In: Psychology of Decision Making in Education
Editor: Jeanine A. Elsworth
ISBN 1-60021-933-7
© 2006 Nova Science Publishers, Inc.
Chapter 15
DEVELOPMENTAL TRENDS IN DECISION MAKING:
THE CASE OF THE MONTY HALL DILEMMA
Wim De Neys1
University of Leuven, Tiensestraat 102
3000 Leuven, Belgium
ABSTRACT
The Monty Hall Dilemma (MHD) is a notorious brain-teaser where people have to
decide whether switching to another option in a game is advantageous. Most adults
erroneously believe that chances of winning remain equal or that they should stick to
their original choice. The present study tested the impact of cognitive development on
MHD reasoning to examine possible differences in the nature of the erroneous intuitions.
Twelve to seventeen year old high school students were presented the MHD and selected
one of three responses (switch, stick, or chances equal). Results showed that whereas
maturation decreased adherence to the erroneous “stick with your first pick” belief, the
“chances are equal” belief became more dominant with increasing age. Consistent with
predictions, children who selected the latter response also scored better on a syllogistic
reasoning task. Results further showed that twelve year old eighth graders selected the
correct switching response more frequently than senior high school students. Implications
for popular reasoning and decision making theories are discussed.
1
Correspondence to: Lab Experimentele Psychologie
K.U. Leuven Tiensestraat 102 3000 Leuven Belgium
E-mail: Wim.Deneys@psy.kuleuven.be
Homepage : http://ppw.kuleuven.be/reason/wim/
Fax: +32 16 326099 Tel: +32 16 326143
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Wim De Neys
DEVELOPMENTAL TRENDS IN MONTY HALL DILEMMA REASONING
Imagine you’re the final guest in a TV quiz. Monty, the show host, is asking you to
choose one of three doors. One of the doors conceals a BMW sports car but the two other
doors only contain a bunch of toilet paper. If you choose the right door you will be the proud
owner of the fancy BMW. However, after you finally select one of the doors, host Monty
does not open it immediately. First, he opens one of the doors you did not choose to reveal it
contained toilet paper. Monty now actually offers you the possibility to chance your mind and
pick the other unopened door. What should you do to have most chance of winning the car of
your dreams? Stay with your first choice, switch to the other unopened door, or doesn’t it
matter whether you switch or not?
The above switching problem is known as the “Monty Hall Dilemma” (MHD) after the
host of the American TV show “Let’s make a deal” where it was introduced. Contrary to
most people’s intuition the correct answer is that switching to the other door will actually
increase the chance of winning. The counter-intuitive solution hinges on the crucial fact that
Monty will never open the door concealing the prize, and obviously, he will not open the door
the guest initially picked either. The probability that you initially select the correct door is one
out of three. In this case it would be better not to switch. However, in the other two thirds of
the cases the non-chosen closed door will hide the prize and switching is advantageous.
Hence, switching yields a 2/3 chance of winning.
If you failed to solve the problem correctly you may find some comfort in the fact that
you’re in good company. Empirical studies show that typically less than 10% of educated
adults give the correct switching response (e.g., Burns & Wieth, 2004; Friedman, 1998;
Granberg & Brown, 1995; Krauss & Wang, 2003; Tubau & Alonso, 2003) and even ace
mathematicians do not seem to be immune to MHD errors (e.g., Burns & Wieth; vos Savant,
1997). Most people have the strong intuition that whether they switch or not the probability of
winning remains 50% either way. Research indicates that this powerful intuition is based on
the so called number-of-cases heuristic (“if the number of alternatives is N, then the
probability of each one is 1/N”, see Shimojo & Ichikawa, 1989, and Falk, 1992). Since only
two doors remain people will automatically assign a 50% chance to each door and fail to take
the “knowledgeable host” information into account. The “equal chance” heuristic is so selfevident that it will literally dominate our thinking.
