pre-
alternative by taking the sum of the cue values multiplied
diction that coincides with the individual’s choice, it is
by the cues’ validities and ?nally selects the alternative
assumed that It?1(i) equals the probability with which
with the largest sum.
the model predicts the selection of these strategies. By
Newell et al. argued that despite the psychological
de?nition, if qt(i) falls below a minimum value ? due to
plausibility of simple heuristics it is necessary to demon-
negative payoffs, qt(i) is set to ?; for the following stud-
strate “that people do indeed use these heuristics in
ies ? = 0.0001 was used.
the environments in which they are claimed to operate”
Finally, the SSL theory assumes that people make er-
(2003, p. 83). For the empirical test of the heuristic they
rors when applying a strategy, so that, by mistake, they
conducted two experiments, of which I have reconsidered
deviate from the strategy’s prediction. Let p(a|i) denote
the ?rst one. In this experiment, the information search
the probability of choosing alternative a out of the set of
preceding participants’ inferences led to high monetary
alternatives when strategy i is selected, so that the prob-
costs. Due to these high costs it should have been obvi-
ability of choosing alternative a given strategy i and an
ous from the beginning of the experiment that a strategy
application error ? is
that requires a lot of information would perform badly.
?
Therefore when following the cost-bene?t approach to
pt(a|i, ?) = (1 ? ?) · pt(a|i) +
· p
strategy selection, participants should have selected an
k ? 1
t(¯
a|i)
(4)
information-frugal strategy right from the beginning of
where pt(¯a|i, ?) denotes the probability of choosing any
the experiment. In contrast, a learning approach predicts
other alternative than a from the available k alternatives,
that a well-performing strategy that requires little infor-
given strategy i was selected. For simplicity, the applica-
mation will be selected more frequently after gaining ex-
tion error is assumed to be the same across strategies (for
perience with the inference problem.
psychologically more plausible error concepts see Mata,
Schooler, & Rieskamp, 2007; Rieskamp, Busemeyer, &
2.1 Procedure
Mellers, 2006). The probability of choosing alternative
a depends on the probabilities of selecting the strategies
Newell et al.’s (2003) participants repeatedly had to in-
and the corresponding choice probabilities of the strate-
fer which stock shares of a company would perform most
gies, so that
pro?tably in the future. Each share was described by six
dichotomous cues, with validities of the cues ranging be-
N
tween .65 and .90. With six dichotomous cue values for
pt(a) =
pt(i) · pt(a|i, ?)
(5)
six cues, 64 different cue pro?les result, leading to 2,016
i=1
possible pair comparisons, of which 180 were selected
The SSL theory is similar to other recent learning mod-
randomly for each participant. The information of the
els that assume a learning process of strategy selection
cues could be acquired with a cost of 1 pence (U.K.) for
(see Busemeyer & Myung, 1992; Erev & Barron, 2005;
each cue. After the participants made an inference they
Siegler & Shipley, 1995; Stahl, 1996). These “cousins”
received feedback on whether their decision was correct
differ from the SSL theory in the exact learning mecha-
and earned 7 pence for each correct decision. The par-
nisms they assume.
ticipants were not informed about the cues’ different va-

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
264
lidities. However, to facilitate learning the objective cue
perfectly, whereas for the remaining 50% TTB could still
validities, after 60 and 120 trials the rank order of the
predict on average 80% of the choices. In sum, Newell
cues according to the validities was provided to the par-
et al. concluded that 11 of the 24 participants most likely
ticipants.
selected TTB, whereas 9 participants used a “weight of
The imposed information search cost of 1 pence for
evidence” strategy, such as WADD.
