On the nature of the conjunction fallacy

In a seminal work, Tversky and Kahneman showed that in some contexts people tend to believe that a conjunction of events (e.g., Linda is a bank teller and is active in the feminist movement) is more likely to occur than one of the conjuncts (e.g., Linda is a bank teller). This belief violates the conjunction rule in probability theory. Tversky and Kahneman called this phenomenon the “conjunction fallacy”. Since the discovery of the phenomenon in 1983, researchers in psychology and philosophy have engaged in important controversies around the conjunction fallacy. The goal of this paper is to explore the most important of these controversies, namely, the controversy about the nature of the conjunction fallacy. Is the conjunction fallacy mainly due to a misunderstanding of the problem by participants (misunderstanding hypothesis) or is it mainly due to a genuine reasoning bias (reasoning bias hypothesis)? A substantial portion of research on the topic has been directed to test the misunderstanding hypothesis. I review this literature and argue that a stronger case can be made against the misunderstanding hypothesis. Thus, I indirectly provide support for the reasoning bias hypothesis.     

A pattern recognition account of decision making

In the domain of pattern recognition, experiments have shown that perceivers integrate multiple sources of information in an optimal manner. In contrast, other research has been interpreted to mean that decision making is nonoptimal. As an example, Tversky and Kahneman (1983) have shown that subjects commit a conjunction fallacy because they judge it more likely that a fictitious person named Linda is a bank teller and a feminist than just a bank teller. This judgment supposedly violates probability theory, because the probability of two events can never be greater than the probability of either event alone. The present research tests the hypothesis that subjects interpret this judgment task as a pattern recognition task. If this hypothesis is correct, subjects’ judgments should be described accurately by the fuzzy logical model of perception (FLMP)—a successful model of pattern recognition. In the first experiment, the Linda task was extended to an expanded factorial design with five vocations and five avocations. The probability ratings were described well by the FLMP and described poorly by a simple probability model. The second experiment included (1) two fictitious people, Linda and Joan, as response alternatives and (2) both ratings and categorization judgments. Although the ratings were accurately described by both the FLMP and an averaging of the sources of information, the categorization judgments were described better by the FLMP. These results reveal important similarities in recognizing patterns and in decision making. Given that the FLMP is an optimal method for combining multiple sources of information, the probability judgments appear to be optimal in the same manner as pattern-recognition judgments.     

认知需要和警告类型对合取谬误的影响

合取谬误是一种常见的判断偏差,它指的是在不确定条件下,个体评估合取事件及其简单事件发生的概率时,对合取规则系统性偏离的一种现象.实验1 就认知需要类型对合取谬误的影响进行探讨,结果发现高认知需要的被试较不易表现出双重合取谬误和单合取谬误.实验2 探讨了警告类型对合取谬误的影响,结果发现无警告时个体最易表现出单合取谬误,其次是间接警告,最后是直接警告;此外,认知需要与警告类型的交互作用显著,高认知需要的被试在直接警告和间接警告时更少表现出双重合取谬误,在直接警告时更少表现出单合取谬误.     

How probability theory explains the conjunction fallacy

The conjunction fallacy occurs when people judge a conjunctive statement B‐and‐A to be more probable than a constituent B, in contrast to the law of probability that P(B ∧ A) cannot exceed P(B) or P(A). Researchers see this fallacy as demonstrating that people do not follow probability theory when judging conjunctive probability. This paper shows that the conjunction fallacy can be explained by the standard probability theory equation for conjunction if we assume random variation in the constituent probabilities used in that equation. The mathematical structure of this equation is such that random variation will be most likely to produce the fallacy when one constituent has high probability and the other low, when there is positive conditional support between the constituents, when there are two rather than three constituents, and when people rank probabilities rather than give numerical estimates. The conjunction fallacy has been found to occur most frequently in exactly these situations. Copyright © 2008 John Wiley & Sons, Ltd.     

