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

Abstract

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.

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

“…We need to be aware, however, that our systematic review provided a sequential analysis of the MHD specifically and its related causes explaining humans’ suboptimal MHD performance. When another (posterior) probability problem would be investigated, another overview of causes for suboptimal performance could be found (see Tubau et al, 2015). …”

“…People hold several misconceptions about (posterior) probabilities (e.g., Batanero & Sanchez, 2005; Garfield & Ahlgren, 1988; Shaughnessy, 1992) of which some play an important role in the MHD as well (e.g., equiprobability bias). In their review, Tubau, Aguilar-Lleyda, and Johnson (2015) pointed to similitudes and differences between the MHD and other Bayesian problems. They stated that “differing from most Bayesian problems, prior and conditional probabilities in the MHD have to be inferred” (p. 1).…”

“…The aim of the current review is to identify all known causes for people’s suboptimal performance on the MHD for which at least some empirical evidence is found. By providing such review, we aim to complement the more theoretical review of Tubau et al (2015) who compared the MHD to other Bayesian tasks.…”

“…The review provides a structured overview of all known erroneous reasoning processes and misconceptions people have about the MHD and which thus may – but do not necessarily (see Tubau, Aguilar-Lleyda, & Johnson, 2015) – play a role in all type of areas in which posterior probabilities are involved (e.g., medical decision making). This is important for the field of statistical education as well, because “the success of any probability curriculum for developing students’ probabilistic reasoning depends greatly on teachers’ understanding of probability as well as a much deeper understanding of issues such as students’ misconceptions [emphasis added]” (Stohl, 2005, p. 351).…”

The Monty Hall Problem as a Bayesian Game

“…The typical justification put forward by the subjects is that the likelihood of success if they were to switch curtains is the same as if they were to stay with their original choice of curtain. This mistake is often cited as an example of the equiprobability bias heuristic ( [14], [15], [19], [9]) as described in Lecoutre (1992) [7]. [6].…”