Baumann on the Monty Hall problem and single-case probabilities

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. See Moser and Mulder (1994, pp. 115–116, 118).     

Monty Hall Dilemma的困难原因探讨

采用Monty Hall Dilemma的经典研究范式,探讨了大学生被试在不同提示条件下对MHD问题的解决情况,结果发现只有在条件关系提示下,MHD的作业成绩才得到了明显的改善,但是正确率也只有35.7%。这表明MHD问题困难的原因很可能是多方面的,如认为主持人的行为完全是随机事件,很难意识到参与者的最初选择与主持人打开特定盒子间的三种条件关系;多数被试作出选择的理由是非理性的,如“坚持第一选择,相信自己的直觉”;不能正确判断MHD问题的内在概率关系以及缺乏认真思考的动力等。     

The psychology of the Monty Hall problem: discovering psychological mechanisms for solving a tenacious brain teaser

The Monty Hall problem (or three-door problem) is a famous example of a "cognitive illusion," often used to demonstrate people's resistance and deficiency in dealing with uncertainty. The authors formulated the problem using manipulations in 4 cognitive aspects, namely, natural frequencies, mental models, perspective change, and the less-is-more effect. These manipulations combined led to a significant increase in the proportion of correct answers given by novice participants, largely because of the synergy of frequency-based formulation and perspective change (Experiments 1, 2). In a raining study (Experiment 3) frequency formulation and mental models, but not Bayes's rule training, showed significant positive transfer in solving related problems.     

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.     

Probability,rational single-case decisions and the Monty Hall Problem

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.     

A Logical Analysis of Monty Hall and Sleeping Beauty

Hintikka and Sandu’s independence-friendly (IF) 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.     

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?     

Working memory capacity and a notorious brain teaser: the case of the Monty Hall Dilemma

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.     

动机的概率意义——概率归并定理与Monty Hall问题的非贝叶斯解

本文探讨Monty Hall问题认知困惑的形成因素。通过对该问题数学结构的分析,导出一个与该问题的解密切相关的定理——概率归并定理;并依据该定理给出了Monty Hall问题的一种更直观的数学解。     

Implications of counterfactual structure for creative generation and analytical problem solving

In the present research, the authors hypothesized that additive counterfactual thinking mind-sets, activated by adding new antecedent elements to reconstruct reality, promote an expansive processing style that broadens conceptual attention and facilitates performance on creative generation tasks, whereas subtractive counter-factual thinking mind-sets, activated by removing antecedent elements to reconstruct reality, promote a relational processing style that enhances tendencies to consider relationships and associations and facilitates performance on analytical problem-solving tasks. A reanalysis of a published data set suggested that the counterfactual mind-set primes previously used in the literature tend to evoke subtractive counterfactuals. Studies 1 and 2 then demonstrated that subtractive counterfactual mind-sets enhanced performance on analytical problem-solving tasks relative to additive counterfactual mind-sets, whereas Studies 3 and 4 found that additive counterfactual mind-sets enhanced performance on creative generation tasks relative to subtractive counterfactual mind-sets.     

Spoilt for choice: The role of counterfactual thinking in the excess choice and reversibility paradoxes

Contrary to popular belief many choice options and the ability to reverse one's initial choice are sometimes associated with decreased chooser satisfaction. Two studies investigated the role of counterfactual thinking in explaining these paradoxes. Participants chose drawing implements from either a limited (6) or extensive (24) choice set (Study 1), or an expected reversible/non-reversible selection (Study 2). Following a drawing task, satisfaction with their chosen implement was rated under either high or low cognitive load to manipulate the availability of counterfactual alternatives. In Study 1 satisfaction was higher with limited vs. extensive choice under low load. The number of counterfactuals generated mediated this effect. Under high load the pattern was reversed. Participants in Study 2 generated more counterfactuals when reversibility was expected under low but not high load and this partially mediated the impact of expected reversibility on revealed satisfaction. Implications for theoretical understanding of these paradoxes are discussed.     

The role of thought-content and mood in the preparative benefits of upward counterfactual thinking

Upward counterfactual thoughts identify how a prior outcome could have been better and have been shown to improve subsequent performance. Both the identification of corrective actions (content-specific effects) and the more general mobilization of effort as a result of negative affect (content-neutral effects) have been suggested to underlie performance benefits. The results of three experiments presented here indicate that counterfactual thoughts have broad benefits for performance, independent of their content and beyond the effects of planning. These benefits were consistently dependent upon the experience of negative affect, but were eliminated when negative affect could be (mis)attributed to an intervening task. This misattribution effect is consistent with the operation of a mood-as-input process in which affect informs judgments of goal progress. Overall, the findings suggest that the benefits of upward counterfactual thinking reflect more broad attempts to improve following a subjectively unsatisfactory performance.     