each cue was very high relative to the gain of 7 pence
for a correct inference. A random choice strategy with
2.3 Reanalysis: Estimating the SSL the-
no search costs would have led to an expected gain of
ory’s parameters
3.5 pence. Thus, by acquiring all available information
it was in principle not possible to outperform the ran-
Newell et al. (2003) reported that the participants im-
dom choice strategy, even when the acquired informa-
proved their inferences over time. Because they found
tion enabled 100% accuracy leading to a payoff of only
substantial learning effects, they focused their analysis
1 pence. This clearly illustrates that if participants antic-
on the behavior at the end of the experiment. Could the
ipated the strategies’ costs and bene?ts from the begin-
strategies the participants selected at the end of the exper-
ning of the experiment as predicted by the cost-bene?t ap-
iment be the result of a learning process, and could this
proach, they should have selected an information-frugal
learning process be described by the SSL theory? To an-
strategy right from the beginning.
swer this question I reanalyzed participants’ behavior for
the whole experiment. The parameters of the SSL theory
2.2 Results
were estimated separately for each individual’s learning
data as follows: The model predicted the probability with
Newell et al. (2003) stated that participants’ inferences
which a participant would choose each of the available
were in?uenced by a learning process, so that they only
alternatives for each trial conditioned on past choices and
examined the last third of the experiment to infer the
feedback. As a goodness-of-?t criterion, the G 2 mea-
strategies participants selected. To classify participants’
surement was used (Burnham & Anderson, 1998), de-
inference strategies, Newell et al. looked at participants’
?ned in Equation 6, for which the likelihood function
search behavior and their ?nal choices. The authors re-
f (y|?, t ? 1) denotes the probability of choice y in trial
ported that at the end of the experiment the vast majority
t given the model’s parameter set ? and all information
of participants searched for the cues in the order of their
from the preceding trial t ? 1:
validities, in particular acquiring the most valid cue ?rst.
t
This search behavior is consistent with TTB, but even for
G 2 = ?2 ·
ln(f (y|?, t ? 1))
(6)
a strategy such as WADD, search according to the cues’
t=1
validities is plausible. Due to the high search costs, it
is reasonable to assume that participants with a prefer-
The parameter values that minimized G 2 were searched
ence for WADD would try to restrict their search. This
for; the initial association parameter was restricted to 1
can be achieved by comparing the alternatives cue-wise,
?w?100.
determining the difference of the cue values weighted by
Three parameters were estimated, resulting in an aver-
the cues’ validities, and updating the difference with each
age estimated association parameter value of 18. The as-
new cue considered. Search stops whenever the present
sociation parameter determines the strategies’ initial ex-
difference cannot be reversed by the outstanding cues not
pectancies, and the larger the value the more time it will
considered yet. With this implementation of WADD, cues
take to develop a preference for another strategy. A value
would also be searched according to their validity, so that
of 18 implies that the payoff for a correct decision of 7
search behavior does not allow unambiguous identi?ca-
pence is multiplied by 18, and this “stock of association”
tion of selected strategies.
is then divided between the two strategies the decision
For this reason it is more revealing to look at when
maker considers (cf. Equation 2). Thus, if the decision
participants stopped their information search (see also
maker showed a strong preference for one strategy by,
Dieckmann & Rieskamp, 2007). One third of the partic-
for instance, dividing up the stock with a ratio of 15 to 3,
ipants stopped search in a manner consistent with TTB,
then this decision maker would need to apply the less-
that is, they always stopped search after they found a dis-
preferred strategy successfully at least 12 times to de-
criminating cue. All other participants stopped their in-
velop an equal preference for both strategies.
formation search in 70% of all inferences after they found
In fact, the initial preferences were not that different.
the ?rst discriminating cue. In addition to looking at par-
The average estimated initial preference parameter was
ticipants’ information search, Newell et al. (2003) also
?TTB = .41 for TTB, implying an initial preference of
considered whether TTB could predict people’s choices.
