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Behavioral and Brain Sciences

Why quantum probability does not explain the conjunction fallacy

Katya Tentori, Vincenzo Crupi

Published May 14, 2013• 13 citations
Publisher

Abstract

We agree with Pothos & Busemeyer (P&B) that formal tools can be fruitfully employed to model human judgment under uncertainty, including well-known departures from principles of classical probability. However, existing findings either contradict P&B's quantum probability approach or support it to a limited extent. The conjunction fallacy serves as a key illustration of both kinds of problems.

DOI
10.1017/s0140525x12003123

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We agree with Pothos & Busemeyer (P&B) that formal tools can be fruitfully employed to model human judgment under uncertainty, including well-known departures from principles of classical probability. However, existing findings either contradict P&B's quantum probability approach or support it to a limited extent. The conjunction fallacy serves as a key illustration of both kinds of problems.
How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B , which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.
A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction, disjunction, inverse, and conditional fallacies, as well as unpacking effects and partitioning effects. Quantum probability theory is a general and coherent theory based on a set of (von Neumann) axioms which relax some of the constraints underlying classic (Kolmogorov) probability theory. The quantum model is compared and contrasted with other competing explanations for these judgment errors including the representativeness heuristic, the averaging model, and a memory retrieval model for probability judgments. The quantum model also provides ways to extend Bayesian, fuzzy set, and fuzzy trace theories. We conclude that quantum information processing principles provide a viable and promising new way to understand human judgment and reasoning.
The conjunction fallacy refers to situations when a person judges a conjunction to be more likely than one of the individual conjuncts, which is a violation of a key property of classical probability theory. Recently, quantum probability (QP) theory has been proposed as a coherent account of these and many other findings on probability judgment "errors" that violate classical probability rules, including the conjunction fallacy. Tentori, Crupi, and Russo (2013) presented an alternative account of the conjunction fallacy based on the concept of inductive confirmation. They presented new empirical findings consistent with their account, and they also claimed that these results were inconsistent with the QP theory account. This comment proved that our QP model for the conjunction fallacy is completely consistent with the main empirical results from Tentori et al. (2013). Furthermore, we discuss experimental tests that can distinguish the 2 alternative accounts.
Human agents happen to judge that a conjunction of two terms is more probable than one of the terms, in contradiction with the rules of classical probabilities—this is the conjunction fallacy. One of the most discussed accounts of this fallacy is currently the quantum-like explanation, which relies on models exploiting the mathematics of quantum mechanics. The aim of this paper is to investigate the empirical adequacy of major quantum-like models which represent beliefs with quantum states. We first argue that they can be tested in three different ways, in a question order effect configuration which is different from the traditional conjunction fallacy experiment. We then carry out our proposed experiment, with varied methodologies from experimental economics. The experimental results we get are at odds with the predictions of the quantum-like models. This strongly suggests that this quantum-like account of the conjunction fallacy fails. Future possible research paths are discussed.