Over the last decades numerous studies have demonstrated that similar intuitive
responses are biasing people’s performance in a wide range of reasoning tasks (e.g., Evans &
Over, 1996; Kahneman, Slovic, & Tversky, 1982). Influential dual process theories of
reasoning and decision making have explained this “rational thinking failure” by positing two
different human reasoning systems (e.g., Epstein, 1994; Evans, 2003; Evans & Over, 1996;
Sloman, 1996; Stanovich & West, 2000). Dual process theories come in many flavours but
generally they assume that a first system (often called the heuristic system) will tend to solve
a problem by relying on intuitions and prior beliefs whereas a second system (often called the
analytic system) allows reasoning according to normative standards. The heuristic default
system is assumed to operate fast and automatically whereas the operations of the analytic
system would be slow and heavily demanding of people’s computational resources. Although
the fast and undemanding heuristics can provide us with useful responses in many daily
situations they can bias reasoning in tasks that require more elaborate, analytic processing
Developmental Trends in Decision Making
57
(e.g., Sloman, 1996; Stanovich & West, 2000; Tversky & Kahneman, 1983). That is, both
systems will sometimes conflict and cue different responses. In these cases the analytic
system will need to override the automatically generated intuitive response. Since the
inhibition of the heuristic intuitions and the computations of the analytic system draw heavily
on people’s limited cognitive resources, most people will be tempted to stick to mere
intuitive reasoning. Therefore, correct analytic reasoning would be characteristic of those
highest in cognitive capacity (Stanovich & West, 2000).
De Neys and Verschueren (2006) recently examined the relation between people’s
cognitive capacities and Monty Hall reasoning. Over 200 participants were presented the
MHD and a test to measure their working memory capacity. Consistent with Stanovich and
West’s (2000) dual process predictions participants who did manage to solve the MHD
correctly were specifically those highest in working memory span. Experimentally burdening
the cognitive resources with a secondary task also decreased the rate of correct switching
responses. These findings supported the basic claim that the analytic override of heuristic
thinking draws on people’s cognitive working memory resources.
Although the dual process framework has been quite influential in the reasoning and
decision making community it has also been criticized severely (e.g., Gigerenzer & Regier,
1996; see commentaries on Evans & Over, 1997, or Stanovich & West, 2000, for an
overview). One of the critiques focuses on the the framework’s postulation of a single
heuristic system that unitarily handles all intuitive processing. It has been argued that
different kinds of heuristics, with a different processing nature, need to be differentiated (e.g.,
De Neys, 2006a; Gigerenzer & Regier, 1996; Moshman, 2000; Newton & Roberts, 2003). De
Neys and Verschueren’s (2006) study pointed to an MHD trend that is specifically interesting
in this respect. Note that one can distinguish different erroneous MHD responses. Whereas
the vast majority of reasoners is biased by the number-of-cases heuristic and beliefs that
switching or sticking does not matter, a small group of reasoners is convinced that they
should stick to the initially selected door to win the prize. These people are biased by the
general belief that when making a decision one should always stick to one’s first choice. Such
a bias has long been noted in responses to multiple-choice exams (e.g., Geiger, 1997).
Gilovich, Medvec, and Chen (1995) clarified that this “stick with your pick” intuition would
be based on an anticipation of regret.
De Neys and Verschueren (2006) found that reasoners in the group of erroneous
responders who selected the “equal chances” response tended to have a slightly higher
working memory span than those who believed that “sticking” was advantageous. Moreover,
the secondary task load specifically boosted the rate of “sticking” responses whereas the
selection of “chances equal” responses was hardly affected. Although the trends were not
significant, De Neys and Verschueren noted that the two types of heuristic beliefs might
indeed have a different processing nature. The number-of-cases heuristic would be based on a
cognitive probability estimation (albeit a simple one) whereas the “stick with your pick”
heuristic would have a more elementary, affective basis. Hence, the “chances are even”
heuristic would be computationally more complex than the “stick with your pick” heuristic.
Therefore, the more elementary, least demanding “stick with your pick” response would be
the preferred answer under conditions of cognitive load.
De Neys and Verschueren (2006) suggested that in their sample of university students the
trends presumably failed to reach significance because the basic number-of-cases
computation would be completely automatic for educated adults. Consequently, the
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Wim De Neys
distinction with the alleged less demanding “sticking” responses would be blurred. The
present study adopts a developmental approach to clarify the issue. The MHD was presented
to twelve to seventeen year old adolescents. A first prediction concerned the selection rates of
the different MHD responses in the different age groups. Younger reasoners have smaller
cognitive resource pools and still lack an important part of the mathematical training that
familiarized university students with fractions and probabilities. Therefore, one can expect
that computation of the number-of-cases heuristic will be less automated and more
demanding in the younger age groups. This should result in a less frequent selection of “equal
chances” responses by younger reasoners. Under the assumption that the “stick with your
pick” heuristic is more basic one predicts that younger reasoners will show a stronger
preference for the “stick” responses. The alleged differential demands of the “stick” and
“equal” heuristics should thus result in a specific developmental trend: The rate of “chances
equal” responses should increase with age whereas the “sticking” responses should decrease.
Given the high computational demands of the correct switching response for adult university
students, correct MHD reasoning in the younger age groups was not expected.
