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Running Head: MONTY HALL EXPERIMENT 1

Please cite this article as:

Saenen, L., Van Dooren, W., Onghena, P. (2015). A randomized Monty Hall experiment: The positive effect of conditional frequency feedback. Thinking & Reasoning, 21 (2), 176-192. doi:10.1080/13546783.2014.918562

A Randomized Monty Hall Experiment:

The Positive Effect of Conditional Frequency Feedback

Lore Saenena, Wim Van Doorenb, & Patrick Onghenaa

a Methodology of Educational Sciences Research Group, KU Leuven, Leuven, Belgium

b Centre for Instructional Psychology and Technology, KU Leuven, Leuven, Belgium

Correspondence concerning this article can be addressed to Lore Saenen, Methodology of

Educational Sciences Research Group, Andreas Vesaliusstraat 2 - Box 3762, B-3000 Leuven,

Belgium. Phone +32 16 373041. Fax +32 16 326200.

E-mail [email protected]

Acknowledgement: This work was supported by the Concerted Research Action of the KU

Leuven under Grant GOA/12/010.

Total word count: 6425

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Abstract

The Monty Hall dilemma (MHD) is a notorious probability problem with a counterintuitive

solution. There is a strong tendency to stay with the initial choice, despite the fact that

switching doubles the probability of winning. The current randomized experiment investigates

whether feedback in a series of trials improves behavioural performance on the MHD and

increases the level of understanding of the problem. Feedback was either conditional or non-

conditional, and was given either in frequency format or in percentage format. Results show

that people learn to switch most when receiving conditional feedback in frequency format.

However, problem understanding does not improve as a consequence of receiving feedback.

Our study confirms the dissociation between, on the one hand, behavioural performance on

the MHD and, on the other hand, actual understanding of the MHD. We discuss how this

dissociation can be understood.

Keywords: Monty Hall dilemma, probability, conditional feedback, frequencies,

decision making

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The Monty Hall dilemma (MHD) is a notoriously difficult conditional probability

problem with a counterintuitive solution (e.g., Baratgin & Politzer, 2010; Howard, Lambdin,

& Datteri, 2007). The classic version of the MHD involves a contestant who is asked to pick

one of three identical doors. Behind one of the doors, there is a valuable prize, usually a car.

The other two doors conceal goats. Once the contestant picks a door, the host, who knows the

location of the prize, opens a door not selected by the contestant, always revealing a goat. The

dilemma then offered to the contestant is whether he stays with his initial choice, or wants to

switch to the other remaining door. The optimal solution to the MHD is to switch, as

switching doors yields a probability to win the prize that is twice as large as the probability

obtained by staying. Nevertheless, there is a strong (cross-cultural) tendency to stay with the

initial decision (Granberg, 1999; Granberg & Brown, 1995; Granberg & Dorr, 1998).

Various studies have shown that this tendency is malleable. First, there is strong

evidence that repeated experience with the MHD leads to increased switching rates across

trials (Franco-Watkins, Derks, & Dougherty, 2003; Granberg & Brown, 1995; Granberg &

Dorr, 1998; Herbranson & Schroeder, 2010; Klein, Evans, Schultz, & Beran, 2013; Petrocelli

& Harris, 2011; Slembeck & Tyran, 2004; Tubau & Alonso, 2003). Second, also the

numerical format in which the MHD is presented to participants seems to influence

participants’ performances. Aaron and Spivey-Knowlton (1998) showed that a MHD

presentation in terms of natural frequencies resulted in higher switching rates than the

situation in which the MHD is presented in terms of probabilities. Although the difference in

switching rates was not statistically significant, the difference in the answers on the

mathematical questions did reach statistical significance: Participants who solved the MHD

stated in natural frequencies, reported significantly more often the correct posterior winning

probabilities compared to the participants who were asked to solve the MHD stated in

probabilities (proportions or percentages). Research of Tubau (2008) confirmed these results:

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Participants who completed the MHD in natural frequencies, showed both statistically

significant higher switching rates and more correct posterior probability judgements than

participants who completed the MHD presented in terms of relative frequencies. Also Krauss

and Wang (2003) included a natural frequencies manipulation in their experiment.

Participants were asked in how many of the three possible MHD situations a participant

would win the game when staying and when switching. When this manipulation was

combined with the manipulation in which the participant was asked to imagine that (s)he was

the host of the MHD, switching behaviour was facilitated.