?WADD = 1 - ?TTB = .59 for the WADD strategy. Thus,
For 50% of the participants, TTB predicted their choices
on average, the decision makers had a slight preference

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
265
0.9
0.8
0.7
TTB
0.6
WADD
0.5
p(TTB I SSL Theory)
p(WADD I SSL Theory)
0.4
0.3
0.2
0.1
Proportion / Probability
0.0
0
1
2
3
4
5
6
Trial blocks
Figure 1: The proportion of choices predicted by Take The Best (TTB) and a weighted additive strategy (WADD)
in the ?rst experimental study by Newell et al. (2003). Also shown is the strategy selection learning (SSL) theory’s
predicted probability of selecting the two strategies. Only discriminating trials where the strategies make different
predictions are shown, and each trial block consists of 30 trials.
to integrate the available information at the beginning of
validities. Yet for only four participants was a relatively
the experiment. This result is surprising if a cost-bene?t
high application error larger than .30 estimated. These
approach to strategy selection is being followed: If partic-
participants apparently made their inferences more or less
ipants were considering the high search costs right from
randomly. Newell et al. also concluded from participants’
the beginning of the experiment they should have pre-
search behavior that four participants made their infer-
ferred a strategy of focusing on single cues from the
ences randomly, and consistently for the SSL theory high
beginning. However, there were large individual differ-
application errors were estimated for three of these same
ences: Although for the majority of participants (n = 14)
four participants.
an initial preference for WADD was found (i.e., ?WADD
How well did the SSL theory describe participants’ in-
> .50), for a substantial number of participants (n = 10)
ferences? The SSL theory was able to predict partici-
an initial preference for TTB was indeed estimated (i.e.,
pants’ choices with an average probability of .71 (with
?TTB > .50).
SD = 0.05), which is slightly lower than, for instance, that
The third parameter estimated was the application er-
found in the similar studies by Rieskamp and Otto (2006),
ror, for which an average value of .18 resulted, which is
where the choices were predicted with average probabil-
relatively high in comparison to average application er-
ities of .74, .75, and .79. More interesting is whether
rors ranging between .05 and .07 estimated across several
the SSL theory can also describe the adaptive selection
experiments by Rieskamp and Otto (2006). A high ap-
of cognitive strategies. The percentage of choices pre-
plication error implies, according to the SSL theory, that
dicted by TTB and WADD, respectively, can be taken as
the participants often deviated from their strategies. For
an approximation of participants’ strategy selections and
the experiment of Newell et al. (2003) these deviations
can be compared to the probability with which the SSL
can be easily explained by the cue validities that are re-
theory predicts this selection per trial block.
quired by the strategies. To determine the SSL theory’s
Figure 1 shows the percentage of inferences that are
prediction, I used the objective cue validities. However,
predicted by TTB and by WADD, restricted to those trials
this is, of course, only an approximation, because the par-
where the strategies make different predictions. In addi-
ticipants had to learn the validities during the experiment
tion, the ?gure shows the SSL theory’s predicted proba-
and only hints about the rank order of the validities were
bilities of selecting TTB and WADD. Newell et al. (2003)
provided. Thus, when the subjective cue validities differ
provided feedback starting with the very ?rst inference,
from the objective cue validities this can easily explain
so that participants’ initial strategy preferences could not
why a participant selecting TTB would frequently devi-
be shown. The ?ts of the strategies in the ?rst trial block
ate from the prediction of TTB using the objective cue
are already affected by learning. Therefore, the ?gure

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Learning and inferences
266
additionally shows the initial probability with which the
3 Study 2: Learning in dynamic en-
SSL theory predicts the selection of the strategy at the be-
vironments
ginning of the experiment before any inference has been
made (i.e., trial block 0).
