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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.

References

  • Aerts D (2009) Quantum structure in cognition. J Math Psychol 53:314–348

    Article  Google Scholar 

  • Aerts D, Gabora L (2005a) A theory of concepts and their combinations I: the structure of the space sets of contexts and properties. Kybernetes 34:167–191

    Article  Google Scholar 

  • Aerts D, Gabora L (2005b) A theory of concepts and their combinations II: a Hilbert space representation. Kybernetes 34:192–221

    Article  Google Scholar 

  • Aerts D, Aerts Arguëlles J, Beltran L, Beltran L, Sassoli de Bianchi M, Sozzo S, Veloz T (2016) Testing quantum models of conjunction fallacy on the world wide web. arXiv:1609.07721v2

  • Allais M (1953a) La psychologie de l’homme rationnel devant le risque: La théorie et l’expérience. Journal de la Société de Statistique de Paris, Janvier 1953:47–72

    Google Scholar 

  • Allais M (1953b) Le comportement de l’homme rationnel devant le risque. Critique des postulats et des axiomes de l’école américaine. Econometrica 21:503–546

    Article  Google Scholar 

  • Allais M (1954) Puissance et dangers de l’utilisation de l’outil mathématique en économique. Econometrica 22:58–71

    Article  Google Scholar 

  • Arndt M, Nairz O, Vos-Andreae J, Keller C, van der Zouw G, Zeilinger A (1999) Wave-particle duality of C60 molecules. Nature 401(October):680–682

    Article  Google Scholar 

  • Atmanspacher H (2012) Dual-aspect Monism à la Pauli and Jung. J Conscious Stud 19(9/10):96–120

    Google Scholar 

  • Atmanspacher H, Fach W (2013) A structural-phenomenological typology of mind-matter correlations. J Anal Psychol 58:219–244

    Article  Google Scholar 

  • Atmanspacher H, Primas H (eds) (2009) Recasting reality. Wolfgang Pauli’s philosophical ideas and contemporary science. Springer, Heidelberg

    Google Scholar 

  • Atmanspacher H, Rotter S (2008) Interpreting neurodynamics: concepts and facts. Cogn Neurodyn 2:297–318

    Article  Google Scholar 

  • Atmanspacher H, Römer H, Walach H (2002) Weak quantum theory: complementarity and entanglement in physics and beyond. Found Phys 32:379–406

    Article  Google Scholar 

  • Bagassi M, Macchi L (2006) Pragmatic approach to decision making under uncertainty: the case of the disjunction effect. Think Reason 12(3):329–350

    Article  Google Scholar 

  • Bagassi M, Macchi L (2007) The ‘vanishing’ of the disjunction effect by sensible procrastination. Mind Soc 6:41–52

    Article  Google Scholar 

  • Barrett JA (1999) The quantum mechanics of minds and worlds. Oxford University Press, Oxford

    Google Scholar 

  • Barrett JA (2006) A quantum-mechanical argument for mind-body dualism. Erkenntnis 65:97–115

    Article  Google Scholar 

  • Bertuglia CS, Vaio F (2003) Non linearità, caos, complessità. Bollati Boringhieri, Torino

    Google Scholar 

  • Bertuglia CS, Vaio F (2005) Non linearity, chaos and complexity. Oxford University Press, Oxford

    Google Scholar 

  • Bertuglia CS, Vaio F (2011) Complessità e modelli. Un nuovo quadro interpretativo per la modellizzazio-ne nelle scienze della natura e della società. Bollati Boringhieri, Torino

    Google Scholar 

  • Bertuglia CS, Vaio F (2019) Il fenomeno urbano e la complessità. Concezioni sociologiche, antropologi-che ed economiche di un sistema complesso territoriale. Bollati Boringhieri, Torino

    Google Scholar 

  • Bohm D, Hiley B (1993) The undivided universe. An ontological interpretation of quantum theory. Routledge, London

    Google Scholar 

  • Bohr N (1948) On the notions of causality and complementarity. Dialectica 2(3–4):312–319

    Article  Google Scholar 

  • Born M (1926) Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37:863–867

    Article  Google Scholar 

  • Bruza PD, Kitto K, Ramm BJ, Sitbon L (2014) A probabilistic framework for analysing the compositionality of conceptual combinations. J Math Psychol 67:26–38

    Article  Google Scholar 

  • Busemeyer JR, Bruza PD (2012) Quantum models of cognition and decision. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Busemeyer JR, Franco R (2010) What is the evidence for quantum-like interference effects in human judgments and decision behavior? NeuroQuantology 8(4):S48–S62

    Article  Google Scholar 

  • Busemeyer JR, Wang Zh (2015) What is quantum cognition, and how is it applied to psychology? Curr Dir Psychol Sci 24:165–169

