Monty Hall Problem

A review of Matt Cook's book Sleight of Mind: 75 Ingenious Paradoxes in Mathematics, Physics, and Philosophy.

We conduct a laboratory experiment using the Monty Hall problem to study how simplified examples improve learning behavior and correct irrational choices in probabilistic situations. In particular, we show that after experiencing a simplified version of the MHP, subjects perform better in the MHP, compared to the control group who only experienced the 3-door version. Our results suggest that simplified examples strongly induces learning.

Inspired by the Monty Hall Problem and a popular simple solution to it, we present a number of game-show puzzles that are analogous to the notorious Sleeping Beauty Problem (and variations on it), but much easier to solve. We replace the awakenings of Sleeping Beauty by contestants on a game show, like Monty Hall’s, and increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to (...) see what the solution to the problem is. We show that these game-show proxies for the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and-true strategy for addressing problems in philosophy of science, i.e., develop a simple model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful. (shrink)

In this paper I show that Elga’s argument for a restricted principle of indifference for self-locating belief relies on the kind of mistaken reasoning that recommends the ‘staying’ strategy in the Monty Hall problem.

Hintikka and Sandu’s independence-friendly logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Finally, we use stochastic IF logic to analyze the Sleeping Beauty problem, leading to (...) the conclusion that the thirders are correct while identifying the main error in the halfers’ argument. (shrink)

This thesis addresses several questions regarding what rational agents ought to believe and how they ought to act. In the first part I begin by discussing how scientists contemplating several mutually exclusive theories, models or hypotheses can reach a rational decision regarding which one to endorse. In response to a recent argument that they cannot, I employ the tools of social choice theory to offer a ‘possibility result’ for rational theory choice. Then I utilize the tools of judgment aggregation to (...) investigate how scientists from across fields can pool their expertise together. I identify an impossibility result threatening such a procedure and prove a possibility result which requires that some scientists sometimes waive their expertise over some propositions. In the second part I first discuss the existing justifications for a restricted principle of indifference that mandates that two agents whose experiences are subjectively indistinguishable should be indifferent with respect to their identities. I argue that all existing justifications rely on the same mistaken reasoning behind the ‘staying’ strategy in the Monty Hall problem. Secondly, I show this mistake is more widespread and I identify it in arguments purporting to show the failure of two reflection-like principles. In the third part I look at a recent argument that fair policy makers face a dilemma when trying to correct a biased distributive process. I show the dilemma only holds if the correction has to happen in one-shot. Finally, I look at how we ought to design public restrooms so that we reduce the discrimination faced by minority groups. I make the case for opening our public restrooms to all genders. (shrink)

The Monty-Hall Problem ($MHP$) has been used to argue against a subjectivist view of Bayesianism in two ways. First, psychologists have used it to illustrate that people do not revise their degrees of belief in line with experimenters' application of Bayes' rule. Second, philosophers view $MHP$ and its two-player extension ($MHP2$) as evidence that probabilities cannot be applied to single cases. Both arguments neglect the Bayesian standpoint, which requires that $MHP2$ (studied here) be described in different terms than usually applied (...) and that the initial set of possibilities be stable (i.e., a focusing situation). This article corrects these errors and reasserts the Bayesian standpoint; namely, that the subjective probability of an event is always conditional on a belief reviser's specific current state of knowledge. (shrink)

The Monty Hall dilemma 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 randomised 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 behavioural performance on the MHD, on one hand, and actual understanding of the MHD, on the other. We discuss how this dissociation can be understood. (shrink)

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. (shrink)

