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The quantum-like approach to modeling classical rationality violations: an introduction

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Abstract

Psychological empirical research has shown that human choice behavior often violates the assumptions of classical rational choice models. In the last few decades a new research field has emerged which aims to account for the observed choice behavior by resorting to the concepts and mathematical techniques developed in the realm of quantum physics, such as the “mental state vector” defined in a Hilbert space and the interference of quantum probability. This article is a short introduction to the quantum-like approach to the description of cognitive processes. I argue that the mathematical apparatus of quantum physics can account for the observed violations of classical logic and can help develop effective models of psychological and cognitive phenomena. This is illustrated through the so-called conjunction and disjunction fallacies by providing an alternative interpretation of the results of Linda test and Hawaii test. No-fallacy configurations are made possible in the quantum-like approach by sequential modeling of mental states transitions.

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Notes

  1. James was the first to speak about the existence of some sort of complementarity intrinsic in human knowledge: “It must be admitted, therefore, that in certain persons, at least, the total possible consciousness may be split into parts which coexist but mutually ignore each other, and share the objects of knowledge between them. More remarkable still, they are complementary” (James 1890, p. 206, italics original).

  2. Actually, the formalism of quantum physics has been applied to a number of scientific domains different from the microworld: mainly to psychology, cognition, decision making, economics, operational research, management, game theory, language and artificial intelligence (partial reviews are in: Busemeyer and Bruza 2012; Haven and Khrennikov 2013). Among the most promising researches, there are the attempts under way to construct quantum-like theories for the World Wide Web and Google as cognitive systems (see e.g.: Aerts et al. 2016).

  3. Allais (1953a, b; see also: 1954) was the first to show such incoherences in risky situations through a field study. Allais remarked that people feel differently and inconsistently about equal risks, depending on how identical situations are presented and perceived, as Kahneman and Tversky (1979) showed later (Allais’ conclusion was in fact an anticipation of the prospect theory of Tversky and Kahneman 1974, 1981). Additionally, Ellsberg (1961) theoretically analyzed choice under uncertainty. His considerations were later confirmed by empirical observations (e.g.: Chow and Sarin 2001; Dominiak et al. 2009; Fox and Tversky 1995; Heath and Tversky 1991). Uncertainty aversion was first put forward by Knight (1921), it was touched upon by Keynes (1921) and later theoretically explored in the Ellsberg paradox. Risk and uncertainty are both causes of contradictory choices, as individuals evaluate risk by connecting unknown probabilities to risk. In both cases, risk and uncertainty, mental processes take place that violate classical rationality assumptions and in turn result in incoherence of choices, commonly referred to as “irrational” behavior.

  4. The authors repeated the test in different conditions, with slightly different numbers, confirming the overall conclusion. Criticisms to Tversky and Kahneman’s explanation have been raised. Hertwig and Gigerenzer (1999) argued that ‘probability’ is polysemous in the general population, ranging from the mathematical meanings to ‘possibility’, ‘acceptability’, ‘believability’, ‘likeliness’, and the like. People are not irrational, as rationality for them does not mean mathematical logic: they normally perform intelligent semantic inferences based on some “social rationality” derived from communication experience (e.g., the maxims of Grice 1975), that can give the same utility to the individual at a lower cost. Mosconi and Macchi (2001) found that, given mild incentives, the proportion of individuals violating the conjunction principle is significantly lower than that reported by Kahneman and Tversky. Moreover, when subjects are allowed to consult with other subjects, this proportion falls dramatically (Charness et al. 2010). Science aims to explain empirical data through deduction from general theories. This should apply also to the cognitive sciences. Prospect theory accounts only for the cognitive bias of overestimating small probabilities and underestimating large probabilities. The empirical findings of judgmental heuristics, e.g. the results of Gigerenzer and Selten (2001) on the “toolbox” idea, lay the ground for a general theory for cognitive sciences, but such a theory is still missing at the moment.

  5. The author’s interpretation of the disjunction fallacy has been questioned. For example, Bagassi and Macchi (2006, 2007) repeated the experiment in different conditions and observed that the disjunction fallacy in some cases is less evident or even disappears and concluded that it does not depend on uncertainty, but on the introduction into the text-problem of a non-relevant goal, e.g., ‘paying to know’.

  6. The Copenhagen School has provided, since 1930 s, the most commonly accepted interpretation of the quantum theory. Paraphrasing Herbert Simon, we could say that it is not the optimal (i.e. logical, complete or, trivially said, ‘the true’) interpretation of quantum mechanics, but it is just the most satisfactory one, at least up to now.

  7. An order effect in psychology occurs e.g. when p(Ay, Bn) ≠ p(Bn, Ay), where y and n stand for answers ‘yes’ and ‘no’ to dichotomous questions A and B. Even if order effects occur, it can be shown that the quantum question order model predicts the equality:

    $$ [p\left( {{\text{A}}y,{\text{B}}n} \right) - p\left( {{\text{B}}n,{\text{A}}y} \right)] = - [p\left( {{\text{A}}n,{\text{B}}y} \right) - p\left( {{\text{B}}y,{\text{A}}n} \right)]. $$

  8. Stated differently, the Born rule says that the probability density of finding a particle at a given point is proportional to the square of the magnitude of the particle’s wave-function computed at that point.

  9. This is analogous to what happens in the well-known Schrödinger’s cat thought experiment: the act of measuring by opening the box causes the cat’s state to collapse onto one of the states ‘cat alive’ or ‘cat dead’.

  10. The quantum-like model of Aerts (2009) (see also Aerts and Gabora 2005a, b) was especially developed to give an account of the conjunctions and disjunctions fallacies emerged in the huge amount of data collected in a field research by James Hampton (1988a, b). Hampton carried out a series of interviews on the perceived typicality (in the sense of Eleanor Rosch 1973) of a long list of items to either of the two proposed classes (concepts), to their conjunctions and their disjunctions. The disjunction and conjunction of concepts A and B are both represented in Aert’s model as the normalized superposition. Aerts presents also a more sophisticated model for Hampton’s data based on Fock spaces, which incorporates both classical and emergent concepts within the same Hilbert space.

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    Franco Vaio

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Vaio, F. The quantum-like approach to modeling classical rationality violations: an introduction. Mind Soc 18, 105–123 (2019). https://doi.org/10.1007/s11299-019-00212-5

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