The experimental results of Newell et al. (2003) illustrate
Overall, the SSL theory’s predicted probabilities of se-
that people do quite well in adapting their inference pro-
lecting the two strategies nicely matched the proportion
cesses to the inference situation. Although some partici-
of predicted inferences by the two strategies. Only at
pants did search for too much information, incurring high
the beginning of the experiment does the SSL theory de-
costs for search, the majority made their inferences in ac-
scribe a faster learning process than is actually observed;
cordance with simple inference processes that were most
that is, the probability of selecting TTB is larger than
suitable for the high-search-costs situation. The reanaly-
one would infer from the percentage of inferences pre-
sis shows that the adaptive behavior Newell et al. reported
dicted by TTB. This is presumably because the SSL the-
for the end of the experiment is the result of a learning
ory makes use of the objective cue validities that had to
process that can be described by the SSL theory.
be learned. The process of learning the validities appar-
Do people also accurately adapt their inference pro-
ently slowed down the process of learning to select an
cess in a dynamic inference situation in which the best-
adaptive strategy. However, after the second trial block
performing strategy changes over time? This question
participants were familiar with the objective rank order
was addressed in a study by Bröder and Schiffer (2006).
of the cues’ validities and the strongest learning process
In the ?rst half of their experiment participants encoun-
was observed, for which the SSL theory provides a good
tered either an environment in which it was best to focus
account.
on single pieces of information (i.e., TTB led to a better
performance than WADD) or an environment in which
2.4 Discussion
WADD outperformed TTB. Thus, the central question
was whether the participants would select the most adap-
The reanalysis of the data of Newell et al. (2003) illus-
tive strategy according to the environment. Moreover, for
trates two points: First, the cost-bene?t approach to strat-
half of the participants the environment changed after the
egy selection does not easily explain the experimental
?rst half of the experiment, that is, they were confronted
?ndings when assuming that people anticipate strategies’
with a second, not previously encountered environment.
costs and bene?ts when encountering an inference prob-
Would the participants also be able to adapt to the new
lem. Second, the selection of strategies is strongly in?u-
environment?
enced by a learning process.
Due to high search costs, the cost-bene?t approach pre-
3.1 Procedure
dicts that participants will favor strategies that rely on lit-
tle information. Contrary to this prediction, at the be-
In the ?rst of Bröder and Schiffer’s (2006) experiments
ginning of the experiment the WADD strategy, which re-
the participants had to infer which of three companies’
quires a lot of information, predicted participants’ infer-
shares of stock would perform most pro?tably in the
ences best. Thus, apparently people did not select a well-
future. The experiment had two phases with 80 infer-
performing strategy from the beginning. Instead, the se-
ences each. The participants encountered either a com-
lection of strategies appears to have been strongly in?u-
pensatory or a noncompensatory environment in the ?rst
enced by a learning process, so that only at the end of
phase, in which WADD or TTB performed best, respec-
the experiment was the most appropriate strategy for the
tively. Thereafter, for half of the participants the environ-
problem (i.e., TTB), on average, the best strategy for pre-
ment was changed so that the environment they had not
dicting participants’ inferences.
previously encountered was used for the second phase
The SSL theory was suitable to describe the observed
(and half of the participants for whom the environment
learning process. However, because no subjective validi-
changed got a hint that the environment had changed).
ties were elicited in the study by Newell et al. (2003), the
The payoff that each share produced was a function of
predictions of the SSL theory were based on the objective
the four cues and the amount of information the partici-
cue validities. To the extent that these objective validities
pants acquired (each acquired cue value led to a reduction
do not correspond to the subjective ones, the SSL theory
of the ?nal gain of 4%). The three shares led to a speci?c
will have trouble predicting the inferences. Overall, when
payoff and in principle all shares could lead to a similar
analyzing the data of the whole experiment, it becomes
positive payoff. Thus, whereas in the study by Newell
clear that the behavior observed at the end of the exper-
et al. (2003) there was one single correct choice with a
iment — on which Newell et al. based their conclusions
gain, in the study by Bröder and Schiffer the options led
— was strongly in?uenced by a learning process that can
to diverse outcomes and these were presented to the par-
be accurately described with the SSL theory.