On the reality of the conjunction fallacy

Attributing higher "probability" to a sentence of form p-and-q, relative to p, is a reasoning fallacy only if (1) the word probability carries its modern, technical meaning and (2) the sentence p is interpreted as a conjunct of the conjunction p-and-q. Legitimate doubts arise about both conditions in classic demonstrations of the conjunction fallacy. We used betting paradigms and unambiguously conjunctive statements to reduce these sources of ambiguity about conjunctive reasoning. Despite the precautions, conjunction fallacies were as frequent under betting instructions as under standard probability instructions.     

Towards a pattern-based logic of probability judgements and logical inclusion “fallacies”

Probability judgements entail a conjunction fallacy (CF) if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic (BL) provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies (IFs). A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 uses data where BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs (e.g. inverse probability, confirmation). IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based probability judgements about logical relations may constitute a basic class of intuitive but potentially rational probability judgements.     

Walter the banker: the conjunction fallacy reconsidered

In a famous experiment by Tversky and Kahneman (Psychol Rev 90:293–315, 1983), featuring Linda the bank teller, the participants assign a higher probability to a conjunction of propositions than to one of the conjuncts, thereby seemingly committing a probabilistic fallacy. In this paper, we discuss a slightly different example featuring someone named Walter, who also happens to work at a bank, and argue that, in this example, it is rational to assign a higher probability to the conjunction of suitably chosen propositions than to one of the conjuncts. By pointing out the similarities between Tversky and Kahneman’s experiment and our example, we argue that the participants in the experiment may assign probabilities to the propositions in question in such a way that it is also rational for them to give the conjunction a higher probability than one of the conjuncts.     

Source reliability and the conjunction fallacy

Information generally comes from less than fully reliable sources. Rationality, it seems, requires that one take source reliability into account when reasoning on the basis of such information. Recently, Bovens and Hartmann (2003) proposed an account of the conjunction fallacy based on this idea. They show that, when statements in conjunction fallacy scenarios are perceived as coming from such sources, probability theory prescribes that the "fallacy" be committed in certain situations. Here, the empirical validity of their model was assessed. The model predicts that statements added to standard conjunction problems will change the incidence of the fallacy. It also predicts that statements from reliable sources should yield an increase in fallacy rates (relative to unreliable sources). Neither the former (Experiment 1) nor the latter prediction (Experiment 3) was confirmed, although Experiment 2 showed that people can derive source reliability estimates from the likelihood of statements in a manner consistent with the tested model. In line with the experimental results, model fits and sensitivity analyses also provided very little evidence in favor of the model. This suggests that Bovens and Hartmann's present model fails to explain fully people's judgements in standard conjunction fallacy tasks.     

The conjunction fallacy?

Tversky and Kahneman (1983) showed that when subjects are asked to rate the likelihood of several alternatives, including single and joint events, they often make a "conjunction fallacy." That is, they rate the conjunction of two events as being more likely than one of the constituent events. This, they claim, is a fallacy, since the conjunction of two events can never be more probable than either of the component events. In addition, they found that prior training in probability theory does not decrease the likelihood of making this fallacy. We argue that in some contexts, an alternative that contains the conjunction of two events can be more probable than an alternative that contains only one of the conjunction's constituent events. We carried out four experiments in which we manipulated this context. The frequency of making a conjunction fallacy was affected by the manipulation of context. Furthermore, when the context was clearly specified, prior training in statistics influenced the ratings.     

Fallacies in probability judgments for conjunctions and disjunctions of everyday events

The conjunction fallacy occurs when people judge a conjunction B‐and‐A as more probable than a constituent B, contrary to probability theory's ‘conjunction rule’ that a conjunction cannot be more probable than either constituent. Many studies have demonstrated this fallacy in people's reasoning about various experimental materials. Gigerenzer objects that from a ‘frequentist’ standpoint probability theory is not valid for these materials, and so failure to follow the conjunction rule is not a fallacy. This paper describes three experiments showing that the conjunction fallacy occurs as consistently for conjunctions where frequentist probability theory is valid (conjunctions of everyday weather events) as for other conjunctions. These experiments also demonstrate a reliable correlation between the occurrence of the conjunction fallacy and the disjunction fallacy (which arises when a disjunction B‐or‐A is judged less probable than a constituent B). This supports a probability theory + random variation account of probabilistic reasoning. Copyright © 2008 John Wiley & Sons, Ltd.     

