Overcoming illusory inferences in a probabilistic counterintuitive problem: the role of explicit representations

Why Humans Fail in Solving the Monty Hall Dilemma: A Systematic Review

The Monty Hall dilemma (MHD) is a difficult brain teaser. We present a systematic review of literature published between January 2000 and February 2018 addressing why humans systematically fail to react optimally to the MHD or fail to understand it. Based on a sequential analysis of the phases in the MHD, we first review causes in each of these phases that may prohibit humans to react optimally and to fully understand the problem. Next, we address the question whether humans' performance, in terms of choice behaviour and (probability) understanding, can be improved. Finally, we discuss individual differences related to people's suboptimal performance. This review provides novel insights by means of its holistic approach of the MHD: At each phase, there are reasons to expect that people respond suboptimally. Given that the occurrence of only one cause is sufficient, it is not surprising that suboptimal responses are so widespread and people rarely understand the MHD.

Testing the limits of optimality: the effect of base rates in the Monty Hall dilemma

The Monty Hall dilemma is a probability puzzle in which a player tries to guess which of three doors conceals a desirable prize. After an initial selection, one of the nonchosen doors is opened, revealing that it is not a winner, and the player is given the choice of staying with the initial selection or switching to the other remaining door. Pigeons and humans were tested on two variants of the Monty Hall dilemma, in which one of the three doors had either a higher or a lower chance of containing the prize than did the other two options. The optimal strategy in both cases was to initially choose the lowest-probability door available and then switch away from it. Whereas pigeons learned to approximate the optimal strategy, humans failed to do so on both accounts: They did not show a preference for low-probability options, and they did not consistently switch. An analysis of performance over the course of training indicated that pigeons learned to perform a sequence of responses on each trial, and that sequence was one that yielded the highest possible rate of reinforcement. Humans, in contrast, continued to vary their responses throughout the experiment, possibly in search of a more complex strategy that would exceed the maximum possible win rate.

Stimulus-preceding negativity represents a conservative response tendency

Humans tend to be conservative and typically will retain their initial decision even if an option to change is provided. We investigated whether the stimulus-preceding negativity (SPN), an event-related potential associated with the affective-motivational anticipation of feedback in gambling tasks, represents the strong response tendency to retain an initial decision. We compared SPNs in three different card-gambling tasks wherein the participants were given the opportunity to change their initial decision after they chose one of three cards. In two of these tasks, the winning probability was equiprobable (1/3 and 1/2, respectively) whether or not the participants changed their initial decision. However, in the Monty Hall dilemma task, changing the initial decision stochastically doubled the probability of winning (2/3) compared with retaining (1/3). In this counterintuitive probabilistic dilemma task, after the participant chose an option among three cards, a nonreward (losing) option is revealed. Then, the participants are offered a chance to change their mind and asked to make their final decision: to retain their initial choice or change to the alternate option. In all tasks, maintenance of previous behaviors was observed, although the rate of retaining earlier choices tended to be lower in the Monty Hall dilemma task than in the other two tasks. The SPNs were larger on retain trials than on change trials irrespective of task. These results suggest that underlying brain activities associated with the strong tendency to retain the initial decision can be observed by the SPN and thus it reflects expectancy of outcomes in terms of self-chosen behaviors.

Reasoning and choice in the Monty Hall Dilemma (MHD): implications for improving Bayesian reasoning

The Monty Hall Dilemma (MHD) is a two-step decision problem involving counterintuitive conditional probabilities. The first choice is made among three equally probable options, whereas the second choice takes place after the elimination of one of the non-selected options which does not hide the prize. Differing from most Bayesian problems, statistical information in the MHD has to be inferred, either by learning outcome probabilities or by reasoning from the presented sequence of events. This often leads to suboptimal decisions and erroneous probability judgments. Specifically, decision makers commonly develop a wrong intuition that final probabilities are equally distributed, together with a preference for their first choice. Several studies have shown that repeated practice enhances sensitivity to the different reward probabilities, but does not facilitate correct Bayesian reasoning. However, modest improvements in probability judgments have been observed after guided explanations. To explain these dissociations, the present review focuses on two types of causes producing the observed biases: Emotional-based choice biases and cognitive limitations in understanding probabilistic information. Among the latter, we identify a crucial cause for the universal difficulty in overcoming the equiprobability illusion: Incomplete representation of prior and conditional probabilities. We conclude that repeated practice and/or high incentives can be effective for overcoming choice biases, but promoting an adequate partitioning of possibilities seems to be necessary for overcoming cognitive illusions and improving Bayesian reasoning.