ticipants after each choice. This experimental aspect is

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
267
important, because it de?nes the task’s incentive structure
For the noncompensatory environment the majority of
and the feedback that induces a learning process. Ac-
participants (79%) were classi?ed as using the noncom-
cording to the SSL theory, a strategy’s reinforcement is
pensatory TTB heuristic, whereas in the compensatory
de?ned by the monetary gains it produces. Therefore, if
environment 58% of the participants were classi?ed as
TTB and WADD lead to different choices, but the payoffs
using the compensatory strategy WADD (ignoring three
attached to these choices do not differ substantially, then
unclassi?ed participants across all conditions). Thus, the
this would not induce a strong learning effect in compar-
participants reacted sensitively and adaptively to the two
ison to a learning situation in which only one choice is
environments.
rewarded.
What happened in the second phase of the experiment?
As in the study of Newell et al. (2003), the participants
For those participants who continued with the same envi-
had to learn the importance of the cues and no cue va-
ronment, results were similar to those in the ?rst phase
lidities were provided. This is important because again
of the experiment; that is, 77% of the participants were
the strategies’ ?ts in predicting participants’ inferences
classi?ed as using TTB in the noncompensatory environ-
were based on the objective cue validities. This implies
ment and 70% were classi?ed as using a compensatory
that a low ?t of TTB could mean either that the strat-
strategy in the compensatory environment. Thus, partic-
egy was not selected at all or that the strategy was ap-
ipants apparently just continued to make their inferences
plied differently by using different subjective cue validi-
in the same way as they had in the ?rst half of the exper-
ties. To determine the ?t of the compensatory strategy
iment, and in the compensatory environment participants
WADD Bröder and Schiffer (2006) assumed that the rank
apparently selected WADD more frequently.
order of WADD’s weights would correspond to the rank
Most interesting is what happened when the partici-
order of the objective validities. Furthermore, three dif-
pants encountered a new environment. Did they adap-
ferent variants of WADD were examined, with different
tively switch their strategy? An adaptive switch was only
weighting schemes for the weights of the cues, and the
partly observed: Only 27% of the participants who en-
variant with the highest ?t was assigned to the partici-
countered the compensatory environment in the second
pants. Naturally, this implies that the participants had a
phase after seeing the noncompensatory environment in
priori a higher chance of being classi?ed as using WADD.
the ?rst phase were classi?ed as using a compensatory
To avoid this a priori advantage, in the following reanaly-
strategy. Thus, apparently the participants only slowly
sis I only used one single weighting schema to determine
adapted to the new environments. However, 65% of the
WADD’s predictions. I selected the schema according to
participants who encountered the noncompensatory en-
which any two less important cues were always suf?cient
vironment in the second phase after seeing the compen-
to compensate for the information of one, more important
satory environment in the ?rst phase were indeed classi-
cue. Moreover, I did not include a compensatory strategy
?ed as using the better performing TTB strategy.
that gave equal weight to each cue as Bröder and Schiffer
These results are somewhat different from the results
did. Thus, I used only one compensatory and one non-
reported by Bröder and Schiffer (2006), because in their
compensatory strategy to reanalyze the data.
analysis three additional compensatory strategies were
included. Accordingly more participants were classi-
3.2 Results and discussion
?ed as using a compensatory strategy and fewer as us-
ing the noncompensatory strategy TTB, and in partic-
Bröder and Schiffer’s (2006) data were reanalyzed by
ular a switch to TTB was not observed when partici-
classifying each participant as using either TTB or
pants encountered the noncompensatory environment af-
WADD.1 For the ?rst phase of the experiment a strong ef-
ter the compensatory environment. It is an open question
fect of the environment on the strategy classi?cation was
whether Bröder and Schiffer classi?ed more participants
observed.
as using a compensatory strategy because they simply
examined more compensatory strategies, or whether the
1Classi?cation was performed on the basis of the quadratic scoring
participants indeed used different types of compensatory
rule (Selten, 1998), according to which a strategy i with application er-
ror ? reaches a ?t QS of
strategies. However, this question is not of particular im-
P
P
QS(i, ?) = 1/T
T
3
portance for the present article.