The critical question that arises when behavioural performance on the MHD improves,

is whether this performance reflects actual understanding of the problem. More specifically, a

decision to switch is not necessarily the consequence of correct probabilistic reasoning. This

proposition is confirmed in previous research: Franco-Watkins, Derks, and Dougherty (2003),

Stibel, Dror, and Ben-Zeev (2009), and Tubau and Alonso (2003) demonstrated that there is a

clear dissociation between choice behaviour on the MHD and the actual understanding of the

problem’s underlying probabilities. Thus, although research demonstrates ways to influence

the use of the optimal strategy to solve the MHD, people still fail to grasp the reason why

switching is beneficial.

The failure to detect the correct underlying probabilities of the MHD can be explained

by the cognitive process of updating (Baratgin & Politzer, 2010) and by a distorted memory

for decision/outcome frequencies when a series of trials is involved (Petrocelli & Harris,

2011). Baratgin and Politzer (2010) showed that when people solve the MHD, they often

erroneously estimate the posterior probability for winning when staying (and switching) as

being 1/2 because they engage in what the authors call ‘updating’, by which they mean that

participants believe that the new situation, with only two remaining doors, is independent

from the conditional information received about the opened non-winning door. As a

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consequence, the posterior winning probabilities for staying and switching are incorrectly

judged equally as 1/2 (Baratgin & Politzer, 2010). For a correct probability understanding,

however, one should engage in what the authors call ‘focusing’, by which they mean that the

situation in which the dilemma occurs is considered as not independent from the situation

before the actual dilemma took place. Thus, when focusing, one takes into account the

conditional information (Baratgin & Politzer, 2010).

Next, Petrocelli and Harris (2011) pointed to the distorted memory for

decision/outcome frequencies that participants have after solving successive MHD trials. In

their research, participants completed several trials of the MHD. Afterwards, they were asked

to estimate how often they had won and lost, depending on the used strategy. Above all,

participants estimated their switch losses being much higher as compared to the actual number

of switch losses during the experiment. Of course, it is difficult to keep in mind the number of

winning and losing trials depending on the used strategy without having any external help. It

requires that the working memory of participants is constantly updating these frequencies

while solving the MHD. The important role of working memory capacity on MHD

performances has indeed been demonstrated, both correlationally and experimentally (De

Neys, 2005; De Neys & Verschueren, 2006).

In sum, when people would solve the MHD repeatedly, we assume that correct

registration of the number of times a participant wins or loses in relation to the choice to

switch or to stay may be extremely difficult because it demands too much from working

memory. There exist several studies in which participants completed successive trials of the

MHD and received feedback about their performances: After each trial, the participant was

informed about whether (s)he won or lost the trial. The results of these studies showed that

participants indeed learned to switch across successive trials, but still did not respond

optimally, which means that they still did not switch on all trials (Herbranson & Schroeder,

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2010; Klein et al., 2013; Mazur & Kahlbaugh, 2012; Slembeck & Tyran, 2004). Although the

influence of feedback was examined in these studies, according to our knowledge no study so

far systematically investigated whether cumulative, constantly updated feedback provided to

the participants during the game, influences performances on the MHD and more generally

improves participants’ understanding of the underlying probabilities.

In the current paper, we will investigate the effect of cumulative, constantly updated,

and available feedback during an experiment in which participants are asked to complete

multiple trials of the MHD. The nature of feedback in the MHD is systematically manipulated

in two different ways.

First, we will provide either conditional or non-conditional feedback that is constantly

updated and shown to the participant. We expect that participants will perform better when

receiving conditional feedback compared to non-conditional feedback, because conditional

feedback will prevent that participants’ memory becomes distorted for decision-outcome

frequencies while this distortion is still possible in the case of unconditional feedback. Thus,

when conditional feedback is provided to the participants while solving multiple trials, their

working memory will be less burdened in comparison with non-conditional feedback.

Second, the feedback will be provided in either a percentage or a frequency format.

