Journal of Cognitive Psychology
ISSN: 2044-5911 (Print) 2044-592X (Online) Journal homepage: http://www.tandfonline.com/loi/pecp21
A belief in trend reversal requires access to
cognitive resources
Tadeusz Tyszka, Łukasz Markiewicz, Elżbieta Kubińska, Katarzyna Gawryluk
& Piotr Zielonka
To cite this article: Tadeusz Tyszka, Łukasz Markiewicz, Elżbieta Kubińska, Katarzyna Gawryluk
& Piotr Zielonka (2016): A belief in trend reversal requires access to cognitive resources, Journal
of Cognitive Psychology, DOI: 10.1080/20445911.2016.1245195
To link to this article: http://dx.doi.org/10.1080/20445911.2016.1245195
Published online: 24 Oct 2016.
Submit your article to this journal
Article views: 9
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=pecp21
Download by: [195.85.252.4]
Date: 14 November 2016, At: 00:25
JOURNAL OF COGNITIVE PSYCHOLOGY, 2016
http://dx.doi.org/10.1080/20445911.2016.1245195
A belief in trend reversal requires access to cognitive resources
Tadeusz Tyszkaa, Łukasz Markiewicza, Elżbieta Kubińskab , Katarzyna Gawryluka and Piotr Zielonkac
a
Center of Economic Psychology and Decision Sciences, Kozminski University, Warsaw, Poland; bDepartment of Financial Markets,
Cracow University of Economics, Kraków, Poland; cDepartment of Physics, Warsaw University of Life Sciences, Warsaw, Poland
ABSTRACT
ARTICLE HISTORY
There are two research traditions studying people’s reactions to random binary events:
one concerns serial choice reaction times, the other concerns predictions of events in a
series. The present studies focused on comparing expectations between these two
approaches. We formed and tested a general hypothesis that, regardless of the type of
task, when an individual faces a sequence of events they initially expect trend
continuation. Only when people assume that a sequence is random might they
override the default and expect trend reversal instead. In a series of experiments we
found that limitation of access to cognitive resources enhances expectations of trend
continuation. Our interpretation of this finding is that an expectation of trend
continuation is the default for the human cognitive system and that a belief in trend
reversal requires access to cognitive resources to overcome the tendency to expect
trend continuation.
Received 5 December 2015
Accepted 1 October 2016
Introduction
There are two research traditions in studying
people’s reactions to random binary events: one
studies serial choice reaction times, the other
studies predictions of events in a series. In a serial
choice reaction time task, participants are shown
one of two different stimuli and are instructed to
respond as quickly as possible by mapping the
stimulus to a corresponding response, say, pressing
the left button for X and the right button for
Y. Response time is recorded and the task is
repeated for the next stimulus in the sequence
(Gökaydin, Navarro, Ma-Wyatt, & Perfors, 2016). In
a prediction task, a sequence of binary events is
shown and participants are asked to predict the
next event (Ayton & Fischer, 2004; Burns & Corpus,
2004). Typically, in these types of experiments the
sequence of binary events presented to participants
is random. The choice reaction time and prediction
tasks may involve the same sequence of events,
though participants face differing requirements.
Researchers have studied the two types of task
quite intensively, and in both of the tasks various
sequential effects are observed. Choice reaction
times and predictions depend on the sequence of
preceding stimuli. One of the goals of the present
CONTACT Łukasz Markiewicz
Random series; choice
reaction time; predictions;
trend continuation; trend
reversal; gambler’s fallacy;
hot-hand fallacy
study was to conduct a systematic comparison
between sequential effects in the two types of
task. A key assumption was that sequential effects
are determined by participants’ expectations:
whether they expect a continuation or a reversal
of a trend. With this assumption, the study focused
on comparing expectations in the reaction time
and prediction tasks.
A large number of experiments on serial choice
reaction time have demonstrated that the reaction
time to a stimulus depends on the sequence of preceding stimuli (Gökaydin et al., 2016). First-order and
higher-order sequential effects have been observed.
The first-order effect is the difference in reaction
times caused by an immediately preceding stimulus.
Higher-order effects, which we focus on in the
current paper, concern longer sequences of earlier
stimuli.
Research shows that expectations have an impact
on reaction times (Gao, Wong-Lin, Holmes, Simen, &
Cohen, 2009). When faced with a sequence of
random events, people respond more quickly to a
signal which repeats a preceding one than when it
is an alternation. The mechanism for this effect
may be explained in terms of expectations. Shorter
reaction times are observed when the same signal
is replicated because people expect trend
[email protected]; Piotr Zielonka
© 2016 Informa UK Limited, trading as Taylor & Francis Group
KEYWORDS
[email protected]
2
T. TYSZKA ET AL.
continuation (Soetens, Boer, & Hueting, 1985). When
runs of repetitions are interrupted, reaction times
increase because expectancies are disconfirmed
(Sommer, Leuthold, & Soetens, 1999).
When people are asked to predict uncertain
events two opposite tendencies are observed:
either a positive or a negative recency effect. The
positive recency effect, associated with a momentum prediction strategy, indicates a tendency to
predict a trend’s continuation. The negative
recency effect, associated with a contrarian prediction strategy, indicates a tendency to predict a
trend’s reversal. It is reasonable to postulate that
when an individual is asked to make predictions of
uncertain events they try to formulate hypotheses
concerning the nature of the source of uncertainty.
As shown by Tyszka, Zielonka, Dacey, and Sawicki
(2008), Ayton and Fischer (2004), and Burns and
Corpus (2004), when an individual assumes that a
sequence of uncertain events is random, in line
with the representativeness heuristic, they tend to
expect trend reversal and therefore follow a contrarian prediction strategy. However, when they
believe that the nature of an observed series is not
random, but deterministic, they expect trend continuation and thus follow a momentum prediction
strategy.
Why is a person’s behaviour in line with a belief in
trend continuation in a serial choice reaction time
task, whereas in prediction tasks it is in line with a
belief in trend reversal? We formed the general
hypothesis that, regardless of task type, when an
individual faces a sequence of binary events they
initially expect trend continuation. Numerous findings appear to support this claim. Huettel, Mack,
and McCarthy (2002), implementing an functional
magnetic resonance imaging method, demonstrated that even when people observe a very
short sequence of one type of event they tend to
expect trend continuation. Also, Blanchard, Wilke,
and Hayden (2014) conducted an experiment
where monkeys made predictions of whether a
stimulus would appear to their left or their right in
a serial choice task. They indicated their choice by
looking in the appropriate direction. The degree of
autocorrelation between trials was manipulated. In
different conditions the same side was rewarded
with a probability of 10–90%. The main question
posed was whether the monkeys would be more
predisposed to pursue the optimal course of action
in clumpy (i.e. positively correlated) or in dispersed
(i.e. negatively correlated) environments. It was
found that they showed better performance in
clumpy than in distributed resource environments.