A second prediction concerns the overall relation between young adolescents’ responses
and their cognitive capacity. After the students had solved the MHD they were also presented
a specific syllogistic reasoning task where they had to inhibit automated responses. Previous
studies established that this task is a good marker of children’s general cognitive ability (e.g.,
Kokis, Macpherson, Toplak, West, & Stanovich, 2004). De Neys and Verschueren (2006)
argued that the completely automated computation of the “equal chances” response in their
sample of university students blurred the distinction with the alleged undemanding “sticking”
responses. The less automated and more demanding nature of the number-of-cases heuristic
for the younger reasoners (vs. adults) should show a clearer distinction. Therefore, it is
predicted that youngsters who manage to give the “chances are equal” response will score
better on the syllogisms than those who believe that mere sticking is the best strategy.
The developmental MHD trends will help clarifying possible differences in the nature of
different heuristics. In addition, examining children’s MHD performance will also have
interesting implications for a fundamental controversy concerning the development of
reasoning and decision making itself. The dual process framework and many developmental
theories (e.g., Case, 1985; Inhelder & Piaget, 1985) share the assumption that children’s
reasoning becomes less heuristic and more logical across the lifespan. Hence, traditionally it
has been assumed that analytic thinking simply replaces heuristic thinking with cognitive
maturation. Recent developmental reasoning studies have argued against this so-called
“illusion of replacement” (e.g., Brainerd & Reyna, 2001; Reyna & Ellis, 1994). Klaczynski
(2001), for example, showed that whereas in some tasks (e.g., reasoning about sunk costs) the
heuristic appeal indeed decreased with age, other tasks showed the opposite pattern and
indicated that heuristic reasoning remained constant or increased with age (e.g., denominator
neglect in statistical reasoning). Such findings are already hard to reconcile with the
traditional view. The present MHD study allows a direct validation of the replacement claim
within one and the same task. Based on the traditional view one would simply expect that
both types of erroneous MHD responses will decrease with age and will be replaced by a
higer number of correct responses. Evidence for the predicted differential developmental
“equal chances” and “sticking” trends will cut the ground under the replacement view.
Developmental Trends in Decision Making
59
METHOD
Participants
A total of 132 high school students in grades 8 to 12 participated in the study. Forty-two
students attended eighth grade (mean age = 12.9, SD = .91), 25 attended ninth grade (mean
age = 14.64, SD = .64), 20 students were in grade 10 (mean age = 15.75, SD = 1.02), 16 were
in grade 11 (mean age = 16.56, SD = .63), and 29 in grade 12 (mean age = 17.66, SD = .69).
Students in grades 9/10 and grades 11/12 were collapsed in two age groups for the analyses.
All participants were recruited from the same suburban school with socially mixed
catchments areas. All spoke Dutch as their first language and had no known behavioral
problems or learning difficulties.
Material
Monty Hall Dilemma. Students were presented a version of the MHD based on Krauss
and Wang (2003, see De Neys & Verschueren, 2006). The formulation tried to avoid possible
ambiguities (e.g., the random placement of the prize and duds behind the doors and the
knowledge of the host were explicitly mentioned). As in Tubau and Alonso (2003),
participants could choose between three answer alternatives (a. Stick – b. Switch – c. Chances
are even). The complete problem format is presented bellow:
Suppose you’re on a game show and you’re given the choice of three doors. Behind one
door is the main prize (a car) and behind the other two doors there are dud prizes (a bunch
of toilet paper). The car and the dud prizes are placed randomly behind the doors before the
show. The rules of the game are as follows: After you have chosen a door, the door remains
closed for the time being. The game show host, Monty Hall, who knows what is behind the
doors, then opens one of the two remaining doors which always reveals a dud. After he has
opened one of the doors with a dud, Monty Hall asks the participants whether they want to
stay with their first choice or to switch to the last remaining door. Suppose that you chose
door 1 and the host opens door 3, which has a dud.
The host now asks you whether you want to switch to door 2. What should you do to have
most chance of winning the main prize?
a.
b.
c.
Stick with your first choice, door 1.
Switch to door 2.
It does not matter. Chances are even.
The MHD was presented on computer. Participants were instructed to carefully read the
basic problem information first. When they were finished reading they pressed the ENTERkey and then the question and answer-alternatives (underscored text) appeared on the screen
(other text remained on the screen). Participants typed their response (a, b, or c) on the
keyboard. Instructions stated there were no time limits.
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Wim De Neys
Syllogistic Reasoning Task. The syllogistic reasoning task was based on Sá, West, and
Stanovich (1999). Participants evaluated four syllogisms taken from the work of Markovits
and Nantel (1989) whereby the logical status of the conclusion conflicted with its
believability (e.g., a valid but unbelievable conclusion like ‘All mammals can walk. Whales
are mammals. Therefore, whales can walk’). Thus, in order to give a correct response the
reasoner has to override the tendency to judge the conclusion based on its believabilty.