    Article  Google Scholar 

  • Busemeyer JR, Pothos EM, Franco R, Trueblood JS (2011) A quantum theoretical explanation for probability judgment errors. Psychol Rev 118(2):193–218

    Article  Google Scholar 

  • Butterfield J (1998) Quantum curiosities of psychophysics. In: Cornwell J (ed) Consciousness and human identity. Oxford University Press, Oxford, pp 122–157

    Google Scholar 

  • Carnal O, Mlynek J (1991) Young’s double-slit experiment with atoms: a simple atom interferometer. Phys Rev Lett 66:2689–2692

    Article  Google Scholar 

  • Charness G, Karni E, Levin D (2010) On the conjunction fallacy in probability judgment: new experimental evidence regarding Linda. Games Econ Behav 68:551–556

    Article  Google Scholar 

  • Chow CCh, Sarin RK (2001) Comparative ignorance and the Ellsberg paradox. J Risk Uncertain 22(2):129–139

    Article  Google Scholar 

  • Dirac PAM (1930) The principles of quantum mechanics. Oxford University Press, Oxford

    Google Scholar 

  • Dominiak A, Dürsch P, Lefort J-P (2009) A dynamic Ellsberg urn experiment. Discussion Paper Series, No. 487, University of Heidelberg, Department of Economics, Heidelberg

  • Ellsberg D (1961) Risk, ambiguity, and the savage axioms. Q J Econ 75:643–669

    Article  Google Scholar 

  • Facchi P, Mariano A, Pascazio P (2002) Mesoscopic interference. Recent Res Dev Phys (Transw Res Netw) 3:1–29

    Google Scholar 

  • Fox CR, Tversky A (1995) Ambiguity aversion and comparative ignorance. Q J Econ 110:585–603

    Article  Google Scholar 

  • Franco R (2009) The conjunctive fallacy and interference effects. J Math Psychol 53:415–422

    Article  Google Scholar 

  • Gigerenzer G, Goldstein DG (1996) Reasoning the fast and frugal way: models of bounded rationality. Psychol Rev 103:650–669

    Article  Google Scholar 

  • Gigerenzer G, Selten R (2001) Bounded rationality. The adaptive toolbox. MIT Press, Cambridge

    Google Scholar 

  • Gigerenzer G, Hertwig R, Pachur Th (eds) (2011) Heuristics. The foundations of adaptive behavior. Oxford University Press, Oxford

    Google Scholar 

  • Grice P (1975) Logic and conversation. In: Cole P, Morgan JL (eds) Syntax and semantics, vol 3. Academic Press, New York, pp 41–58

    Google Scholar 

  • Hackermüller L, Hornberger K, Brezger B, Zeilinger A, Arndt M (2004) Decoherence of matter waves by thermal emission of radiation. Nature 427(February):711–714

    Article  Google Scholar 

  • Hagan S, Hameroff SR, Tuszyński JA (2002) Quantum computation in brain microtubules. Decoherence and biological feasibility. Phys Rev E 65:061901-1–061901-11

    Article  Google Scholar 

  • Hameroff SR (1994) Quantum coherence in microtubules. A neural basis for emergent consciousness? J Conscious Stud 1:91–118

    Google Scholar 

  • Hameroff SR (1998) Quantum computing in brain microtubules? The Penrose–Hameroff “Orch OR” model of consciousness. Philos Trans R Soc A 356:1869–1896

    Article  Google Scholar 

  • Hameroff SR (2007) The brain is both neurocomputer and quantum computer. Cogn Sci 31:1035–1045

    Article  Google Scholar 

  • Hameroff SR (2012) How quantum brain biology can rescue conscious free will. Front Integr Neurosci 6:1–17

    Article  Google Scholar 

  • Hampton JA (1988a) Overextension and conjuntive concepts: evidence for a unitary model for concept typicality and class inclusion. J Exp Psychol Learn Mem Cogn 14:12–32

    Article  Google Scholar 

  • Hampton JA (1988b) Disjunction of natural concepts. Mem Cognit 16:579–591

    Article  Google Scholar 

  • Haven E, Khrennikov AYu (2013) Quantum social science. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Heath C, Tversky A (1991) Preference and belief: ambiguity and competence in choice under uncertainty. J Risk Uncertain 4:5–28

    Article  Google Scholar 

  • Hertwig R, Gigerenzer G (1999) The ‘conjunction fallacy’ revisited: how intelligent inferences look like reasoning errors. J Behav Decis Mak 12:275–305

    Article  Google Scholar 

  • James W (1884) The dilemma of determinism. Unit Rev (Sept 1884) [republished in: (1897/2007) The will to believe and other essays in popular philosophy. Cosimo Classics, New York, pp 145–182]