They were biased to thinking stimuli were more
streaky, or more positively correlated, than they
actually were. Finally, Wilke and Barrett (2009) conducted an experiment where participants played a
computerised sequential foraging game in which
they experienced a sequence of 100 hits and
misses, and after each event in the sequence were
asked to predict whether the next event would be
a hit or miss. The event distributions were equivalent
to a series of coin tosses with a 0.5 probability. Types
of sequences were framed differently: in some cases
they were depicted as actual coin tosses and, in
other cases, as fruits, bird’s nests, parking spots,
and bus stops. With the latter types of framing, at
each step, subjects were asked to predict whether
the next step would be a hit (e.g. a nest) or a miss
(e.g. no nest). There were two groups of participants:
American students and members of a South American indigenous population. It was found that both
groups of participants displayed the positive
recency effect in the majority of situations, but this
effect was diminished in one situation: where the
group of American students was predicting coin
tosses. This result suggests that expectations for
trend reversal occur only where there is a strong
belief in the random nature of events.
Where does this expectation of trend continuation come from? Evolutionary psychologists give
an environmental explanation. Human hunter-gatherers depended upon the resources they were able
to find in nature. The animals and plants they
sought were not randomly scattered: rather, they
were aggregated in order to exploit a natural
habitat or to realise the benefits of living in close
proximity or forming a herd. Since humans were
hunter-gatherers for much of their evolutionary
history, human psychology evolved to expect such
spatial and/or temporal aggregation, that is, to
expect positive correlations.
Thus, existing research strongly suggests that,
irrespective of task type (choice reaction time or prediction), individuals initially form expectations of
trend continuation. At the same time, research
shows that a crucial factor in the prediction task is
whether an individual assumes that a sequence of
uncertain events is random or non-random. When
people assume that a sequence is random they
tend to exhibit a negative recency effect, whereas
if they assume a sequence of events to be deterministic the positive recency effect prevails.
JOURNAL OF COGNITIVE PSYCHOLOGY
One focus of the present research was on
responses to choice reaction time and prediction
tasks when people consider a sequence of binary
events to be random. In this case there is a substantial difference between the tasks, which may cause
participants to behave differently.
On the one hand, in a choice reaction time task a
participant is asked to react to a stimulus as quickly
as possible. Therefore they do not have enough time
to reflect on the nature of a sequence. In effect, participants’ reactions are based on an expectation of
trend continuation. This assumption leads to the
hypothesis that:
H1: In a serial choice reaction time task, response
time decreases as the number of stimuli of the
same type in a sequence increases.
Hypothesis 1 is an alternative wording of the previous finding that people respond more quickly to a
signal when it is a repetition of a preceding signal
than when it is an alternation.
On the other hand, in a standard prediction task
an individual has enough time to consider the
nature of observed events. Thus, they have an
opportunity to change their initial basic expectation
of trend continuation toward the possibility of trend
reversal. Illuminating observations concerning gamblers’ speculations about the nature of events in
roulette can be found in Fyodor Dostoyevsky’s
(1867–2000) novel “The Gambler”:
3
random sequence may be an individual’s ability to
access cognitive resources. We speculated that an
impediment could be introduced into the prediction
task by limiting participants’ access to their cognitive
resources to produce a situation analogous to the
choice reaction time task, where a participant does
not have enough time to reflect on the nature of a
sequence. We reasoned that limiting a participant’s
cognitive resources would make them unable to
comprehend the nature of a sequence. In such a
situation an individual should retain their initial
expectation of trend continuation. Two separate
methods of limiting cognitive resources were
applied in the present research:
(1) adding a cognitive load to the prediction task;
(2) asking children (who are less able than adults to
engage in effective probabilistic thinking) to
perform the prediction task.
Thus, following the claim by DeSteno, Bartlett,
Braverman, and Salovey (2002), we expected that
adding a cognitive load would lead to the inhibition
of deliberative processes in the prediction task,
which would support a belief in trend continuation,
this being manifested by an increase in the number
of participants using the momentum strategy
(resulting in a positive recency effect). Thus, we
formed Hypothesis 2:
H2: In a prediction task involving a random
sequence, imposing a cognitive load leads to an
increase in choices based upon a momentum
strategy.
But, as ever, fortune seemed to be at my back. As
though of set purpose, there came to my aid a circumstance which not infrequently repeats itself in
gaming. The circumstance is that not infrequently
luck attaches itself to, say, the red, and does not
leave it for a space of say, ten, or even fifteen,
rounds in succession. Three days ago I had heard
that, during the previous week there had been a
run of twenty-two coups on the red—an occurrence
never before known at roulette—so that men spoke
of it with astonishment. Naturally enough, many
deserted the red after a dozen rounds, and practically no one could now be found to stake upon it.
(pp. 125–126, The Gambler)
Our second experimental method of limiting cognitive resources was to use young children as experimental participants, with the assumption that their
ability to engage in probabilistic thinking would be
underdeveloped. Thus, Hypothesis 3 was as follows:
The above episode is not simply fictitious. It is
known from Dostoyevsky’s letters to his wife Anna
(Dostoyevsky, 1998) that he himself tried several
times to take advantage of these speculations
when he played in casinos. Interestingly, he called
this strategy his “discovery”, that is, something that
required cognitive effort.
This suggests that a crucial factor determining
strategy in predicting uncertain events in a
Jointly, hypotheses H1–H3 allowed us to examine
whether limiting access to cognitive resources leads
to more frequent expectations of trend continuation
(the positive recency effect). In Study 1 cognitive
resources were limited by time pressure, in Study 2
we limited access to cognitive resources by introducing a cognitive load, and in Study 3 we assumed
limitation of cognitive resources by inviting cognitively underdeveloped participants (children) to
H3: Children base their choices on a momentum
strategy more often than adults in the prediction
task.
4
T. TYSZKA ET AL.
participate in an experiment. Suggesting to participants that the events were random, exactly the
same sequence of binary events was used in all
three experiments.