The Monty Hall problem is consistently misunderstood. Mathematician Jeffrey Rosenthal argues in Monty Hall, Monty Fall, Monty Crawl” and Struck By Lightning that a proportionality principle can solve and explain the Monty Hall problem and its variants like Monty Fall and Monty Crawl better than the classic solution. Rosenthal’s Monty Fall example and solution are examined in detail. I show he has misidentified the crucial assumption in the Monty Hall problem, and his own Monty Fall problem is logically equivalent to (...) the original Monty Hall problem. I then present the Monty Fall* case where the probabilities for which door to pick post tease reveal are actually 50/50 using nothing more than Bayes’ Theorem and the standard rules of probability to prove the results—no proportionality principle is needed. The classic solution prevails as explanatorily more powerful. Finally, I show that Monty Crawl is also better explained and solved with the classic solution rather than with Rosenthal’s proportionality principle. (shrink)

The Bayesian model has been used in psychology as the standard reference for the study of probability revision. In the first part of this paper we show that this traditional choice restricts the scope of the experimental investigation of revision to a stable universe. This is the case of a situation that, technically, is known as focusing. We argue that it is essential for a better understanding of human probability revision to consider another situation called updating (Katsuno & Mendelzon, 1992), (...) in which the universe is evolving. In that case the structure of the universe has definitely been transformed and the revision message conveys information on the resulting universe. The second part of the paper presents four experiments based on the Monty Hall puzzle that aim to show that updating is a natural frame for individuals to revise their beliefs. (shrink)

In “Judy Benjamin is a Sleeping Beauty” (2010) Bovens recognises a certain similarity between the Sleeping Beauty (SB) and the Judy Benjamin (JB). But he does not recognise the dissimilarity between underlying protocols (as spelled out in Shafer (1985). Protocols are expressed in conditional probability tables that spell out the probability of coming to learn various propositions conditional on the actual state of the world. The principle of total evidence requires that we not update on the content of the proposition (...) learned but rather on the fact that we learn the proposition in question. Now attention to protocols drives a wedge between the SB and the JB. We have shown that the solution to a close variant of the SB which involves a clear protocol is P*(Heads) = 1/3 and since Beauty’s has precisely the same information at her disposal in the original SB at the time that she is asked to state her credence for Heads, the same solution should hold. The solution to the JB, on the other hand, is dependent on Judy’s probability distribution over protocols. One reasonable protocol yields P(Red) = 1/2, but Judy could also defend alternative values or a range of values in the interval [1/3, 1/2] depending on her probability distribution over protocols. (shrink)

The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e.g. Horgan, Philos Pap 24:209–222, 1995; Levy, Synthese 158:139–151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42:71–79, 2005; Synthese 162:265–273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning (...) and rational decisions. In particular, I elaborate a flaw in Baumann’s reductio of rational single-case decisions in a modified Monty Hall Problem. (shrink)

Decision making theory in general, and mental models in particular, associate judgment and choice. Decision choice follows probability estimates and errors in choice derive mainly from errors in judgment. In the studies reported here we use the Monty Hall dilemma to illustrate that judgment and choice do not always go together, and that such a dissociation can lead to better decision-making. Specifically, we demonstrate that in certain decision problems, exceeding working memory limitations can actually improve decision choice. We show across (...) four experiments that increasing the number of choice alternatives forces people to collapse choices together, resulting in better decision-making. While choice performance improves, probability judgments do not change, thus demonstrating an important dissociation between choice and probability judgments. We propose the Collapsing Choice Theory (CCT) which explains how working memory capacity, probability estimation, choice alternatives, judgment, and regret all interact and effect decision quality. (shrink)

In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.