t=1
k=1[Dt(k) ? pt(k|i, ?)]2,
with t for the trial, t as the number of trials, and k for the three alterna-
tives; pt(k|i, ?) denotes the probability of choosing alternative k when
using strategy i (see Equation 4), and Dt(k) is an indicator function that
3.2.1 Describing the learning process with the SSL
equals 1 if alternative k was chosen and equals 0 if the alternative was
theory
not chosen. For each strategy and participant the optimum value for the
application error ? was selected that led to the lowest, that is, the best
Bröder and Schiffer (2006) described the lack of adap-
value for QS. Participants were assigned to the strategy with the lowest
QS value and were left unclassi?ed if both strategies reached the same
tivity in the second phase of the experiment as a routine
?t.
effect; that is, they assumed that the strategies the par-

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
268
ticipants selected in the ?rst phase of the experiment be-
= .39 for WADD. Thus, on average, the participants had
came routines, so that when the environment changed the
an initial preference for TTB. This is surprising consider-
participants stuck to their routines. To explain why par-
ing that in similar studies an initial preference for WADD
ticipants in the second phase did not switch to another
has often been observed. One explanation might be the
strategy they referred to a dual-system theory of deci-
high search costs. In Bröder and Schiffer’s (2006) study
sion making (e.g., Betsch, 2005; Evans & Over, 1996;
the participants had to pay 4% of their potential gain for
Sloman, 1996). According to the dual-system approach,
every single cue value, meaning that if they searched for
people can be in either of two cognitive modes when
all the information their gain was reduced by 48%. This
making decisions, a deliberative, top-down mode or an
procedure might have made it more salient that a strategy
associative, experienced-based, bottom-up mode. Bröder
that requires a lot of information cannot perform well,
and Schiffer argued that when participants encountered a
leading to an initial preference for TTB.
novel decision situation they were in a top-down mode for
The third parameter estimated was the application er-
“calculations of long-term payoff advantages” that led to
ror, for which an average value of .26 resulted. This
the selection of an adaptive strategy. Thus, the top-down
value is relatively high in comparison to the results of
mode refers to the cost-bene?t approach, which should
Rieskamp and Otto (2006), with average application er-
lead to the selection of the best-performing strategy in
rors ranging between .05 and .07 across several exper-
an environment. In contrast, after participants selected a
iments. A high application error according to the SSL
successful strategy they “switched to a bottom-up mode
theory implies that the participants often deviated from
by routinely applying this strategy, which avoids testing
their selected strategies. For Bröder and Schiffer’s (2006)
the consequences in each trial” (p. 915). However, nei-
experiment this high value can be explained by the sub-
ther proponents of the two-system approach nor Bröder
jective importance the participants gave to the cues. To
and Schiffer provided a computational model that speci-
determine the predictions of the strategies one particular
?es how these two modes of cognition are activated, de-
compensatory weighting schema was employed. How-
termines how and whether they interact, and describes the
ever, this is only an approximation and the participants
observed learning processes.
might have given rather different subjective importance
In the following I will show how the SSL theory ex-
to the cues. If the subjective importance differed from the
plains the results. Bröder and Schiffer (2006) also re-
objective weights it is not surprising that TTB or WADD
ferred to the SSL theory as a potential description of the
relying on the objective weights do not predict all infer-
bottom-up mode of strategy selection. I will show with
ences, which is re?ected in a high application error. Note,
the reanalysis that the SSL theory is able to describe the
however, that the SSL theory does not require that people
learning process not only for the second half of the ex-
learn speci?c weights; it would only be preferable to elicit
periment but also for the whole experiment by assuming
the subjective importance of the cues, so that the strate-
a learning process of strategy selection. Thus, instead of
gies’ predictions are determined on the basis of subjective
postulating a dual-system theory of decision making, I
importance.