Finally, in Study 4, we followed a reviewer’s suggestion of replicating Experiment 2 with the same
sequence of binary events, but with a different—
non-random—interpretation. Such a non-random
interpretation was available from Tyszka et al.
(2008) in the form of the sequence of a basketball
player’s successful and unsuccessful throws at a
basket. Thus, while in Study 2 arrow up and arrow
down symbols were presented to participants, in
the follow-up study the sequence consisted of a basketball player’s successful and unsuccessful throws
at a basket. As in Study 2, participants performed
the prediction task either with or without a cognitive
load. We tested the hypothesis that when people
assume that a sequence is non-random, regardless
of the availability of cognitive resources, their expectations of trend continuation prevail.
Study 1: the choice reaction time task
Study 1 aimed to test the hypothesis that in a serial
choice reaction time task response time decreases as
the number of stimuli of the same type in a
sequence increases.
Participants
A total of 35 Kozminski University undergraduate
management and finance students (70% females,)
in the age range 20–41 years old (M = 23.3, SD =
3.6) participated in Study 1 for credit points with
no monetary compensation. All were native Polish
speakers, and all materials were prepared in Polish.
Apparatus
The study was conducted in the Computer Lab for
Experimental Research with 35 individual stations
separated by cardboard screens. Participants performed the task individually. Displays were generated by computers attached to 19-inch panoramic
LCD monitors with 1280 × 960 resolution. Participants viewed the LCD displays from a distance of
about 60 cm. Responses were collected via computer keyboards. The experiment was programmed
using Inquisit Millisecond 4.0 software (Inquisit,
2012).
Procedure: the choice reaction time task
Shortly after consent for participation was obtained
and basic sociodemographics were recorded, participants took part in quick response training
(QRT), in which they were asked to press their spacebar as fast as possible after seeing a stimulus (a down
or up arrow) on their screen. In each of 10 trials, feedback with exact reaction times was presented, along
with a request for faster responses if reaction times
exceeded 1000 ms. The purpose of this part of the
study was to familiarise participants with the computer environment and to minimise the variance of
reaction times in the principal task.
Previous research shows that sequential effects
occur when a participant has enough time to formulate expectations about a stimulus which is about to
appear. Researchers have noted that reaction times
depend on the reaction—stimulus interval (RSI),
that is, how quickly the next stimulus occurs after
a respondent’s reaction to a previous one. Participants are able to formulate expectations when the
RSI is long enough, otherwise their reactions are
automatic (Kirby, 1972, 1976). In order to allow participants time to form their expectations, we
implemented an RSI of 2000 ms.
The principal task was divided into four blocks. In
the first block, participants were asked to carefully
track “the randomly generated sequence of up and
down arrows”. Here, no reaction was required from
participants: their task was simply to observe a
sequence of 21 events presented for 500 ms, with
a 750 ms separation between stimulus presentations. This part of the study was used to induce a
sense of randomness with respect to the presented
sequence: studies show that participants learn probabilities better through experience than when they
are simple presented with probabilities expressed
in percentage points (Tyszka & Sawicki, 2011).
In the second block, participants were asked to
carefully track the sequence of up and down
arrows and respond to each event in the sequence.
Thus, the script presented stimuli sequentially, and a
participant’s task was to categorise stimuli by pressing Y for an up arrow and B for a down arrow as
quickly as possible. To control for dominant hand
preference, half of the participants were asked to
put their left forefinger on the Y key and their
right forefinger on the B key, while the other half
were asked to do the opposite. To provide accurate
reaction time measurement, participants were
instructed to keep their forefingers on the B and Y
JOURNAL OF COGNITIVE PSYCHOLOGY
5
Figure 1. Timeline of stimulus presentations in Blocks 2 and 4 of Study 1 (Choice reaction time Task: Panel A) and Study 2
(Prediction task: Panel B).
keys throughout the whole task. Both the nature of
responses (up or down arrow) and response times
(in milliseconds) were recorded. After each response
was made, a mask screen was presented for 2000 ms
before the next stimulus appeared. The timeline of
the task is presented in Figure 1, Panel A. Although
results from Block 2 were recorded in a data file,
the block was treated as a training exercise for the
fourth block with its manipulated sequence.
In the third block, participants were asked to
observe 21 other events in a randomly generated
sequence—similarly to Block 1. Finally, in the
fourth block, participants performed a task identical
to that in Block 2, except that the randomly generated sequence for this section was manipulated.
Only the answers and reaction times for this
section were used for hypothesis verification.
The sequence
The sequence of stimuli (arrows) in the whole task
was predefined and identical for all participants.
The sequences of 21 events in each of four blocks
were created with a symmetric probability of 50%
of generating either an up or a down arrow. While
the first three blocks of random stimuli were used
for Blocks 1–3 as described above, the last block of
21 events used for Block 4 was intentionally modified, starting from the fifth stimulus, to contain a
sequence of eight up arrows, then three random
events, and then five down arrows. The final sequence
of events used in the study is presented in Figure 2.
Following Falk and Konold (1997) we calculated
alternation probabilities for Blocks 1–4, which were
.70, .45, .40, and .35, respectively.
We were particularly interested in studying participants’ behaviour during the presentation of a univariate sequence of events (first sequence: eight up
arrows; second sequence: five down arrows). These
sections were intentionally planned to be longer
than three univariate events since previous research
suggests that the cognitive representation of a
sequence is formed after observing a minimum of
three univariate events (Barron & Leider, 2010;
Carlson & Shu, 2007). On the other hand, there was
little reason to make the sequences longer than
eight elements, due to working memory limitations
(Kareev, 2000). Moreover, while some studies have
researched reactions to a sequence of two to four
elements, as noted by Scheibehenne and Studer
(2014), reactions to longer sequences have received
little research attention. We also expected that an
Figure 2. The final sequence of events used in the studies:
Blocks 1 and 3—observation only; Blocks 2 and 4—choice
reaction time (Study 1) or prediction task (Study 2).
6
T. TYSZKA ET AL.
initial propensity for either positive or negative
recency would become stronger as sequence
length increased (Barron & Leider, 2010; Croson &
Sundali, 2005; Scheibehenne & Studer, 2014;
Sundali & Croson, 2006). To maintain the illusion
that the sequence presented in Block 4 was
random, the first manipulated sequence appeared
after the fifth random event representing the fluctuation pattern (Roney & Trick, 2003).