How should we update our beliefs when we learn new evidence? Bayesian confirmation theory provides a widely accepted and well understood answer – we should conditionalize. But this theory has a problem with self-locating beliefs, beliefs that tell you where you are in the world, as opposed to what the world is like. To see the problem, consider your current belief that it is January. You might be absolutely, 100%, sure that it is January. But you will soon believe it (...) is February. This type of belief change cannot be modelled by conditionalization. We need some new principles of belief change for this kind of case, which I call belief mutation. In part 1, I defend the Relevance-Limiting Thesis, which says that a change in a purely self-locating belief of the kind that results in belief mutation should not shift your degree of belief in a non-self-locating belief, which can only change by conditionalization. My method is to give detailed analyses of the puzzles which threaten this thesis: Duplication, Sleeping Beauty, and The Prisoner. This also requires giving my own theory of observation selection effects. In part 2, I argue that when self-locating evidence is learnt from a position of uncertainty, it should be conditionalized on in the normal way. I defend this position by applying it to various cases where such evidence is found. I defend the Halfer position in Sleeping Beauty, and I defend the Doomsday Argument and the Fine-Tuning Argument. (shrink)

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" (...) between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem. (shrink)

The Monty Hall Problem (MHP), a process of two-stage decision making, was presented in atypical form via a custom software game. Differing from the normal three-box MHP, the game added one additional box on-screen for each game—culminating on game 23 with 25 on-screen boxes to initially choose from. A total of 108 participants played 23 games (trials) in one of four conditions; (1) “Vanish” condition—all non-winning boxes totally removed from the screen; (2) “Empty” condition—all non-winning boxes remain on-screen, but with (...) an “empty” label on them; (3) “Steroids” condition—all non-winning boxes removed from the screen, with initially chosen box becoming 25% larger; (4) “Steroids2” condition—all non-winning boxes removed from the screen, box not currently chosen becomes 25% larger. Results indicate second-stage on-screen presence of boxes influences switching; with their absence having the opposite effect. Size manipulation appears to elicit demand characteristics resulting in indeterminate influence. (shrink)

Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann's particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what (...) justifies the switching strategy is its leading me to win a greater percentage of the time. In fact, what justifies the switching strategy is not any statistical result over the long run but rather the "causal structure" intrinsic to each individual game itself. Finally, I argue that an argument by Hilary Putnam will not help to save Baumann's second conclusion above. (shrink)

The well known Monty Hall-problem has a clear solution if one deals with a long enough series of individual games. However, the situation is different if one switches to probabilities in a single case. This paper presents an argument for Monty Hall situations with two players (not just one, as is usual). It leads to a quite general conclusion: One cannot apply probabilistic considerations (for or against any of the strategies) to isolated single cases. If one does that, one cannot (...) but violate a very plausible non-arbitrariness condition and is led into a Moore-paradoxical incoherence. Even though arguments for switching are correct as applied to series of games, they don’t say anything useful about what rationality demands in a single case. (shrink)

We give an analysis of the Monty Hall problem purely in terms of confirmation, without making any lottery assumptions about priors. Along the way, we show the Monty Hall problem is structurally identical to the Doomsday Argument.

This research examined choice behaviour and probability judgement in a counterintuitive reasoning problem called the Monty Hall problem (MHP). In Experiments 1 and 2 we examined whether learning from a simulated card game similar to the MHP affected how people solved the MHP. Results indicated that the experience with the card game affected participants' choice behaviour, in that participants selected to switch in the MHP. However, it did not affect their understanding of the objective probabilities. This suggests that there is (...) dissociation between implicit knowledge gained from the task and the explicit understanding as to why switching was the best strategy. In Experiment 3, the number of prizes and doors were manipulated to examine how participants construed the problem space of the MHP. Results revealed that participants partition the probability judgement to reflect the number of prizes over the number of unopened doors. (shrink)

Despite the results of David Lewis, Peter Gärdenfors, and others, showing that imaging and classical conditionalization coincide only in the most trivial probabilistic models of belief revision, it turns out that imaging on a proposition A can always be described via Popper function conditionalization on a proposition that entails A. This result generalizes to any method of belief revision meeting certain minimal requirements. The proof is illustrated by an application of imaging in the context of the Monty Hall Problem.

A version of the Monty Hall Problem is presented and the "switch" solution is defended using a method of enumeration rather than Bayes Theorem.

This definitive survey of the hottest issues in inductive logic sets the stage for further classroom discussion.