This latter manipulation is based on research indicating that people perform better on

conditional probability problems when the problem is stated in frequencies rather than in

percentages or probabilities (e.g., Gigerenzer, 1991, 1994; Gigerenzer & Hoffrage, 1995;

Moro, Bodanza, & Freidin, 2011). As already mentioned above, this has also been

demonstrated for the MHD by the research of Aaron and Spivey-Knowlton (1998), Krauss

and Wang (2003), and Tubau (2008). However, in these studies the numerical format of the

problem presentation of the MHD was manipulated, whereas in our study we will investigate

whether the numerical presentation of the feedback makes a difference in how people solve

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the MHD. We expect that participants perform better on the MHD when feedback is given in

frequency format rather than in percentage format, because humans’ cognitive systems are

considered to be more suitable for natural frequencies, but less suitable for proportions and/or

percentages (see Cosmides & Tooby, 1996). The latter statement does not necessarily imply

that the frequentist format is the causal explanation for the reported research results on the

MHD (Aaron & Spivey-Knowlton, 1998; Krauss & Wang, 2003; Tubau, 2008). An

alternative plausible explanation is that natural frequencies favour the correct focusing

interpretation of the MHD, instead of the misleading updating interpretation (for an overview,

see Baratgin, 2009).

Notice that the previously mentioned expectations imply that we expect participants to

perform best when they receive conditional feedback presented in frequencies, because they

would benefit from both manipulations.

To the best of our knowledge, no previous study has systematically investigated the

effect of cumulative, constantly updated, and available feedback, nor has it been investigated

which format of such feedback (i.e., the numerical and conditional format in which the

feedback is provided) is most effective. In our experiment, we will investigate the influence of

cumulative, constantly updated, and available feedback on people’s performances on the

MHD. More specifically, we will investigate whether such feedback leads to both more

optimal behavioural responses on the MHD and an increased understanding of the MHD.

Methods

Participants and Design

Seventy-seven (under)graduate students of the University of Leuven (Belgium)

volunteered to participate in the experiment, in return for a small financial reward. Nine

participants were eliminated due to prior familiarity with the MHD, leaving data of 68

participants (18 males, 50 females; Mage = 21.60, SDage = 4.36) used for further analyses.

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The participants were randomly assigned to one of four treatments created by a 2 × 2

between-subjects design. Both independent variables considered the nature of the constantly

updated feedback on the MHD. The first independent variable was the conditional format of

the feedback (conditional vs. non-conditional). Conditional feedback was given by providing

information about the number of wins and losses for staying and switching separately. When

non-conditional feedback was given, only information about the number of times the

participant won or lost was provided. The second independent variable was the numerical

format of the feedback (in frequencies vs. in percentages). This led to the following four

treatments: non-conditional feedback in frequencies (NCF), non-conditional feedback in

percentages (NCP), conditional feedback in frequencies (CF), and conditional feedback in

percentages (CP). Figure 1 shows the feedback as given to the participants in the different

treatments.

(Figure 1 about here)

Two dependent variables were included in the study. The first dependent variable was

the behavioural response on the MHD (staying vs. switching). The second dependent variable

was the probability judgement of winning for the final decision that was made by the

participant (staying vs. switching). This means that when the participant chose to stay with the

initial choice, (s)he was asked to give the posterior winning probability of staying, while the

choice to switch resulted in a question about the posterior winning probability of switching.

The study protocol was approved by the Ethical Committee of the University of

Leuven.

Materials and Procedure

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The MHD was completed repeatedly (80 times) and feedback on the performance (see

Figure 1) was constantly updated on a computer screen that was placed in an angle of 45

degrees and at a distance of approximately 45 centimetres at the left front of the participant.

Participants came individually to the laboratory. Upon arrival, the participant took

place in front of the experimenter. Three piles of 80 cards laid face down on the table. For

each trial of the MHD, the three cards that laid on top of the piles were used. These three

cards always included two black cards and one red (winning) card. The participant received

the instructions for the MHD task on a sheet of paper (see the Appendix). After reading the

instructions, the experimenter summarized the procedure to the participant. The participant

was told that (s)he would participate in a game of chance that consisted of 80 trials and that

for each trial, the aim was to locate the red winning card. For each trial, the participant made

an initial choice by mentioning a number from one to three (number one being the card on the

left, number three being the card on the right). Next, the experimenter turned face up another

card than the initial choice of the participant. This card was always black, while the

participant knew that the experimenter was aware of the location of the red winning card.