Results
According to Hypothesis 1, as the number of stimuli
of the same type in a sequence increased, the
response time should have decreased. Figure 3 presents means of logarithmic transformations of reaction times for stimuli sequentially presented on
screen (reaction times were log transformed in
accordance with research practice, to minimise variance and reduce the influence of outliers arising
from external factors). The averaged reaction times
for the 11th, 12th and 13th stimuli were significantly
lower, t(34) = 3.83; p = .001, than averaged reactions
times for the 7th, 8th, and 9th stimuli as H1 stated.
Similarly, the averaged reaction times for the 19th
to 21st stimuli were lower than averaged reaction
times for the 17th and 18th stimulus, t(34) = 3.74;
p = .001. Also, repeated measures ANOVA performed
on every second trial (in the interests of data
reduction) in the up sequence (four levels: 7th, 9th,
11th and 13th trials) showed that log reaction
times were significantly affected by the presented
univariate sequence, F(3,102) = 4.96, p = .003, and
similarly the log reaction times of every second
trial in the down sequence: (17th, 19th, 21st trials)
were significantly affected by the presented
sequence, F(1.23, 41.72) = 8.11, p = .004. All of these
results support Hypothesis 1.
Study 2: the consequences of adding a
cognitive load to a prediction task
Study 2 tested the hypothesis that imposing a cognitive load to a prediction task involving random
sequences leads to an increase in choices based
upon a momentum strategy.
Participants
Altogether, 68 Cracow University of Economics
undergraduate finance students, all native Polish
speakers, (57% male) in the age range 23–35 years
(M = 24.49, SD = 2.08) participated in the study for
credit points with no monetary compensation.
Apparatus
Study 2 was conducted over the course of several inlab sessions (between 10 and 35 participants for
each session). As in Study 1, participants performed
the task individually in front of a computer, the task
being programmed using Inquisit Millisecond 4.0
software. The task was similar to the choice reaction
Figure 3. Study 1: Mean logarithmic reaction times for choice reaction time to stimuli. The line graph in the background
illustrates the sequence development over the task. The error bar represents 95% Confidence Intervals.
JOURNAL OF COGNITIVE PSYCHOLOGY
time task in Study 1. However, in the second and
fourth blocks participants’ task was not to recognise
each following event (an up or down stimulus), but
rather, to predict it. The prediction task used the
same sequence of stimuli as the Study 1 choice reaction time task.
Procedure: the prediction task
As in Study 1, participants first performed QRT
before moving on to a principal task divided into
four blocks. In the first and third blocks, which
aroused a sense of randomness about the presented
sequence, participants were asked to carefully track
a randomly generated sequence of up and down
arrows with no reaction required of them.
In Block 2, participants were asked to not only
carefully track the sequence of arrows but also to
predict each following event in the sequence.
Thus, before presenting each of 21 stimuli (up or
down arrows), a question mark (“?”) was presented,
and participants were required to make predictions
(by pressing the Y key for an up arrow or the B key
for a down arrow) with their left or right forefinger
(again, to control for dominant hand preference,
half of the participants were asked to put their left
forefinger on the Y key and their right forefinger
on the B key, while the other half were asked to
do the opposite). Both the nature of the response
and the response time (in milliseconds) were
recorded. Immediately after the response, a stimulus
was presented for 2000 ms, and after a 1000 ms
mask screen the question mark was presented
again. Thus, participants’ responses (i.e. predictions)
did not influence the forthcoming stimulus (up or
down arrow), and participants could only observe
whether their prediction was correct or incorrect
after seeing the stimulus (however, no special
message containing this information appeared).
The timeline of the task is presented in Figure 1,
Panel B. Although results from Block 2 were
recorded in a data file, again, this block was
treated as a training exercise for the fourth block
with its manipulated sequence.
As in Block 1, in the third block, participants were
again asked to observe the third part (21 events) of a
randomly generated sequence. Finally, in the fourth
block, participants performed a task identical to that
in Block 2, except that the randomly generated
sequence for this block was manipulated as presented in the “Sequence” section of this article.
The answers and reaction times for this block only
were used for hypothesis verification.
7
Procedure and design: the manipulation
All participants performed the whole task wearing
headphones, but only an experimental group
heard sound through the headphones. Participants
were randomly assigned to one of two conditions.
The control group (n = 40) performed the task with
no additional impediments, while the experimental
group (n = 28) performed Block 4 of the task
under cognitive load. Although the random
assignment created unequal groups, no relationships were observed between group membership
and gender, χ 2(1) = 2.32; p = .128, or age, t(66) =
1.85, p = .070.
The additional cognitive load took the form of
participants listening to a wildlife talk read by an
actor while they performed the prediction task in
Block 4. As a manipulation check for this (experimental) condition, participants were informed that
shortly after the task they would participate in a
short quiz related to the story that they had heard.
On completing this quiz, the participants correctly
answered M = 3.5, SD = 1.29 out of six control questions focusing upon details in the story. Bearing in
mind the non-trivial nature of the questions, we
interpreted this as being indicative of an effective
manipulation. With respect to the different types
of cognitive load discussed by Block, Hancock, and
Zakay (2010) the cognitive load manipulation
employed can be classified as one involving high
load attentional demands, requiring participants to
divide their attention between two sources (the
story heard through the headphones and the
visual stimulus presented on the screen). In our
opinion, this type of cognitive load manipulation is
more ecologically valid and reflects real-life situations in which people often need to predict patterns while performing other activities. It could
also be argued that other types of cognitive load
manipulation (e.g. making demands on memory by
asking participants to memorise 2–3 digits) could
impair participants’ ability to recall the subsequent
elements of a response sequence because of the
need to count previous elements; it is possible that
any subsequent effect could result not from an
enhanced inclination to adopt a momentum strategy but from impaired memory of the sequence.
Results
The raw distribution of up predictions for the 21
trials (horizontal axis) of Block 4 is presented in
Figure 4, with the black bars representing the
8
T. TYSZKA ET AL.
experimental group and the white bars representing
the control group. The line graph in the background
presents the development of the sequence encountered by participants in Block 4.