The illusion of replacement in research into the development of thinking biases: the case of the conjunction fallacy

The conjunction fallacy is a violation of a very basic rule of probability. Interestingly, although committing the fallacy seems irrational, adults are no less susceptible to the fallacy than young children. In Experiment 1, by employing tasks where the conjunctive response option involved two non-representative items, we found a large reduction in fallacy rates as compared to traditional conjunction fallacy problems. Nevertheless, fallacy rates remained relatively high in both adolescents and adults, although adults showed more consistency in their normative responses. In Experiment 2, we demonstrated that children’s relatively good performance on the task was not the consequence of their missing knowledge of social stereotypes. Additionally, children were more strongly affected by explicitly presented frequency information than adults. Indeed, adults only took frequency information into account when frequencies were made relevant by a training in probabilistic reasoning. Overall, the results suggest that whereas the potential for normative reasoning increases with development, this potential is often overshadowed by a pervasive tendency in adolescence and adulthood to rely on contextual information, knowledge, and beliefs, even when conflicting information is available. By contrast, children are more strongly influenced by explicitly presented information than relevant knowledge cued by the tasks.     

The dependence of the conjunction fallacy on subtle linguistic factors

Summary The three experiments reported in this article are concerned with moderating conditions of the so-called conjunction fallacy. Although the conjunction of two events (A&B) is necessarily less probable than one event alone, intelligent people's judgments sometimes violate this logical principle when it is easier to think about or imagine the conjunction A&B than the component event A. It was hypothesized that the fallacy might be reduced either by a priming treatment (directing the participants' attention to the logical rule prior to the conjunctions tasks) or by a linguistic manipulation (replacing the ambiguous term probability by the term frequency. Experiment 1 shows that the bias is drastically reduced by the subtle linguistic manipulation and suggests that the fallacy is partly due to a common misunderstanding of the probability concept. The lack of a priming effect seems to imply that cueing or activation of logical schemata is not a critical factor. Experiment 2 replicates the linguistic influence under slightly modified task conditions, and Experiment 3 corroborates the ineffectiveness of the priming factor using a stronger priming treatment.     

Explaining High Conjunction Fallacy Rates: The Probability Theory Plus Noise Account

The conjunction fallacy occurs when people judge the conjunctive probability P(A ∧ B) to be greater than a constituent probability P(A), contrary to the norms of probability theory. This fallacy is a reliable, consistent and systematic part of people's probability judgements, attested in many studies over at least 40 years. For some events, these fallacies occur very frequently in people's judgements (at rates of 80% or more), while for other events, the fallacies are very rare (occurring at rates of 10% or less). This wide range of fallacy rates presents a challenge for current theories of the conjunction fallacy. We show how this wide range of observed fallacy rates can be explained by a simple model where people reason according to probability theory but are subject to random noise in the reasoning process. Copyright © 2016 John Wiley & Sons, Ltd.     

Noncategorical approaches to feature prediction with uncertain categories

In four experiments, we investigated how people make feature predictions about objects whose category membership is uncertain. Artificial visual categories were presented and remained in view while a novel instance with a known feature, but uncertain category membership was presented. All four experiments showed that feature predictions about the test instance were most often based on feature correlations (referred to as feature conjunction reasoning). Experiment 1 showed that feature conjunction reasoning was generally preferred to category-based induction in a feature prediction task. Experiment 2 showed that people used all available exemplars to make feature conjunction predictions. Experiments 3 and 4 showed that the preference for predictions based on feature conjunction persisted even when category-level information was made more salient and inferences involving a larger number of categories were required. Little evidence of reasoning based on the consideration of multiple categories (e.g., Anderson, (Psychological Review, 98:409–429, 1991)) or the single, most probable category (e.g., Murphy & Ross, (Cognitive Psychology, 27:148–193, 1994)) was found.     