Next, the participant was asked to decide to either stay with the initial choice or to switch to

the other remaining card. After this final choice, the other two remaining cards were turned

faced up so that the participant could see whether (s)he had won or lost this trial (see Figure

2). Before starting the next trial, the information on the computer screen was updated.


Two practice trials of the MHD were included in order to make sure the participant

understood the procedure of the game. These practice trials were analogous to the

experimental trials. After these two practice trials, the MHD was completed 80 times. The

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reason to include 80 trials was both practical and theoretical. From a practical point of view, a

pilot study showed that 80 trials were easily completed in a one hour experimental session.

From a theoretical point of view, 80 trials is considered to be sufficient to investigate the

influence of feedback as the majority of learning is known to happen in approximately the

first 50 trials (Herbranson & Schroeder, 2010). After every 10 trials, the participant was

explicitly encouraged by the experimenter to look at the feedback on the computer screen (see

Figure 1). The participant could look at the computer screen more often if (s)he wanted to,

because the feedback on the computer screen was constantly updated and remained available

during the entire experiment.

During trial 1, trial 40, and trial 80, after the experimenter revealed a non-winning

card, the participant was also asked to make a probability judgement with respect to winning,

given the final decision that (s)he had made. This was done in order to detect whether his/her

understanding of the underlying probabilities of the MHD improved.

The participant was motivated to perform well by receiving the instruction that for

every winning trial (s)he would gain 10 eurocents. At the end of the experiment, the

participant received the actual amount of money (s)he had obtained during the experiment.

Results

Choice Behaviour

Frequencies of switching were counted for every block of 20 trials. A repeated

measures analysis of variance was performed with one within-subjects variable ‘block’ (four

blocks of 20 trials) and two between-subjects variables ‘conditional format’ (conditional vs.

non-conditional feedback) and ‘numerical format’ (feedback in frequencies vs. percentages)

and with frequency of switching per block as the dependent variable. Mauchly’s test indicated

that the assumption of sphericity was violated, ²(5) = 34.13, p < .01. Therefore, corrections

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for the degrees of freedom using the Greenhouse-Geisser estimates of sphericity were used.

The significance level was determined at α = .05.

As expected, after having experienced more trials of the MHD, participants switched

more often (first block: M = 7.57, SD = 5.08; second block: M = 9.31, SD = 5.73; third block:

M = 11.03, SD = 6.27; fourth block: M = 11.59, SD = 6.69), F(2.16, 138.04) = 26.90, p < .01,

Cohen’s f = .65, which indicated a main learning effect. In line with our expectations, an

interaction effect between conditional format and numerical format was also observed, F(1,

64) = 5.69, p = .02, Cohen’s f = .30. Post-hoc pairwise contrasts between the four treatments

were tested across blocks, using Tukey’s Honestly Significant Difference (HSD) test. This

test revealed only one statistically significant pairwise contrast: Participants in the conditional

feedback in frequencies condition (CF) switched significantly more often than participants in

the non-conditional feedback in frequencies condition (NCF), HSD = 23.24, p < .01. The

other five pairwise contrasts did not reach statistical significance. See Table 1 for an overview

of all post-hoc pairwise contrasts.

(Table 1 about here)

The above mentioned main effect of block and the two-way interaction effect between

conditional format and numerical format were further qualified by a three-way (block by

conditional format by numerical format) interaction effect, F(2.16, 138.04) = 8.92, p < .01,

Cohen’s f = .37. Figure 3 clarifies this three-way interaction effect by showing how the two-

way interaction effect (conditional format by numerical format) differs from block to block.


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To interpret this three-way interaction effect, we examined how the two-way

interaction between conditional format and numerical format differed for each block. A

Bonferroni-Holm correction was used to control the Familywise Error Rate at 5%. For blocks

1 and 2, the interaction between conditional format and numerical format was not statistically

significant, F(1, 64) = 0.69, p = .41, Cohen’s f = .11, and F(1, 64) = 0.85, p = .36, Cohen’s f =

.11, respectively. This interaction, however, reached statistical significance in blocks 3 and 4,

F(1, 64) = 8.77, p < .01, Cohen’s f = .37, and F(1, 64) = 13.24, p < .01, Cohen’s f = .45,

respectively. These two-way interactions reveal that participants in the last 40 trials switched

more often when receiving conditional feedback in frequencies than when receiving non-

conditional feedback in frequencies, while the switch rates for participants receiving either

conditional or non-conditional feedback in percentages fell in between.