We aimed to research participants’ behaviour
after the impression of a sequence emerged
(Carlson & Shu, 2007). Thus, we investigated the
two sequences after the occurrence of a third univariate stimulus, and were therefore interested in
predictions for the 8th–13th stimuli for the sequence
of up arrows, and for the 19th–21st stimuli for the
sequence of down arrows. Both the experimental
and the control participants increased their momentum approach as the sequence of up arrows
emerged. The distribution of up predictions for the
streak of up arrows had an inverted U shape in the
control group, while in the experimental group,
after its initial growth, the momentum strategy
also became dominant in subsequent choices. This
is not surprising: Altmann and Burns (2005) reported
that
On the first of a streak of heads, participants showed
positive recency, meaning that they predicted
heads for the next outcome with a greater-thanbaseline probability. As streak length increased,
positive recency first decreased but then increased
again, producing a quadratic trend. (p. 5)
Also, other studies have shown that, when
observing a streak, decision-makers, exhibit positive
recency for short runs and negative recency for long
runs (Jarvik, 1951). Thus, we expected the finding of
a more intense momentum tendency at the beginning of the trend, as well as a decrease in this tendency with the development of the sequence in a
long run. On the other hand, Scheibehenne and
Studer (2014) showed that as the size of runs
increases most people exhibit either positive or
negative recency. This was also observed by
Barron and Leider (2010), and by Croson and
Sundali (2005).
As stated in H2, we assumed that cognitive load
would increase momentum strategy usage in the
prediction task. Thus, the black bars (representing
the experimental group) should be higher than the
white bars (representing the control group) for
the streak of up arrows (8th–13th stimuli), and the
black bars should be lower than the white bars for
the streak of down arrows (19th–21st stimuli). An
initial chi-square test showed that respondents in
the experimental group used a momentum
approach significantly more often for predictions
in the case of the 8th, χ 2(1) = 6.07, p = .014, 11th,
χ 2(1) = 4.73, p = .030, and 12th, χ 2(1) = 4.39, p = .036
stimuli, thereby supporting H2 for these stimuli,
but the remaining differences were non-significant.
However, rather than suggesting that each single
round would be influenced, H2 predicted a greater
tendency to adopt a momentum approach on
average when acting under cognitive load. Therefore to facilitate a more direct test of H2 we calculated a numeric ratio by counting how many
momentum predictions were made by participants
for the streak of up arrows (0–6) and down arrows
(0–3). The results are presented in Table 1.
In line with H2, for the streak of up arrows, we
observed that the experimental group more frequently predicted that the next event would be
identical to the observed series of the most recent
events (M = 3.96) compared to the control group
(M = 3.05), thus revealing a momentum strategy, t
(66) = 2.42, p = .018. A similar calculation for the
other sequence of four down events (18th–21st),
with the expected value of two up events for
random respondents, showed that members of the
control group expected an average of M = 1.42 up
events, while the experimental group members
expected M = 1.11. Although the direction of this
difference shows that experimental group
members expected fewer ups (and thus more
downs), showing greater belief in momentum, the
Table 1. Study 2: A random frame sequence with adult participants. Number of up predictions in different
parts of the four block random sequence.
Control group
(no load), N = 40
2–7
8–13
(streak of up arrows)
14–18
19–21
(streak of down arrows)
Experimental group
(cognitive load), N =
28
M
SD
M
SD
t
p
3.08
3.05
1.61
1.62
3.18
3.96
1.19
1.40
−.290
−2.422
.773
.018
2.50
0.93
1.22
1.05
2.29
0.82
0.81
0.94
.870
.418
.387
.678
JOURNAL OF COGNITIVE PSYCHOLOGY
9
Figure 4. Study 2: The raw distribution of up predictions in the control (white, left bar) and experimental (black, right bar)
groups for a random trend perspective. The line graph in the background illustrates the sequence development over the task.
The error bar represents 95% Confidence Intervals.
result lacked statistical significance, t(66) = .418, p
= .678. We believe that extending the second
down sequence (which was two events shorter
than the previous up sequence) could make the
effect more visible, and similar to the result observed
in the up sequence discussed above. Also, the fact
that just before the second sequence the participants were exposed to a rising “up” sequence
could potentially have diminished the expected
effect in the second sequence.
We concluded that cognitive load increased
momentum strategy usage, thus supporting H2.
Compared to the control group, we found that participants under cognitive load more frequently predicted that the next event would be identical to the
observed series. These results support Hypothesis 2.
Study 3: children’s prediction of uncertain
events
Study 3 aimed to test the hypothesis that children
base their choices upon a momentum strategy more
often than adults when performing a prediction task.
Participants
Altogether, 71 pupils (62% girls) of a Warsaw primary
school, all native Polish speakers in the age range 7–
14 years (M = 9.97, SD = 2.47), participated in the
study. Parents gave written informed consent for
their children to participate.
Apparatus
Study 3 was conducted in individual, one-to-one sessions (one child was in a room with a female experimenter who helped the child to navigate their way
through the study). The same Inquisit script with
simplified instructions was used as for the Study 2
control group (the prediction task with no impediments). In the second and fourth blocks, participants’ task was to predict a following event (an up
or down stimulus). The script used the same
sequence of stimuli as Studies 1 and 2, and the children followed exactly the same procedure as Study
2 participants. Therefore, as previously, we were
interested in child participants’ predictions in Block
4 sequences (more than three events involving a
stimulus in the same direction: 8th to 13th, and
19th to 21st event).
Results
The raw distribution of up predictions for the 21
trials (horizontal axis) of Block 4 is presented in
Figure 5, with the black bars representing the child
group compared with the adult control group from
Study 2 (white bars). As previously, the line graph
in the background presents the development of
the sequence encountered by participants in
Block 4.
We assumed that both a cognitive load (H2: Study
2) and being a child (H3: Study 3) would increase
momentum strategy usage in the prediction task.
Thus, the black bars (representing the child
10
T. TYSZKA ET AL.
Figure 5. Study 3: The raw distribution of up predictions for the adult control group (white, left bar) and children (black, right
bar) for a random trend perspective. The line graph in the background illustrates the sequence development over the task.
The error bar represents 95% Confidence Intervals.
experimental group) should be higher than the white
bars (representing the adult control group) for the
streak of up arrows (8th to 13th stimuli), and the
black bars should be lower than the white bars for
the streak of down arrows (19th to 21st stimuli). An
initial chi-square test showed that children used a
momentum approach significantly more often for predictions than the adult control group for both the
11th, χ 2(1) = 4.80, p = .032, 12th, χ 2(1) = 12.06, p
= .001, and 17th, χ 2(1) = 5.63, p = .022, stimuli,
thereby supporting H3 for these stimuli. However,
we believed that children, who lack deliberative abilities, are generally more prone to using momentum
approaches. Thus, rather than suggesting that their
predictions would be different from adults in each
single round, we predicted a greater tendency for children to adopt a momentum approach on average
when compared to adults. Therefore, as in Study 2,
we calculated a numeric ratio by counting how
many momentum predictions were made by
participants for the streak of up arrows (0–6) and
down arrows (0–3). The results are presented in
Table 2.
