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arXiv:1606.03430 · Theory and Decision

Quantum-like models cannot account for the conjunction fallacy

Thomas Boyer‐Kassem, Sébastien Duchêne, Éric Guerci

Published Apr 22, 2016• 40 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s11238-016-9549-9

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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.
In classical probability theory, the probability of the conjunction of two events is smaller than the probability of only one of these events.?Yet, agents do not always empirically judge in this way: this is the traditional conjunction fallacy.?One of the currently promising accounts of this paradox relies on so-called quantum-like models, which have been developed from mathematical tools used in quantum theory.?But are these models empirically adequate??Which versions of these models can be used??In particular, can the simplest versions, the non-degenerate ones, be sufficient??We propose here an original experimental protocol to test the quantum-like models for the conjunction fallacy in the lab.?The results we obtain suggest that the non-degenerate models are not empirically adequate, and that future research on quantum-like models should consider degenerate ones. Classification JEL : C60, C91, D03.
© 2017 Springer Science+Business Media New YorkThe ‘conjunction fallacy’ has been extensively debated by scholars in cognitive science and, in recent times, the discussion has been enriched by the proposal of modeling the fallacy using the quantum formalism. Two major quantum approaches have been put forward: the first assumes that respondents use a two-step sequential reasoning and that the fallacy results from the presence of ‘question order effects’; the second assumes that respondents evaluate the cognitive situation as a whole and that the fallacy results from the ‘emergence of new meanings’, as an ‘effect of overextension’ in the conceptual conjunction. Thus, the question arises as to determine whether and to what extent conjunction fallacies would result from ‘order effects’ or, instead, from ‘emergence effects’. To help clarify this situation, we propose to use the World Wide Web as an ‘information space’ that can be interrogated both in a sequential and non-sequential way, to test these two quantum approaches. We find that ‘emergence effects’, and not ‘order effects’, should be considered the main cognitive mechanism producing the observed conjunction fallacies.
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.
In the present article we consider the conjunction fallacy, a well known cognitive heuristic experimentally tested in cognitive science, which occurs for intuitive judgments in situations of bounded rationality. We show that the quantum formalism can be used to describe in a very simple way this fallacy in terms of interference effect. We evidence that the quantum formalism leads quite naturally to violations of Bayes' rule when considering the probability of the conjunction of two events. Thus we suggest that in cognitive science the formalism of quantum mechanics can be used to describe a \textit{quantum regime}, the bounded-rationality regime, where the cognitive heuristics are valid.
Lately, so-called 'quantum models' based on parts of the mathematics of quantum mechanics, have been developed in decision theory and cognitive sciences to account for seemingly irrational or paradoxical human judgments. In this paper, we limit ourselves to such quantum-like models that address order effects. It has been argued that such models are able to account for existing and new empirical data, and meet some a priori predictions. From the quantum law of reciprocity, we derive new empirical predictions that we call the Grand Reciprocity equations, that must be satisfied by quantum-like models on the condition that they are non-degenerate. We show that existing non-degenerate quantum-like models for order effects fail this test on several existing data sets. We take it to suggest that degenerate quantum-like models should be the focus of forthcoming research in the area.