Probability Judgements

During trial 1, trial 40, and trial 80, after the experimenter revealed a non-winning

card and the participant made a final choice to either stay with his/her initial decision or to

switch, (s)he was asked to estimate the probability of winning. The probability judgements of

the participants are summarized in Table 2. Answers were classified as correct when the

participant answered “1/3” and had chosen to stay with the initial choice, or “2/3” when the

participant switched to the other remaining card. All other answers were classified as

incorrect and were further subdivided in the typical “1/2” answer category, or the category

“other incorrect answers”.

(Table 2 about here)

The results show that in all conditions and for all three investigated trials, the majority

of participants judged the posterior probability associated with the winning card incorrectly as

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1/2. Thus, most participants’ understanding of the probabilities underlying the MHD did not

improve through the experiment. Only two participants – one participant in the non-

conditional feedback in percentages condition (NCP) and one participant in the conditional

feedback in frequencies condition (CF) – judged the posterior winning probabilities correctly

for all three investigated trials.

When considering the category of “other” incorrect answers, the following answers

were reported: Besides one nonsensical answer, three other students gave an answer that

possibly also expressed some understanding that switching was advantageous, but the

justifications could not unequivocally be considered correct.

Three Fisher-Freeman-Halton Tests were performed – one for each trial that involved

the probability judgement (i.e., trial 1, trial 40, and trial 80) – to test for statistically

significant differences in the number of times a particular probability judgement was given

between the four treatments. A Bonferroni-Holm correction was used to keep the Familywise

Error Rate under control at 5%. The results of these tests revealed that the type of probability

judgement (i.e., correct, “1/2”, or “other”) participants gave, was not systematically related to

the treatment participants were assigned to (trial 1: p = .38; trial 40: p = .54; and trial 80: p

= .64).

Discussion

The present study focused on the notorious MHD. The MHD was solved repeatedly

and the constantly updated feedback on the outcomes was manipulated in two different ways.

First, the feedback was either conditional or non-conditional. Second, the feedback was given

either in percentages or in frequencies. To the best of our knowledge, this is the first study

that directly addressed the differential effect of these forms of feedback on the MHD

performances of participants in a controlled and randomized experimental setting. We

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investigated the effect of feedback on both the behavioural level and the level of

understanding.

As in previous MHD studies, participants of our experiment learned to switch by

solving the MHD repeatedly (Franco-Watkins et al., 2003; Granberg & Brown, 1995;

Granberg & Dorr, 1998; Herbranson & Schroeder, 2010; Klein et al., 2013; Petrocelli &

Harris, 2011; Slembeck & Tyran, 2004; Tubau & Alsono, 2003). Our results showed that

behavioural performance on the MHD is related to the feedback that is provided to the

participants. More specifically and in line with our expectations, at the behavioural level

participants learned most when they received conditional feedback in frequencies. In order to

keep the load on participants’ working memory capacity low, we provided the participants

with constantly updated conditional feedback. The conditional format of the feedback most

likely helped to avoid the distortion of participants’ memory for decision/outcome frequencies

(see Petrocelli & Harris, 2011). Furthermore, the results of the behavioural performances on

the MHD show that when giving conditional feedback, the numerical representation of

feedback is important. Previous research on the MHD (Aaron & Spivey-Knowlton, 1998;

Krauss & Wang, 2003; Tubau, 2008) and other conditional probability problems (e.g.,

Gigerenzer, 1991, 1994; Gigerenzer & Hoffrage, 1995; Moro et al., 2011) have shown that

participants perform better when the problems are presented in frequencies rather than in

percentages or proportions. In our study, we did not replicate this overall superiority of the

frequency manipulation as such.

Although the results of participants’ choices on the MHD indicated that cumulative,

constantly updated, and available feedback enhances participants’ behavioural performances

on the MHD, especially when this feedback is provided in conditional frequency format, the

results of the probability judgements clearly demonstrate that feedback does not help to

improve participants’ level of understanding of the underlying probabilities of the MHD. The

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majority of participants still judged their posterior winning probability as 1/2. In sum, our

results provide further support for the earlier demonstrated dissociation between behavioural

performance on the MHD and the level of actual problem understanding (Franco-Watkins et

al., 2003; Stibel, Dror, & Ben-Zeev, 2009; Tubau & Alonso, 2003).