can illustrate that a single “cognitive mode” of learning
How well did the SSL theory describe participants’ in-
is suf?cient to explain the experimental ?ndings. I will
ferences? The SSL theory was able to predict partici-
argue that the SSL theory provides a more parsimonious
pants’ choices with an average probability of .58 (with SD
explanation of the maladaptive selection of strategies de-
= 0.05), which is much larger than the .33 one would pre-
scribed by Bröder and Schiffer than the dual-system ap-
dict by random chance. Nevertheless, it is lower than the
proach does.
average predicted probability of .75 found in Rieskamp
The parameters of the SSL theory were estimated sep-
and Otto’s (2006) experiment, which also examined a
arately for each individual’s learning data as follows: The
three-alternative inference problem. The lower ?t is pre-
model predicted the probability with which a participant
sumably due to the employed weights for the cues that
would choose each of the available alternatives for each
only partly correspond with the subjective importance the
trial conditioned on past choices and feedback (using
participants gave to the cues.
maximum likelihood as a goodness-of-?t criterion, cf.
Can the SSL theory also describe the adaptive selec-
Equation 6). The predictions of TTB and WADD were
tion of cognitive strategies? Figure 2 shows for the two
determined on the basis of the objective weights of the
stable environments the percentage of choices predicted
cues as described above. For the SSL theory three param-
by TTB and WADD across eight trial blocks of 20 tri-
eters were estimated, resulting in an average estimated as-
als each, restricted to those trials where the two strate-
sociation parameter value of 36; this is similar to values
gies make different predictions. Additionally, the ?gure
estimated in Rieskamp and Otto (2006). The average esti-
shows with what probability the SSL theory predicts the
mated initial preference parameter was ?TTB = .61 for for
selection of TTB and WADD. Figure 2A shows the re-
TTB, implying an initial preference of ?WADD = 1 - ?TTB
sults for the participants only facing a noncompensatory

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
269
A
Noncompensatory - noncompensatory environment
0.8
0.7
0.6
0.5
0.4
0.3
Proportion / Probability
0.2
TTB
WADD
0.1
p(TTB I SSL Theory)
p(WADD I SSL Theory)
0.0
0
1
2
3
4
5
6
7
8
Trial blocks
B
Compensatory - compensatory environment
0.8
0.7
0.6
0.5
0.4
0.3
Proportion / Probability
0.2
TTB
WADD
0.1
p(TTB I SSL Theory)
p(WADD I SSL Theory)
0.0
0
1
2
3
4
5
6
7
8
Trial blocks
Figure 2: The proportion of choices predicted by TTB and WADD in the stable noncompensatory (A) and stable
compensatory (B) environment conditions of the ?rst experimental study of Bröder and Schiffer (2006). Also shown
is the SSL theory’s predicted probability of selecting the two strategies. Only discriminating trials where the strategies
make different predictions are shown, and each trial block consists of 20 trials.

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Judgment and Decision Making, Vol. 3, No. 3, March 2008
Learning and inferences
270
environment and illustrates a strong learning effect: The
performance compared to TTB. Second, as a result of the
proportion of choices that could be best predicted by TTB
?rst phase of the experiment, the strategies’ expectancies,
steadily increases across the eight blocks of trials. This
in particular TTB’s expectancies, have grown substan-
learning effect explains why the majority of participants
tially, so that it will take a considerable number of trials
were classi?ed as using TTB in both halves of the exper-
with positive reinforcement for WADD’s expectancies to
iment.
exceed TTB’s expectancies again.