Individual differences studies (e.g., De Neys, 2006b; Stanovich & West, 2000; Newstead,
Handley, Harley, Wright, Farrelly, 2004) and developmental research (Kokis et al., 2004)
suggest that performance on this task is a good marker of cognitive capacity as measured with
classic intelligence or working memory tests2.
Procedure
Participants were tested in groups during a course break. The MHD was presented before
the syllogistic reasoning task.
RESULTS
Figure 1 shows the selection rates of the three possible MHD responses in the different
age groups. For completeness, rates are shown for every grade separately. Performance of
university students in De Neys and Verschueren’s (2006) study is also plotted for
comparison.
As Figure 1 shows, twelve year old eighth graders have no clear preference for the
“chances are equal” or “stick with your pick” response. Both are selected by about 45% of the
eighth graders. As expected, however, the number-of-cases heuristic is becoming more and
more dominant in the older age groups. The preference for the “stick” response, on the other
hand, decreases with age. To test these trends statistically students in grades 9/10 and 11/12
were collapsed in two age groups. Selection rates in these two groups were compared with
selection rates in grade 8. This resulted in an approximately equal n in the three age groups.
Cochran’s Q test showed that the increase in the proportion of “chances are equal” responses
from grade 8, over grade 9/10, to grade 11/12 was significant, Q(2)= 21.44, p < .0001. The
decrease in “sticking” responses over the three age groups also reached significance, Q(2)=
16.91, p <.001. These findings are consistent with the claim that the “stick with your pick”
belief is based on a more basic and computationally less complex heuristic than the “chances
are equal” belief. Cognitive maturation and mathematical training during the high school
years seem to be especially boosting the number-of-cases heuristic.
2
Time-constraints and school policy prevented the administration of more classic ability tests.
Developmental Trends in Decision Making
61
-
90
Proportion of responses
100
80
70
60
Equal
Stick
Switch
50
40
30
20
10
0
Grade 8
Grade 9 Grade 10 Grade 11 Grade 12 University
Age Level
Figure 1. Proportion of the three possible Monty Hall Dilemma responses in different age groups.
None of the students in grades 9 to 12 managed to give the correct switching response on
the standard MHD. However, the youngest participants in our study, the twelve to thirteen
year old eighth graders, showed about 10% switching responses. The decrease in switching
responses over the three high school age levels also reached significance, Cochran Q(2) = 10,
p < .01. The finding that twelve year olds outperform senior high school (and even university)
students might seem surprising at first sight. However, we believe that the higher rate of
correct responding for the eighth graders can be described as a “Less Is More Effect”. Adults’
reasoning is impeded precisely because of the automated nature of the “50%” response. For
adults the “equal chance” heuristic is so self-evident that it literally dominates their reasoning.
For a twelve year old the conclusion that since two doors remain both will have a 50% chance
of hiding the prize will be less evident and will require active, resource demanding
computation. Bluntly put, for younger reasoners the switching response will be less counterintuitive than for adults. In this sense children thus actually benefit from a lack of knowledge.
This issue is further addressed in the General Discussion.
Students’ performance on the syllogistic reasoning task was taken as a measure of their
general cognitive ability. It was predicted that for less mature reasoners computation of the
“chances are equal” response would be less automated than for adult, university students.
Therefore, in the present younger age group the two response types should show a better
differentiation in terms of cognitive capacity. The “chances are equal” heuristic is assumed to
be more demanding than the “stick with your pick” intuition. Consequently it was predicted
that children with a larger resource pool would be more likely to arrive at the “equal”
response. This hypothesis was confirmed. Overall, students who selected the “equal” response
solved 43% of the syllogisms correctly (mean score = 1.70, SD = 1.27) whereas students who
believed that sticking was beneficial only solved 28% correctly (mean score = 1.12, SD =
1.08), t(125) = 2.50, n1 = 86, n2 = 41, p < .015, d = .48.
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Wim De Neys
The eighth graders were the only students who gave switching responses. For
completeness we compared the syllogistic reasoning performance of eighth graders who gave
correct and incorrect responses. There were no significant differences between the two
groups. Interestingly, contrary to the university students in De Neys and Verschueren’s
(2006) study, eighth graders who selected the switching response actually tended to score
lower on the measure of executive capacities (19% correct conclusions) than the eighth
graders who erroneously selected the “sticking” or “chances equal” responses (28% correct
conclusions). This is consistent with the idea that selecting the switching response for eighth
graders is based on an intuitive rather than an executive resource demanding reasoning
process.