In line with H3, for the streak of up arrows, we
observed that the child group predicted that the
next event would be identical to a series of
recently observed events (M = 3.85) more frequently than the adult control group (M = 3.05),
thus revealing a clear momentum strategy,
t(65.54) = 2.690, p = .009. Interestingly, this tendency applied (p < .05) only for the long
sequence of identical events (8th to 13th) and
none of the other Block 4 sequences (Table 2).
This leads to the conclusion that the natural
inability of children to deliberate about the
nature of an event generator (Study 3) works
similarly to impeding this ability by introducing
a cognitive load in adults (Study 2) In both
cases usage of a momentum strategy is increased,
supporting both H2 and H3 (see Figure 6).
Table 2. Study 3: A random frame sequence with child participants. Number of up predictions in different parts
of the four block sequence.
Control group
(no load), N = 40
2–7
8–13
(streak of up arrows)
14–18
19–21
(streak of down arrows)
Experimental group
(children), N = 71
M
SD
M
SD
t
p
3.08
3.05
1.61
1.62
3.45
3.85
1.12
1.25
−1.310
−2.691
.195
.009
2.50
0.93
1.22
1.05
2.46
1.24
0.77
0.85
.165
−1.715
.870
.089
JOURNAL OF COGNITIVE PSYCHOLOGY
11
Study 2 procedure was repeated with only one
notable change. In Study 2 subjects were informed
that a random sequence would be observed and
predicted (arrow up and arrow down symbols were
presented), but in Study 4 we informed participants
that they were going to watch/predict a sequence of
throws made by an unnamed basketball player (with
the arrow up symbol replaced by a picture of a successful throw, and the arrow down symbol by a
picture of an unsuccessful throw—see Figure 1 for
details). As in Study 2, only Block 4 answers involving
the longest sequences of similarly valenced events
were used for hypothesis verification.
Figure 6. Differences in momentum strategy usage (in the
Block 4 sequence) among three groups: predicting without
load, predicting under cognitive load, and children’s predictions. The error bar represents 95% Confidence Intervals.
Study 4: a follow-up study of the prediction
task when sequences are perceived as nonrandom
In Studies 1–3 we focused on sequences of random
binary events. Consequently, participants in the
experiments were informed that the sequences of
events they were presented with were random.
However, in response to reviews, we tested
whether the effect of cognitive load found in
Study 2 would also occur when the nature of
events suggests that sequences are non-random.
For this study it was, of course, not possible to use
two equivalent groups to form a 2 × 2 experimental
setup: (random vs. non-random interpretation of the
sequence) × (predictions with no cognitive load vs.
predictions under cognitive load). Nevertheless, we
followed-up Study 2 with Study 4 as a preliminary
test of the above considerations, albeit that the
populations used to conduct Studies 2 and 4 could
hardly be considered equivalent.
Participants
Altogether, 107 Cracow University of Economics
undergraduate finance students (17% male), all
native Polish speakers in the age range 23–35
years (M = 24.17, SD = 1.68), participated in the
study for credit points with no monetary
compensation.
Apparatus and procedure
Because Study 4 was designed as a follow-up to
provide further insight into the Study 2 results, the
Procedure and design: manipulation
All participants performed the whole task wearing
headphones as in Study 2, but only an experimental
group heard sound through the headphones. Participants were randomly assigned to one of two conditions. The control group (n = 50) performed the
task with no additional impediments, while the
experimental group (n = 57) performed Block 4 of
the task under cognitive load (listening to a wildlife
talk on headphones, similarly to Study 2 participants). No relationships were observed between
group membership and gender, χ 2(1) = 1.80, p
= .180, but despite the random assignment procedure an age difference was identified, t(71.23) =
2.66, p = .010, the experimental group (M = 24.54,
SD = 2.14) being slightly older than the control
group (M = 23.74, SD = 0.75). As a manipulation
check for the experimental condition, participants
were informed that shortly after the task they
would participate in a short quiz related to the
story that they had heard (the control group heard
the story separately from the prediction task). On
completing the quiz, experimental group participants made fewer correct choices (M = 3.05, SD =
1.04) than the control group (M = 3.72, SD = 1.31), t
(93.35) = 2.89, p = .005. This indicated that the
manipulation was effective.
Results
The raw distribution of up predictions for the 21
trials (horizontal axis) of Block 4 is presented in
Figure 7, with the black bars representing the experimental group and the white bars representing the
control group. The line graph in the background presents the development of the sequence encountered by participants in Block 4.
Since we aimed to research participants’ behaviour after the impression of a sequence emerged
12
T. TYSZKA ET AL.
Figure 7. Study 4: The raw distribution of up predictions in the control (white, left bar) and experimental (black, right bar)
groups for a deterministic trend perspective (basketball). The line graph in the background illustrates the sequence development over the task. The error bar represents 95% Confidence Intervals.
(Carlson & Shu, 2007), as in Study 2, we considered
the two sequences after the occurrence of a third
univariate stimulus: the predictions for the 8th–
13th stimuli for the sequence of up arrows, and predictions for the 19th–21st stimuli for the sequence of
down arrows.
In Study 2 we investigated whether a cognitive
load would increase momentum strategy usage in
the prediction task based on a random event
sequence. In Study 4 we used the same event
sequence framed as a non-random sequence. Two
effects of this manipulation were expected: (1) an
increase of choices based upon a momentum strategy for non-random framing compared to random
framing and (2) no difference in the number of
choices based upon a momentum strategy
between the control (prediction task without cognitive load) and experimental (prediction task under
cognitive load) groups. In support of the latter
expectation, as shown in Table 3, there was no difference in trend continuation expectations between
the control and experimental groups, t(105) =
1.389, p = .168.
To test the first expected outcome of the manipulation, analysis of covariance (ANCOVA, GLM2) was
performed, with momentum strategy usage as the
dependent variable, and two factors: random vs.
non-random framing of the sequence, and presence
vs. absence of a cognitive load. There was a significant effect of the cognitive load manipulation on
momentum strategy usage, F(1, 171) = 7.55, p
= .006, partial η 2 = .04, but a non-significant effect
for the framing manipulation, F(1, 171) = 0.35, p
= .557, partial η 2 = .002, and no significant interaction, F(1, 171) = 1.04, p = .31, partial η 2 = .006.