The question that this dissociation raises is what exactly is going on in terms of

participants’ underlying reasoning processes that can explain why understanding does not

improve.

A first possibility that can explain why the posterior probability judgements were

rarely correct in our research is participants’ probability understanding in general: Previous

MHD research showed that participants sometimes even have difficulties with understanding

the initial probabilities (Tubau, 2008), thus, before a door is eliminated and the actual

dilemma takes place. Of course, in that case it is very unlikely that a participant would give a

correct winning posterior probability judgement. Note that we did not manipulate the way in

which the posterior probability judgement questions were posed: We always formulated these

questions in terms of probabilities (i.e., “What is your chance of winning?”). However,

formulating the posterior probability judgement questions in terms of frequencies could have

positively influenced participants’ correct answer rates. Therefore, future research should

investigate whether this is the case.

A second explanation for the lack of problem understanding in our research refers to

the structure of the MHD itself, because previous research already showed that people do not

take into account the fact that the behaviour of the host is dependent on the initial choice of

the participant. Thus, participants fail to take into account the structure of the MHD (Burns &

Wieth, 2003, 2004; Idson et al., 2004; Krauss & Wang, 2003; Tubau & Alonso, 2003).

Feedback does not influence the structure of the problem, and does not make the underlying

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structure of the MHD more explicit,. This again explains why feedback may not have helped

to improve participants’ understanding of the MHD and its underlying probabilities.

Third, the finding that the majority of our participants overwhelmingly estimated the

posterior winning probabilities as 1/2 can be explained by two related cognitive processes,

being partitioning-editing-counting (Fox & Levav, 2004) and updating (Baratgin, 2009;

Baratgin & Politzer, 2010). When the actual dilemma of the MHD occurs, one door has been

eliminated by the host and only two doors remain. At this point, a participant may involve in

the cognitive strategy of partitioning-editing-counting to deal with the probabilities, as

described by Fox and Levav (2004). Partitioning-editing-counting in the MHD means that in

order to calculate the posterior winning probabilities, the number of prizes (i.e., one) is

divided by the number of remaining options (i.e., two). As a consequence, the posterior

winning probability for both staying and switching are considered to be equal (i.e., 1/2). The

partition-edit-count strategy (Fox & Levav, 2004) is a specific example of the cognitive

strategy of updating (Baratgin, 2009; Baratgin & Politzer, 2010) which we already described

earlier. Thus, updating is a valid alternative explanation for the majority of 1/2 answers which

were obtained on the posterior probability judgement questions of our experiment.

Besides the fact that after our study, there remain questions about the reasons for the

dissociation between behavioural performance on the MHD and the level of understanding of

the problem, our study has some other limitations. One limitation of the current study

concerns the physical implementation of the experiment. First, the posterior winning

probabilities for both staying and switching could differ from the theoretical 1/3 and 2/3

probabilities. Although the completion of 80 trials resulted in posterior winning probabilities

of approximately 1/3 and 2/3, the posterior winning probabilities could deviate substantially

for smaller subsets of the 80 trials, for example for the first 10 trials. Furthermore, the

physical implementation led to (rather small) differences in posterior winning probabilities

16


between participants. Second, because of the physical implementation of the experiment, it

was not possible to use eye tracking methodology in order to control for participants’ number

of fixations and fixation lengths on the screen that provided the feedback. Despite these

methodological issues, we were convinced of the strength of a physical implementation of the

experiment because computerized versions may lead participants to believe that the computer

has been manipulated to change the winning card after the initial choice has been made,

whereas manipulating the outcomes of an MHD experiment is not possible in the case of a

physical implementation. Although an effect of the physical versus computerized

implementation on participants’ MHD performances is speculative because no prior research

has been done on this topic, we decided to rule out the degree of participants’ suspiciousness

about the experiment by choosing for a physical implementation.

Another limitation of the current research is that posterior probability judgements of

participants were included as a dependent variable, but these mathematical answers do not

provide further insight in the reasoning process that led to the (in)correct answers.