The role and treatment of irresponsibility in dysfunctional relationships

This article explores the role of personal responsibility in dysfunctional relations. The premise is that the stances one or both partners take with regard to their relationship responsibilities predict the success or failure of that relationship. The four parts of this article include a definition of responsibility, a delineation of four crucial areas of relationship responsibility, a listing of irrational beliefs that block responsibility, and an outline of irresponsibility treatment.Responsibility is a unique concept. It can only reside and inhere in a single individual. You may share it with others, but your portion is not diminished. You may delegate it, but it is still with you. You may disclaim it, but you cannot divest yourself of it. Admiral Hyman Rickover     

Learning harmony: the role of serial statistics

How do listeners learn about the statistical regularities underlying musical harmony? In traditional Western music, certain chords predict the occurrence of other chords: Given a particular chord, not all chords are equally likely to follow. In Experiments 1 and 2, we investigated whether adults make use of statistical information when learning new musical structures. Listeners were exposed to a novel musical system containing phrases generated using an artificial grammar. This new system contained statistical structure quite different from Western tonal music. Our results suggest that learners take advantage of the statistical patterning of chords to acquire new musical structures, similar to learning processes previously observed for language learning.     

The myth of the categorical counterfactual

I aim to show that standard theories of counterfactuals are mistaken, not in detail, but in principle, and I aim to say what form a tenable theory must take. Standard theories entail a categorical interpretation of counterfactuals, on which to state that, if it were that A, it would be that C is to state something, not relative to any supposition or hypothesis, but categorically. On the rival suppositional interpretation, to state that, if it were that A, it would be that C is to state that it would be that C relative to the supposition that it were that A. The two interpretations make incompatible predictions concerning the correct evaluation of counterfactuals. I argue that the suppositional interpretation makes the correct prediction.

If the real world were irrelevant, so to speak: The role of propositional truth-value in counterfactual sentence comprehension

Propositional truth-value can be a defining feature of a sentence's relevance to the unfolding discourse, and establishing propositional truth-value in context can be key to successful interpretation. In the current study, we investigate its role in the comprehension of counterfactual conditionals, which describe imaginary consequences of hypothetical events, and are thought to require keeping in mind both what is true and what is false. Pre-stored real-world knowledge may therefore intrude upon and delay counterfactual comprehension, which is predicted by some accounts of discourse comprehension, and has been observed during online comprehension. The impact of propositional truth-value may thus be delayed in counterfactual conditionals, as also claimed for sentences containing other types of logical operators (e.g., negation, scalar quantifiers). In an event-related potential (ERP) experiment, we investigated the impact of propositional truth-value when described consequences are both true and predictable given the counterfactual premise. False words elicited larger N400 ERPs than true words, in negated counterfactual sentences (e.g., "If N.A.S.A. had not developed its Apollo Project, the first country to land on the moon would have been Russia/America") and real-world sentences (e.g., "Because N.A.S.A. developed its Apollo Project, the first country to land on the moon was America/Russia") alike. These indistinguishable N400 effects of propositional truth-value, elicited by opposite word pairs, argue against disruptions by real-world knowledge during counterfactual comprehension, and suggest that incoming words are mapped onto the counterfactual context without any delay. Thus, provided a sufficiently constraining context, propositional truth-value rapidly impacts ongoing semantic processing, be the proposition factual or counterfactual.     

Learning the unlearnable: the role of missing evidence

Syntactic knowledge is widely held to be partially innate, rather than learned. In a classic example, it is sometimes argued that children know the proper use of anaphoric one, although that knowledge could not have been learned from experience. Lidz et al. [Lidz, J., Waxman, S., & Freedman, J. (2003). What infants know about syntax but couldn't have learned: Experimental evidence for syntactic structure at 18 months. Cognition, 89, B65-B73.] pursue this argument, and present corpus and experimental evidence that appears to support it; they conclude that specific aspects of this knowledge must be innate. We demonstrate, contra Lidz et al., that this knowledge may in fact be acquired from the input, through a simple Bayesian learning procedure. The learning procedure succeeds because it is sensitive to the absence of particular input patterns--an aspect of learning that is apparently overlooked by Lidz et al. More generally, we suggest that a prominent form of the "argument from poverty of the stimulus" suffers from the same oversight, and is as a result logically unsound.     

The relationship between media exposure and antifat attitudes: the role of dysfunctional appearance beliefs

This study examined the relationship between media exposure, antifat attitudes, and body dissatisfaction, as well as the mediating effect of dysfunctional appearance beliefs. A sample of 112 women completed surveys measuring media exposure, antifat attitudes, body dissatisfaction, and dysfunctional beliefs about appearance. It was found that time spent reading fashion magazines was positively correlated with antifat attitudes and that this relationship was mediated by dysfunctional beliefs about appearance. Measures of antifat attitudes and body dissatisfaction were both found to be correlated with endorsement of dysfunctional beliefs about appearance and body mass index. Results suggest that time spent reading fashion magazines may be related to antifat attitudes through dysfunctional appearance beliefs.