Linda问题：“齐当别”抉择模型的解释

大量有关人类归因判断的研究表明,人类经常违反理性概率公理。Tversky和Kahneman（1983）使用Linda问题等特定场景的研究发现,人们系统性地表现出违反理性推断标准,判断合取事件发生概率大于其组成事件发生概率,称之为合取谬误,并用人们使用代表性启发式判断概率来解释该现象产生的原因。然而使用启发式观点对合取谬误现象进行解释过于模糊不清。该文首先介绍了合取谬误现象及其解释模型,然后应用Li（1994,2004）提出的不确定情形下决策理论——“齐当别”抉择模型对Linda问题中合取谬误产生的原因进行了新的解释     

Reasoning about conjunctive probabilistic concepts in childhood.

While adults are known to exhibit biases when making conjunctive probability judgments, little is known about childhood competencies in this area. Participants (aged between four and five years, eight and ten years, and a group of young adults) attempted to select the more likely of two events, a single event, and a conjunctive event containing, as one of its components, the single event. The problems were such that the objective probabilities of the component events were potentially available. Children in both age groups were generally successful when the single event was likely. However, when it was unlikely, a majority of children rejected it, choosing the conjunctive event instead, thereby committing the conjunction fallacy. A substantial minority of adults also committed the fallacy under equivalent conditions. It is concluded that under certain conditions children are capable of normative conjunctive judgments but that the mechanisms underpinning this capacity remain to be fully understood.     

Reasoning about complex probabilistic concepts in childhood

The competencies of children, particularly their understanding of the more complex probabilistic concepts, have not been thoroughly investigated. In the present study participants were required to choose the more likely of two events, a single event, and a joint event (conjunctive or disjunctive). It was predicted that the operation of the representativeness heuristic would result in erroneous judgements when children compared an unlikely component event with a likely-unlikely conjunction (the conjunction fallacy) and when a likely component event was compared to a likely-unlikely disjunction. The results supported the first prediction with both older children aged between 9 and 10 years and younger children aged between 4 and 5 committing the conjunction fallacy. However, the second prediction was not confirmed. It is proposed that the basis of representativeness judgements may differ between the conjunctive and disjunctive cases with absolute frequency information possibly playing a differential role.     

Representativeness and conjoint probability

People commonly violate a basic rule of probability, judging a conjunction of events to be more probable than at least 1 of its component events. Many manifestations of this conjunction fallacy have been ascribed to people's reliance on the representativeness heuristic for judging probability. Some conjunction fallacies, however, have been ascribed to the incorrect rules people use to combine probabilities. In 2 experiments, representativeness was pitted against probability combination to determine the contributions of each to the fallacy. Even for exemplar representativeness problems, the fallacy stemmed primarily from the application of incorrect combination rules. Representativeness seemed to be involved only insofar as it influenced the probabilities of a conjunction's component events. Implications of these findings are discussed for the representativeness account of judgmental errors and the relation between similarity and probability.     

Unitization during category learning

Five experiments explored the question of whether new perceptual units can be developed if they are useful for a category learning task, and if so, what the constraints on this unitization process are. During category learning, participants were required to attend either a single component or a conjunction of 5 components. Consistent with unitization, the conjunctive task became much easier with practice; this improvement was not found for the single-component task or for conjunctive tasks in which the components could not be unitized. Influences of component organization (Experiment 1), component contiguity (Experiment 2), component proximity (Experiment 3), and number of components (Experiment 4) on practice effects were found. Deconvolved response times (Experiment 5) showed that prolonged practice yielded faster responses than predicted by an analytic model that integrates evidence from independently perceived components.