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?     

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。     

A randomised Monty Hall experiment: The positive effect of conditional frequency feedback

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 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.     

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.     

Nonstable concurrent choice in pigeons

Six pigeons were trained on concurrent variable-interval schedules in which the arranged reinforcer ratios changed from session to session according to a 31-step pseudorandom binary sequence. This procedure allows a quantitative analysis of the degree to which performance in an experimental session is affected by conditions in previous sessions. Two experiments were carried out. In each, the size of the reinforcer ratios arranged between the two concurrent schedules was varied between 31-step conditions. In Experiment 1, the concurrent schedules were arranged independently, and in Experiment 2 they were arranged nonindependently. An extended form of the generalized matching law described the relative contribution of past and present events to present-session behavior. Total performance in sessions was mostly determined by the reinforcer ratio in that session and partially by reinforcers that had been obtained in previous sessions. However, the initial exposure to the random sequence produced a lower sensitivity to current-session reinforcers but no difference in overall sensitivity to reinforcement. There was no evidence that the size of the reinforcer ratios available on the concurrent schedules affected either overall sensitivity to reinforcement or the sensitivity to reinforcement in the current session. There was also no evidence of any different performance between independent and nonindependent scheduling. Because of these invariances, this experiment validates the use of the pseudorandom sequence for the fast determination of sensitivity to reinforcement.     

Self-control by pigeons in the prisoner's dilemma

Pigeons played a repeated prisoner's dilemma game against a computer that reflected their choices: If a pigeon cooperated on trial n, the computer cooperated on trial n + 1; if the pigeon defected on trial n, the computer defected on trial n + 1. Cooperation thus maximized reinforcement in the long term, but defection was worth more on the current trial. Under these circumstances, pigeons normally defect. However, when a signal correlated with the pigeon's previous choice immediately followed each current trial choice, some pigeons learned to cooperate. Furthermore, cooperation was higher when trials were close together in time than when they were separated by long intertrial intervals.     

Discrete-trial choice in pigeons: Effects of reinforcer magnitude

The preference of pigeons for large reinforcers which occasionally followed a response versus small reinforcers which invariably followed a response was studied in a discrete-trial situation. Two differently colored keys were associated with the two reinforcement alternatives, and preference was measured as the proportion of choice trials on which the key associated with uncertain reinforcement was pecked. A combination of choice and guidance trials insured that received distributions of reinforcement equalled the scheduled distributions. For five of six subjects, preference for the uncertain reinforcer appeared to be a linear function of the magnitude of the certain reinforcer. In addition, there was greater preference for the response alternative associated with uncertain reinforcement than would be expected on the basis of net reinforcer value.     

The cognitive structure of choice dilemma decisions

We tried to find out why Ss chose a specific risk level. We, therefore, constructed a questionnaire with 30 statements about aspects of the contents of the choice dilemmas. A factor analysis of the 30 statements over ten choice dilemmas and 61 field-officers led to a structure of four cognitive elements. With these four factors as predictors we could explain a main part of the variance of the individual risk level of each choice dilemma. This regression analysis was done with normal product-moment-correlations and with scalar-products. Both analyses show great similarity, although it seemed to be better to use scalar-products because one can use the information of the means. The four cognitive elements could be interpreted as: (1) Decision-making strategy; (2) Responsibility for others; (3) Reputation; (4) Socially valued riskiness.     

Preference for free choice over forced choice in pigeons

Eight severely retarded young men learned color and line-tilt discrimination. After 95% accuracy was achieved for both dimensions, they were combined to form "conflict-compound" stimuli in which prior reinforcement history was reversed for one element of the compound and unchanged for the other. When responding to the compound was 95% accurate, control exerted by each element was measured. The unchanged element consistently exercised control in agreement with the reinforcement contingencies associated with the compound, regardless of whether it was color or line orientation. The reversed element, which had a conflicting prior history or reinforcement, most often exerted control associated wtih original training, or no control, suggesting that it had been "ignored" during the compound. Conflict compounds produced selective attention. When elements were combined to form "compatible-compound" stimuli, both exercised control in agreement with the compound in post-tests. Selective attention was not produced by compatible compounds.     

