The conjunction fallacy and interference effects

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Abstract

In the present article we consider the conjunction fallacy, a well known cognitive fallacy experimentally tested in cognitive science, which occurs for intuitive judgments. We show that the quantum formalism can be used to describe, in a very simple way, this fallacy, in terms of interference effect. We provide evidence that the quantum formalism leads quite naturally to violations of the joint probability rule of classic probability when considering the estimated probability of the conjunction of two events. By defining the concept of maximal conjunction error, we find a good agreement with experimental results. Thus we suggest that in cognitive science the formalism of quantum mechanics can be used to describe a quantum regime, where the cognitive fallacies and heuristics are valid.

Introduction

This article addresses two main directions of research: the investigation of how the quantum formalism is compatible with the joint probability rule of classic probability theory, and the attempt to describe systems and situations with the quantum formalism that are very different from microscopic particles. A number of attempts have been made to apply the formalism of quantum mechanics to domains of science different from the micro-world with applications to economics (Baaquie, 2004, Bagarello, 2006, Haven, 2005, Schaden, 2002), operations research and management sciences (Bordley, 1998, Bordley and Kadane, 1999, Franco, 2007a), psychology and cognition (Aerts, submitted for publication, Busemeyer, Wang et al., 2006, Gabora and Aerts, 2002, Grossberg, 2000), game theory (Eisert et al., 1999, Piotrowski and Sladkowski, 2003), and language and artificial intelligence (Aerts and Czachor, 2004, Bruza and Cole, 2005, Widdows, 2003, Widdows and Peters, 2003). The present article is part of a research topic which can be named Quantum Decision Theory (for more details, see Busemeyer, Matthew, and Wang (2006), Franco (2007b), Franco (2008a), Khrennikov (2007), La Mura (2005), Mogiliansky, Zamir, and Zwirn (2006), Site Group of Quantum Decision Theory (2006), and Yukalov and Sornette (2008).

Quantum mechanics, for its counterintuitive predictions, seems to provide a good formalism to describe puzzling effects of contextuality. In the present article, we try to describe within the quantum formalism an important cognitive fallacy, the conjunction fallacy (Gigerenzer, 2000, Miyamoto et al., 1995, Shafir et al., 1990, Shafir and Tversky, 1992, Tentori et al., 2004, Tentori et al., in press, Tversky and Kahneman, 1982, Tversky and Kahneman, 1983, Wedell and Moro, 2008, Wells, 1985, Yates and Carlson, 1986). This fallacy evidences cognitive limitations of both knowledge and cognitive capacity, similarly to the bounded rationality (Simon, 1957), a central theme in behavioral economics and psychology which is concerned with the ways in which the actual decision-making process influences agents’ decisions. Previous attempts to describe features connected with this situation in terms of quantum formalism have been made in Bordley (1998) and in Franco (2007c).

The main results of this article are: (1) the opinion state of an agent for simple questions with only two possible answers can be represented by a two dimensional vector called a qubit; (2) different questions in such situations can be formally written as different sets of measurement operators acting on the cubit state; (3) the explicit answer of an agent to a question can be described as a collapse of the opinion state onto a basis vector corresponding to a measurement operator; (4) different questions represented by noncommuting measurement operators cause violations of the classic probability rule for joint events; (5) the predictions produced by this quantum model are consistent with previous experimental tests of the conjunctive fallacy.

In conclusion, we present a very general and abstract formalism which seems to describe the conjunction fallacy. Other fallacies and heuristics in the meanwhile have been studied and described within the quantum formalism, such as the disjunction fallacy (Busemeyer, Matthew et al., 2006), the inverse fallacy (Franco, 2008b), the framing effect (Khrennikov, 2007, Mogiliansky et al., 2006), the effects of risk and ambiguity (Franco, 2007b, La Mura, 2005), the gambler’s and hot hand fallacies (Franco, 2008a).

Section snippets

The conjunction fallacy

The conjunction fallacy is a well known cognitive fallacy which occurs when some specific conditions are assumed to be more probable than the general ones. More precisely, many people tend to ascribe higher probabilities to the conjunction of two events than to one of the single events. The most often-cited example of this fallacy found by Tversky and Kahneman, 1982, Tversky and Kahneman, 1983 is the case of Linda, which we will consider carefully in the present article. The test is preceded by

Quantum description of opinion states

In cognitive psychology, an important object of study is the degree of belief about an unknown event. In mathematical terms, it can be described as a judged or subjective probability: for intuitive judgements, we make the hypothesis that such probability can be computed with the formalism of quantum mechanics and of quantum probability. Our approach differs from the one of Busemeyer, Matthew et al. (2006) and of Yukalov and Sornette (2008), where the quantum probability is used to compute the

Interference effect and conjunction fallacy

Two dichotomic questions A and B are associated to two orthonormal bases of H, for example the base of feminist (a0,a1) and the base of bank teller (b0,b1). The opinion state can be expressed in one of such bases, which corresponds to the elaboration of a subjective probability relevant to the the question A or B respectively: we say that we have two different representations of the same opinion state. The change from one representation to another is a change of basis, which can be described as

Experimental results

The bibliography about the conjunction fallacy shows many experiments whose main scope is to show that the fallacy is lower when the problem is in frequency terms, or that the fallacy does not depend on a misinterpretation of the problem (Linda is feminist and bankteller, or Linda is feminist and not bankteller).

Here we list the main experimental results about the conjunction fallacy which evidence quantitative data: they are relevant to the mean conjunction error and to the probability of

Conclusions

This article, addressed both to quantum physicists and to experts of cognitive science, shows the incompatibility of quantum formalism with the joint probability rule of classic probability theory, by deriving the violation of Eq. (1) in intuitive judgements. In particular, we use mathematical objects like vector state and density matrix to describe the opinion state of agents, and hermitian operators for the questions: in Section 2 we show that the conjunction fallacy can be explained as an

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