Furthermore, it should be mentioned that correct probability judgements do not necessarily

reflect a real understanding of the MHD problem. For example, a participant might

accidentally give a correct probability judgement on the MHD when involving in the

cognitive process of updating (see Baratgin, 2009). In order to get a better grip on the

reasoning processes participants involve in when confronted with the MHD, further research

should consider to include open questions. Mixed methods research in which both quantitative

and qualitative dependent variables are included may lead us to a deeper understanding of

how people reason when confronted with the MHD (see Heyvaert, Maes, & Onghena, 2013).

More specifically, open questions (i.e., qualitative component) may show the reasoning

process a participant is involved in to give a particular probability judgement (i.e.,

quantitative component).

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In sum, this paper investigated how and which kind of cumulative, constantly updated,

and available feedback influenced people’s performance on the MHD, both on the

behavioural and level of understanding. First, the results showed that participants’ switch

rates increased most when they received conditional feedback in frequency format, which was

in line with our hypotheses. Second, the findings in this paper provided additional evidence

for the dissociation between behavioural performances on the MHD and actual understanding

of the problem (Franco-Watkins et al., 2003; Stibel et al., 2009; Tubau & Alonso, 2003).

Although cumulative, constantly updated, and available feedback, especially in conditional

frequency format, helped the participants to detect that switching was the optimal response on

the MHD, the feedback did not help the participants to improve their actual understanding of

why this is the case.

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Appendix1

Instructions

As you can see, three piles of cards lie on the table. We will always play with the three cards

that lie on top of the piles. Out of these three cards, one card is red and is the “winning card”.

The other two cards are black and are “losing cards”. Your aim is to collect as many winning

cards as possible. For each “winning card” you indicate correctly, you will be rewarded with €

0.10.

Each trial will take place in an identical way, as described below.

For each trial, you will be asked to pick one of the three cards. Make your choice for a card

explicit by mentioning the number of the card to the experimenter (imagine: you choose card

1). The experimenter is aware of the location of the red winning card. After you have

mentioned your choice to the experimenter, the experimenter will reveal one of the black non-

winning cards. This will never be the card that you have initially chosen. When the

experimenter has the choice between two black cards, he will randomly pick one of both

black cards (imagine: card 2 is one of the losing cards and will be turned face up). Next, the

experimenter will give you the opportunity to either stay with your initial choice (card 1), or

to switch to the other remaining card (card 3). Make your final choice explicit by again

mentioning the number of the card to the experimenter. After you have communicated your

final choice for a card, the experimenter will turn face up the two remaining cards in order to

reveal whether you have won or lost the trial.

Each trial you win will be rewarded with € 0.10. Losing a trial will result in nothing (you will

never lose money).

1 Original instructions were in Dutch.

23


You will play this game 80 times. After each trial, you will receive feedback about your

previous performances on the computer screen. Have a look at this screen on a regular basis.

Before starting the game, you will be playing two practice trials.

If you still have any questions about the game after reading the above instructions, please do

not hesitate to communicate your questions to the experimenter.

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Table 1Post-hoc pairwise contrasts between the four treatments using Tukey’s Honestly Significant Difference (HSD)Condition 1 Condition 2 Mean

difference (1 - 2)p

CF CP 13.29 .23

CF NCF 23.24 < .01

CF NCP 13.24 .23

CP NCF 9.94 .48

CP NCP -0.06 1.00

NCF NCP -10.00 .47

Note. CF = conditional feedback in frequencies condition; CP = conditional feedback in percentages condition; NCF = non-conditional feedback in frequencies condition; NCP = non-conditional feedback in percentages condition.

25


Table 2Frequencies of participants per MHD probability judgement category

Probability judgementCorrect “1/2” “other”

Trial 1 NCF 0 16 1NCP 2 15 0CF 2 15 0CP 0 17 0



Note. NCF = non-conditional feedback in frequencies condition; NCP = non-conditional feedback in percentages condition; CF = conditional feedback in frequencies condition; CP = conditional feedback in percentages condition.

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27


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Figure 3. Mean frequencies of switching in the different feedback formats for each block of 20 trials (block 1 = trial 1-20; block 2 = trial 21-40; block 3 = trial 41-60; block 4 = trial 61-80). Error bars represent standard errors. NCP = non-conditional feedback in percentages condition; NCF = non-conditional feedback in frequencies condition; CP = conditional feedback in percentages condition; CF = conditional





























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