Although there was not a significant increase of
choices based upon a momentum strategy for
non-random framing compared to random
framing, the direction of the difference was in
accordance with our expectations (see Figure 8).
The failure to observe a significant effect for
framing is surprising bearing in mind previous
Table 3. Study 4: A non-random frame sequence (basketball sequence). Number of up predictions in different parts of
the four block sequence.
Control group
(no load), N = 50
2–7
8–13
(streak of up arrows)
14–18
19–21
(streak of down arrows)
Experimental group
(cognitive load), N = 57
M
SD
M
SD
t
p
3.28
3.44
1.13
1.58
3.35
3.86
1.11
1.54
−.327
−1.389
.917
.413
2.32
1.02
1.00
0.98
2.18
0.91
0.97
0.89
.760
.595
.716
.442
JOURNAL OF COGNITIVE PSYCHOLOGY
Figure 8. Means for the ANCOVA with momentum strategy
usage as the dependent variable and random vs. nonrandom framing of the sequence, and presence vs.
absence of a cognitive load as factors.
studies showing momentum strategy usage to be
greater for non-random than random sequences
(Tyszka et al., 2008). This may be because our
follow-up study did not implement a full 2 × 2 experimental design and because our experimental
groups were not well matched on some basic sociodemographic characteristics. Here, it should be
noted that the two separate experiments providing
data for the comparison were separated by a gap
of almost one year. Also, although both of the experiments were conducted with Cracow students, they
were from different populations (recruited from
different specialisations and courses), with Experiment 2 involving mostly males (57%) and Experiment 4 involving mostly females (83%).
Discussion
In a series of experiments we confirmed three
hypotheses addressing the question of when
people tend to believe in trend continuation and
when they tend to believe in trend reversal. First,
for a choice reaction time task, we found that
increasing the number of stimuli of the same type
in a sequence leads to shorter response times,
while alternation of stimulus types leads to longer
response times. We interpret this as indicating a
belief in trend continuation: when the number of
stimuli of the same type in a sequence increases,
an individual expects a trend to continue.
Second, in two studies of the prediction of uncertain events we found that the percentage of choices
following a momentum strategy under conditions of
cognitive load was significantly greater than when
13
the same task was performed without a cognitive
load, indicating greater expectations of trend continuation under conditions of cognitive load. When
predicting uncertain events without a cognitive
load an individual has enough time to reflect on
the nature of observed events and can develop an
expectancy of either trend continuation or trend
reversal. On the other hand, when an individual performs a prediction task under cognitive load they
maintain their default expectation of trend continuation. Similarly, we found a higher percentage of
choices following a momentum strategy in a group
of child subjects than in a group of adult subjects.
We interpret this result as indicating an increased
expectancy of trend continuation in participants
who are less able than adults to engage in effective
probabilistic thinking.
Finally, we found that when people assume that a
sequence is non-random, regardless of the availability of cognitive resources, an expectation of
trend continuation prevails.
All the above results fit our general assumption
that people initially form an expectation of trend
continuation, and only when they are able to
engage in deliberative processes can they change
their expectation to trend reversal. Thus, we conclude that when a person is asked to predict uncertain events their initial tendency is to follow a belief
in trend continuation (a positive recency effect
occurs). Expectations of trend reversal (a negative
recency effect) occur only when an individual:
(1) is at a stage of cognitive development which
makes them capable of engaging in effective
probabilistic thinking;
(2) has access to their cognitive resources;
(3) recognises a sequence as random.
One can question whether people from different
expectations depending on the context in which a
sequence’s nature is perceived, or whether the
expectation for trend continuation is the default
expectation regardless of the context, but in some
contexts (when they assume that a sequence is
random), people override the default and form an
expectation of trend reversal. We assume that the
second case holds, and believe that even in roulette,
where advanced players consider a sequence as
random, a novice gambler may expect a positive
recency pattern and follow a momentum strategy.
So, after a streak of seven coups on red they will
predict the eighth coup to be red again. Only
14
T. TYSZKA ET AL.
when they start to speculate about the random character of the sequence and follow a representativeness heuristic will they expect a negative recency
pattern and after a streak of seven coups on red
predict the eighth coup as black.
Naturally, our research does not definitively
support our hypothesis that, regardless of the
context, an expectation for trend continuation is
the default, and then only in some contexts (when
people assume that a sequence is random) individuals override the default and expect trend reversal
instead. However, other observations in the literature support this hypothesis in addition to our evidence. For example, while naïve novice stock
investors tend to expect trend continuation, professional traders tend to put money on trend reversal (De Bondt, 1993; Kubińska & Markiewicz, 2008,
2012). This said, much further research is necessary
to bolster the support for our hypothesis. For
example, it would be interesting to test whether prediction strategies vary depending on whether participants are told that an event generator is a
random mechanism or whether they are told that
a generator’s nature is completely unknown. We
believe that fewer choices would be based upon a
momentum strategy in the first case than in the
second case. One could also perform an experiment
in which initially a participant does not know
whether a presented sequence is random and
during the experiment they form their own hypothesis with respect to randomness or otherwise. We
would expect that participants would initially use a
momentum strategy, but perhaps switch to a contrarian strategy later.
Acknowledgements
All experiments were approved by the Kozminski University Ethics Committee, and the research was conducted
in accordance with APA recommendations after obtaining
participants’ informed consent.
Disclosure statement
No potential conflict of interest was reported by the
authors.
Funding
This work was supported by the National Science Centre
Poland (NCN) under Grant [DEC-2012/04/A/HS6/00614].
References
Altmann, E. M., & Burns, B. D. (2005). Streak biases in
decision making: Data and a memory model. Cognitive
Systems Research, 6(1), 5–16. doi:10.1016/j.cogsys.2004.
09.002
Ayton, P., & Fischer, I. (2004). The hot hand fallacy and the
Gambler’s fallacy: Two faces of subjective randomness.
Memory & Cognition, 32(8), 1369–1378.
Barron, G., & Leider, S. (2010). The role of experience in the
Gambler’s fallacy. Journal of Behavioral Decision Making,
23(1), 117–129. doi:10.1002/bdm.676
Blanchard, T. C., Wilke, A., & Hayden, B. Y. (2014). Hot-hand
bias in rhesus monkeys. Journal of Experimental
Psychology: Animal Learning and Cognition, 40(3), 280–
286. doi:10.1037/xan0000033
Block, R. A., Hancock, P. A., & Zakay, D. (2010). How cognitive load affects duration judgments: A meta-analytic
review. Acta Psychologica, 134(3), 330–343. doi:10.