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.     

Probabilistically reinforced choice behavior in pigeons

A single principle, "momentary maximizing", may account for much of a pigeon's steady-state behavior in both probability learning and concurrent variable interval experiments. The principle states that a pigeon tends to choose the alternative that momentarily has the higher probability of reinforcement. A successive discrimination procedure, which produced matching in an earlier experiment, produced here a tendency to maximize if training were adequately extended. Maximizing was produced also by other procedures, in which no reinforcing event was presented on some trials: one procedure did and two did not provide a bird with information about the availability of reinforcement on a key after an unreinforced response on the other key. The latter two procedures were analogous to concurrent variable interval schedules in two respects: the reinforcement probability on one key increased while a bird responded on the other key; and they produced matching. But sequential statistics suggested that matching resulted from momentary maximizing. Depending on the procedure, the tendency to maximize produced different relative frequencies of pecking a key for a fixed relative frequency of reinforcement. Computer simulation of maximizing behavior in several concurrent variable interval schedules produced matching and sequential statistics similar to those produced by a real bird.     

Handling time and choice in pigeons

According to optimal foraging theory, animals should prefer food items with the highest ratios of energy intake to handling time. When single items have negligible handling times, one large item should be preferred to a collection of small ones of equivalent total weight. However, when pigeons were offered such a choice on equal concurrent variable-interval schedules in a shuttlebox, they preferred the side offering many small items per reinforcement to that offering one or a few relatively large items. This preference was still evident on concurrent fixed-cumulative-duration schedules in which choosing the alternative with longer handling time substantially lowered the rate of food intake.     

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。     

On some determinants of choice in pigeons

A pigeon's pecking at each of two or three simultaneously available red keys was reinforced at different frequencies with a conditioned reinforcer, an orange key, on which 25 pecks resulted in a presentation of grain. Pecking was occasionally punished with a period of no reinforcement during which each key was dark. Both with two and with three keys, the relative frequency of pecking on a key was equal to the relative frequency of reinforcement obtained by pecks on that key. Also, the absolute frequency of pecking on each key was a linear function with zero intercept of the absolute frequency of reinforcement associated with that key. The slope of this function varied with the number of available keys; it was steeper with two than with three. The relative frequency of switching from any key (two successive pecks on different keys) approximated a linear function with zero intercept and slope slightly greater than 1.0 of the total relative frequency of reinforcement associated with the keys to which the bird could switch. However, the relative frequency of switching to a particular key often showed systematic irregularities. The invariance in these data is the equality between the relative frequency of pecks on one of two or three keys and the relative frequency of reinforcement associated with that key.     

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.     

Rapid acquisition of choice and timing in pigeons

Pigeons were trained on a concurrent-chains procedure in which the initial link associated with the shorter terminal-link delay to food changed unpredictably across sessions. In the minimal-variation condition, delays were always 10 s and 20 s, whereas in the maximal-variation condition delays were generated pseudorandomly for each session. On some terminal links, food was withheld to obtain measures of temporal control. Measures of choice (log initial-link response ratios) and timing (start and stop times on no-food trials) showed temporal control and stabilized within the 1st half of each session. In the maximal-variation condition, choice was a nonlinear function of the log delay ratio, consistent with a categorical discrimination but contrary to models based on the matching law. Residuals from separate regressions of log response and log start and stop time ratios on log delay ratios were positively correlated. Overall, results support cognitive models that assume that initial-link choice is based on an all-or-none decision process, and that choice and timing are mediated by a common representation of delay.     

The relationship between riskiness and confidence in choice dilemma decisions

The relationship between riskiness of decisions and confidence in decisions was examined using Choice Dilemma items. Graphs show a clear curvilinear relationship between riskiness and confidence; on both risky items and cautious items greater confidence is associate with extremely risky and extremely cautious decisions than with moderate ones. it was also demonstrated that individual subjects have more confidence in their extreme decisions than in their moderate ones.