The equiprobability bias from a mathematical and psychological perspective

The equiprobability bias (EB) is a tendency to believe that every process in which randomness is involved corresponds to a fair distribution, with equal probabilities for any possible outcome. The EB is known to affect both children and adults, and to increase with probability education. Because it results in probability errors resistant to pedagogical interventions, it has been described as a deep misconception about randomness: the erroneous belief that randomness implies uniformity. In the present paper, we show that the EB is actually not the result of a conceptual error about the definition of randomness. On the contrary, the mathematical theory of randomness does imply uniformity. However, the EB is still a bias, because people tend to assume uniformity even in the case of events that are not random. The pervasiveness of the EB reveals a paradox: The combination of random processes is not necessarily random. The link between the EB and this paradox is discussed, and suggestions are made regarding educational design to overcome difficulties encountered by students as a consequence of the EB.

The Monty Hall dilemma in pigeons: effect of investment in initial choice

In the Monty Hall dilemma, humans are initially given a choice among three alternatives, one of which has a hidden prize. After choosing, but before it is revealed whether they have won the prize, they are shown that one of the remaining alternatives does not have the prize. They are then asked whether they want to stay with their original choice or switch to the remaining alternative. Although switching results in obtaining the prize two thirds of the time, humans consistently fail to adopt the optimal strategy of switching even after considerable training. Interestingly, there is evidence that pigeons show more optimal switching performance with this task than humans. Because humans often view even random choices already made as being more valuable than choices not made, we reasoned that if pigeons made a greater investment, it might produce an endowment or ownership effect resulting in more human-like suboptimal performance. On the other hand, the greater investment in the initial choice by the pigeons might facilitate switching behavior by helping them to better discriminate their staying versus switching behavior. In Experiment 1, we examined the effect of requiring pigeons to make a greater investment in their initial choice (20 pecks rather than the usual 1 peck). We found that the increased response requirement facilitated acquisition of the switching response. In Experiment 2, we showed that facilitation of switching due to the increased response requirement did not result from extinction of responding to the initially chosen location.

Working Memory Capacity and a Notorious Brain Teaser

Abstract. The Monty Hall Dilemma (MHD) is an intriguing example of the discrepancy between people’s intuitions and normative reasoning. This study examines whether the notorious difficulty of the MHD is associated with limitations in working memory resources. Experiment 1 and 2 examined the link between MHD reasoning and working memory capacity. Experiment 3 tested the role of working memory experimentally by burdening the executive resources with a secondary task. Results showed that participants who solved the MHD correctly had a significantly higher working memory capacity than erroneous responders. Correct responding also decreased under secondary task load. Findings indicate that working memory capacity plays a key role in overcoming salient intuitions and selecting the correct switching response during MHD reasoning.

The Collider Principle in Causal Reasoning: Why the Monty Hall Dilemma Is So Hard

The authors tested the thesis that people find the Monty Hall dilemma (MHD) hard because they fail to understand the implications of its causal structure, a collider structure in which 2 independent causal factors influence a single outcome. In 4 experiments, participants performed better in versions of the MHD involving competition, which emphasizes causality. This manipulation resulted in more correct responses to questions about the process in the MHD and a counterfactual that changed its causal structure. Correct responses to these questions were associated with solving the MHD regardless of condition. In addition, training on the collider principle transferred to a standard version of the MHD. The MHD taps a deeper question: When is knowing about one thing informative about another?