1016/j.actpsy.2010.03.006
Burns, B. D., & Corpus, B. (2004). Randomness and inductions from streaks: “Gambler’s fallacy versus “Hot
hand”. Psychonomic Bulletin and Review, 11(1), 179–184.
Carlson, K. A., & Shu, S. B. (2007). The rule of three: How the
third event signals the emergence of a streak.
Organizational Behavior and Human Decision Processes,
104(1), 113–121. doi:10.1016/j.obhdp.2007.03.004
Croson, R., & Sundali, J. (2005). The Gambler’s fallacy and
the Hot hand: Empirical data from casinos. Journal of
Risk and Uncertainty, 30(3), 195–209. doi:10.1007/
s11166-005-1153-2
De Bondt, W. F. M. (1993). Betting on trends: Intuitive forecasts of financial risk and return. International Journal of
Forecasting, 9(3), 355–371.
DeSteno, D., Bartlett, M. Y., Braverman, J., & Salovey, P.
(2002). Sex differences in jealousy: Evolutionary mechanism or artifact of measurement? Journal of
Personality and Social Psychology, 83(5), 1103–1116.
doi:10.1037/0022-3514.83.5.1103
Dostoyevsky, F. M. (1998). Letters to wife 1866-1874. In Z.
Podgórzec (Trans.), Aniele, Stróżu mój … : (Listy do żony)
1866-1874. Z dzieł Fiodora Dostojewskiego. London: Puls.
Dostoyevsky, F. M. (1867/2000). The Gambler (novel). In C.
J. Hogarth (Trans.), The Gambler. Retrieved from http://
www.gutenberg.org/files/2197/2197-h/2197-h.htm
Falk, R., & Konold, C. (1997). Making sense of randomness:
Implicit encoding as a basis for judgment. Psychological
Review, 104(2), 301–318. doi:10.1037/0033-295X.104.2.301
Gao, J., Wong-Lin, K., Holmes, P., Simen, P., & Cohen, J. D.
(2009). Sequential effects in two-choice reaction time
tasks: Decomposition and synthesis of mechanisms.
Neural Computation, 21(9), 2407–2436. doi:10.1162/
neco.2009.09-08-866
Gökaydin, D., Navarro, D. J., Ma-Wyatt, A., & Perfors, A.
(2016). The structure of sequential effects. Journal of
Experimental Psychology: General, 145(1), 110–123.
doi:10.1037/xge0000106
Huettel, S. A., Mack, P. B., & McCarthy, G. (2002). Perceiving
patterns in random series: Dynamic processing of
sequence in prefrontal cortex. Nature Neuroscience, 5
(5), 485–490.
JOURNAL OF COGNITIVE PSYCHOLOGY
Inquisit. (2012). (Version 4.0.2). Seattle, WA: Millisecond
Software.
Jarvik, M. E. (1951). Probability learning and a negative
recency effect in the serial anticipation of alternative
symbols. Journal of Experimental Psychology, 41(4),
291–297. doi:10.1037/h0056878
Kareev, Y. (2000). Seven (indeed, plus or minus two) and
the detection of correlations. Psychological Review, 107
(2), 397–402. doi:10.1037/0033-295X.107.2.397
Kirby, N. H. (1972). Sequential effects of serial reaction
time. Journal of Experimental Psychology, 96(1), 32–36.
doi:10.1037/h0033464
Kirby, N. H. (1976). Sequential effects in two-choice reaction time: Automatic facilitation or subjective expectancy? Journal of Experimental Psychology: Human
Perception and Performance, 2(4), 567–577. doi:10.
1037/0096-1523.2.4.567
Kubińska, E., & Markiewicz, Ł. (2008). Analiza decyzji inwestycyjnych uczestników gry giełdowej – skłonności wirtualnych inwestorów, inwestujących wirtualne środki
[Analysis of investment decisions taken by participants
of the game − susceptibility of virtual investors investing virtual money]. Decyzje, 9, 57–82.
Kubińska, E., & Markiewicz, Ł. (2012). Strategie prognostyczne a preferencja ryzyka u inwestorów giełdowych
[Forecasting strategies and propensity towards risk
among stock market investors].
Psychologia
Ekonomiczna/Polish Journal of Economic Psychology, 1,
40–56. doi:10.14659/PJOEP.2012.01.03
Roney, C. J. R., & Trick, L. M. (2003). Grouping and gambling: A gestalt approach to understanding the gambler’s fallacy. Canadian Journal of Experimental
15
Psychology/Revue
canadienne
de
psychologie
expérimentale, 57(2), 69–75. doi:10.1037/h0087414
Scheibehenne, B., & Studer, B. (2014). A hierarchical
Bayesian model of the influence of run length on
sequential predictions. Psychonomic Bulletin & Review,
21(1), 211–217. doi:10.3758/s13423-013-0469-1
Soetens, E., Boer, L. C., & Hueting, J. E. (1985). Expectancy
or automatic facilitation? Separating sequential
effects in two-choice reaction time. Journal of
Experimental Psychology: Human Perception and
Performance, 11(5), 598–616. doi:10.1037/0096-1523.
11.5.598
Sommer, W., Leuthold, H., & Soetens, E. (1999).
Covert signs of expectancy in serial reaction time
tasks revealed by event-related potentials. Perception
& Psychophysics, 61(2), 342–353. doi:10.3758/
BF03206892
Sundali, J., & Croson, R. (2006). Biases in casino betting: The
hot hand and the gambler’s fallacy. Judgment and
Decision Making, 1(1), 1–12.
Tyszka, T., & Sawicki, P. (2011). Affective and cognitive
factors influencing sensitivity to probabilistic information. Risk Analysis, 31(11), 1832–1845. doi:10.1111/j.
1539-6924.2011.01644.x
Tyszka, T., Zielonka, P., Dacey, R., & Sawicki, P. (2008).
Perception of randomness and predicting uncertain
events. Thinking & Reasoning, 14(1), 83–110. doi:10.
1080/13546780701677669
Wilke, A., & Barrett, H. C. (2009). The hot hand phenomenon as a cognitive adaptation to clumped resources.
Evolution and Human Behavior, 30(3), 161–169. doi:10.
1016/j.evolhumbehav.2008.11.004