A corresponding learning effect was observed for the
Figure 3B shows the results for the condition starting
compensatory environment (Figure 2B). Here, in the ?rst
with the compensatory environment and continuing with
two trial blocks TTB predicted more inferences than
the noncompensatory environment. Here, the results are
WADD, but due to learning, the percentage of inferences
less clear. In the ?rst half of the experiment the percent-
predicted by WADD increased so that in the second half
age of inferences that could be best predicted by WADD,
it predicted the majority of inferences. This learning ef-
the better performing strategy in the compensatory envi-
fect can explain why most participants were classi?ed as
ronment, increased across the ?rst four trial blocks, so
using WADD in the second half of the experiment. The
that in the third and fourth trial block it predicted more
compensatory environment also illustrates that the par-
inferences than TTB. In the second half of the experi-
ticipants did not select the most adaptive strategy right
ment the percentage of inferences predicted by WADD
from the beginning, as a cost-bene?t approach predicts.
decreased with a corresponding increase of the inferences
Instead, a preference for WADD was developed through
predicted by TTB, so that TTB did better in the last three
the ?rst 80 choices. The probabilities with which the SSL
trial blocks. Thus, the proportion of predicted inferences
theory predicts the selection of TTB or WADD nicely
by WADD and TTB changed adaptively depending on the
match the proportion of inferences predicted by the two
environment. However, the differences are rather small
strategies. These results suggest that people do not ini-
and less conclusive, considering that the proportion of
tially select a strategy for an environment and keep using
inferences predicted by TTB or WADD varies between
it without monitoring its success, as suggested by Bröder
approximately 40 and 55%. Nevertheless, the SSL the-
and Schiffer (2006, see p. 915). Instead people’s initial
ory describes this learning process by an increasing prob-
preferences for a speci?c strategy are quickly wiped out
ability with which WADD is selected in the ?rst half
by the experiences they have, so that the strategy they se-
of the experiment, followed by a decreasing probability
lect for a speci?c environment is essentially the result of
in the second half. Thus, the SSL theory can also ex-
the strategies’ success.
plain when an adaptation to a new environment occurred,
for instance, when previous experience did not lead to a
strong preference for a speci?c strategy.
3.2.2 Describing behavior in dynamic environments
with the SSL theory
Bröder and Schiffer’s (2006) experimental results il-
lustrate that people are able to select strategies adaptively
But do people also learn to adapt to dynamic environ-
in a stable environment. However, in a dynamic envi-
ments? Figure 3 shows for the dynamic environments the
ronment in which the best strategy to solve an inference
percentage of choices predicted by TTB and WADD. Fig-
problem changes, people do not always switch to the bet-
ure 3A shows the results for the experimental condition
ter performing strategy. These results can be explained by
starting with the noncompensatory environment and con-
the SSL theory. First, the selection of the best-performing
tinuing with the compensatory environment. For the ?rst
strategy in the stable environment appeared to be the re-
half of the experiment a learning process is observed with
sult of a continuous learning process, rather than of an
an increasing proportion of inferences predicted by TTB.
initial cost-bene?t trade-off process. In fact, the strategy
After the shift of the environment in the ?fth trial block
that predicted the most inferences at the end of the ex-
the percentage of inferences predicted by TTB decreases
periment was not necessarily the strategy that predicted
only slowly. In the seventh trial block a larger decrease
more inferences at the beginning, as illustrated with the
can be observed. Nevertheless, across all trial blocks in
compensatory environment.
the second phase of the experiment when participants en-
The SSL theory also explains the lack of adaptivity:
countered the compensatory environment TTB did much
When people develop a strong preference for a speci?c
better than WADD in predicting the inferences. Can this
strategy based on their experience, as, for instance, for
maladaptive behavior be described by the SSL theory?
TTB in the noncompensatory environment, they will too
The SSL theory predicts an even slower adaptation
rarely select alternative strategies to be able to detect their
process to the new environment, for two reasons: First,
potentially superior performance. However, this does
when TTB is selected with a high probability a person
not mean that they do not change their inferences at all.
will only rarely select the competing compensatory strat-
When considering the proportion of inferences across the
egy and will therefore only rarely experience its better
different trial blocks it becomes clear that the participants

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