GENERAL DISCUSSION
The present study explored developmental trends in children’s reasoning about the Monty
Hall Dilemma (MHD). By comparing the MHD performance in different age groups, possible
differences in the nature of erroneous intuitions were examined. Previous MHD studies
distinguished at least two types of erroneous MHD reasoning (e.g., De Neys & Verschueren,
2006; Gilovich et al., 1995; Tubau & Alonso, 2003). Participants could indicate they believed
that switching and sticking had both a 50% chance of winning or they could indicate they
believed that sticking to the first chosen door was advantageous. Based on trends in the work
of De Neys and Verschueren it was hypothesized that the “stick with your pick” response
would be more elementary and computationally less demanding than the “chances are equal”
response. Consistent with this claim results showed that maturation over the high school years
boosted the dominance of the “chances are equal” heuristic. Children who selected this
response also scored higher on an indirect measure of cognitive capacity than those who
preferred the “sticking” response. These findings support the idea that the two heuristic
responses differ in computational complexity. For educated adults, however, the computation
of the “50%” heuristic would become fully automated. Indeed, it is precisely the automated,
self-evident, and intuitive nature of this heuristic that is supposed to be impeding university
students’ reasoning.
Results further showed that the youngest reasoners in the study outperformed senior high
school or university students. Twelve and thirteen year old eighth graders showed about 10%
correct responses. It is interesting to note that such apparently counter-intuitive
developmental findings have also been observed with other classic reasoning tasks. Jacobs
and Potenza (1991), for example, studied children’s performance on the notorious base-rate
neglect problems (e.g., the lawyer-engineer problem, Kahneman & Tversky, 1973). In these
tasks, salient, stereotypical information is pitted against more reliable statistical base rate
information. When a person is described as an engineer, adults will conclude it is an engineer
although they were told that the person was drawn from a sample were there were twice as
many lawyers than engineers. The vast majority of educated adults typically neglect the
statistical base rate information. Jacobs and Potenza observed that the base rate neglect
decreased with decreasing age. As Kokis et al. (2004) argued, the finding that younger
children err less frequently on these problems is not surprising because stereotype knowledge
Developmental Trends in Decision Making
63
is typically less developed for children. Since children lack knowledge of many social
stereotypes, they may seem to be using base-rate information more, simply because the
potentially biasing information is unavailable to them. As with the present findings, the point
is not that twelve year olds are actively computing the correct base rate or switching response
but rather that twelve year olds are less tempted by the heuristics that are impeding adults’
reasoning.
The present MHD trends support the rejection of the “illusion of replacement” in the
developmental literature (e.g., Brainerd & Reyna, 2001). Traditionally, reasoning and
decision making researchers have characterized cognitive development as a process whereby
children’s reasoning becomes less heuristic and more in line with logical standards. Recent
developmental studies started cutting the ground under this view (e.g., Klaczynski, 2001).
The present study strengthens this critique by showing that different heuristics can follow a
different developmental path. This further dismisses the “illusion of replacement” as a
simplistic idealization.
The work of De Neys and Verschueren (2006) stipulated that limitations in working
memory resources cannot be neglected as cause of erroneous MHD reasoning. For educated
adults, sidestepping salient intuitions and selecting the correct switching response will be
compromised by a lack of cognitive resources. These findings fitted well with dual process
theories’ characterization of the analytic reasoning system. The present study clarifies that at
the same time one must bear in mind that more extreme resource shortages for less mature or
gifted reasoners might also compromise the automated cueing of erroneous heuristics itself.
Moreover, the developmental findings seriously question the framework’s characterization of
the heuristic system as a unitary system that handles all intuitive processing uniformly (e.g.,
De Neys, Schaeken, & d’Ydewalle, 2005). The present MHD results stress that different
kinds of heuristics, with a different processing nature, need to be differentiated. One and the
same heuristic might be differentially computed by different groups of participants: A
heuristic that is automatically triggered for one group of participants might require active
cognitive resource demanding computations for a younger and/or less gifted group of
reasoners. In line with recent critiques, dual process theories need to take this diversity into
account in order to fully characterize the processing specifications of the two reasoning
systems.
ACKNOWLEDGEMENTS
Wim De Neys is post doctoral fellow of the Fund for Scientific Research-Flanders
(FWO-Vlaanderen). I would like to thank Ellen Gillard and Deborah Everaerts for their help
testing the high school students. Parts of this study were presented at the 27th Annual
Conference of the Cognitive Science Society, Stresa, Italy.
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Wim De Neys
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