  • James W (1890) Principles of psychology, 2 volumes. Henry Holt, New York

    Book  Google Scholar 

  • Jung CG, Pauli W (1952) Naturerklärung und Psyche. Rascher, Zurich [English edition: (1955) The interpretation of nature and the psyche; Jung CG, Synchronicity. An acausal connecting principle; Pauli W, The influence of archetypal ideas on the scientific theories of Kepler. Pantheon, New York]

  • Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47:263–291

    Article  Google Scholar 

  • Kaiser D (2011) How the Hippies saved physics. Science, counterculture, and the quantum revival. W.W. Norton & Co, New York

    Google Scholar 

  • Keynes JM (1921) A treatise on probability. Macmillan & Co., London

    Google Scholar 

  • Khrennikov AYu (2010) Ubiquitous quantum structure. From psychology to finance. Springer, Heidelberg

    Book  Google Scholar 

  • Khrennikov AY, Haven E (2008) The importance of probability interference in social science: rationale and experiment. arXiv:0709.2802v1

  • Khrennikov AYu, Haven E (2009) Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts. J Math Psychol 53:378–388

    Article  Google Scholar 

  • Knight FH (1921) Risk, uncertainty and profit. The University of Chicago Press, Chicago

    Google Scholar 

  • Meier CA (ed) (1992) Wolfgang Pauli und C.G. Jung. Ein Briefwechsel, 1932–1958. Springer, Heidelberg

    Google Scholar 

  • Merli GP, Missiroli GF, Pozzi G (1976) On the statistical aspect of electron interference phenomena. Am J Phys 44:306–307

    Article  Google Scholar 

  • Mohrhoff U (2005) The Pondicherry interpretation of quantum mechanics: an overview. PRAMANA J Phys Indian Acad Sci 64(2):171–185

    Article  Google Scholar 

  • Mohrhoff U (2006) Quantum mechanics explained. arXiv:0607005v2

  • Mohrhoff U (2011) The world according to quantum mechanics. Why the laws of physics make perfect sense after all. World Scientific, Singapore

    Book  Google Scholar 

  • Mosconi G, Macchi L (2001) The role of pragmatic rules in the conjunction fallacy. Mind Soc 3:31–57

    Article  Google Scholar 

  • Murdoch D (1987) Niels Bohr’s philosophy of physics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Neth H, Gigerenzer G (2015) Heuristic: tools for an uncertain world. In: Scott R, Kosslyn S (eds) Emerging trends in the social and behavioral sciences. An interdisciplinary, searchable and linkable research. Wiley, New York, pp 1–18

    Google Scholar 

  • Penrose R (1989) The emperor’s new mind. Concerning computers. Minds and the laws of physics. Oxford University Press, Oxford

    Google Scholar 

  • Penrose R (1994) Shadows of the mind. Oxford University Press, Oxford

    Google Scholar 

  • von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Springer, Heidelberg

    Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

  • Petersen A (1963) The philosophy of Niels Bohr. Bull Atomic Sci 19:8–14

    Article  Google Scholar 

  • Pothos EM, Busemeyer JR (2009) A quantum probability explanation for violations of ‘rational’ decision theory. Proc R Soc B 276(1165):2171–2178

    Article  Google Scholar 

  • Pothos EM, Busemeyer JR (2011) Formalizing heuristics in decision-making: a quantum probability perspective. Front Psychol 2:289

    Article  Google Scholar 

  • Pothos EM, Busemeyer JR (2013) Can quantum probability provide a new direction for cognitive modeling? Behav Brain Sci 36:255–327

    Article  Google Scholar 

  • Rosch EH (1973) Natural categories. Cogn Psychol 4:328–350

    Article  Google Scholar 

  • Savage LJ (1954) The foundations of statistics. Wiley, New York

    Google Scholar 

  • Schrödinger E (1944) What is life? The physical aspect of the living cell. Cambridge University Press, Cambridge

    Google Scholar 

  • Squires E (1990) Conscious mind in the physical world. Adam Hilger, Bristol

    Book  Google Scholar 

  • Stapp HP (1979) Whiteheadian approach to quantum theory and the generalized Bell’s theorem. Found Phys 9(1/2):1–25

    Article  Google Scholar 

  • Stapp HP (1993) Mind, matter and quantum mechanics. Springer, Heidelberg

    Book  Google Scholar 

  • Stapp HP (1998) Whiteheadian process and quantum theory of mind. Lawrence Berkeley National Laboratory Report, LBNL-42143

  • Stapp HP (2001) Quantum theory and the role of mind in nature. Found Phys 31:1465–1499

    Article  Google Scholar 

  • Stapp HP (2004) Mind, matter, and quantum mechanics. Springer, Heidelberg

    Book  Google Scholar 

  • Stapp HP (2005) Quantum interactive dualism: an alternative to materialism. J Conscious Stud 12(11):43–58

    Google Scholar 

  • Stapp HP (2007a) Mindful universe: quantum mechanics and the participating observer. Springer, Heidelberg

    Google Scholar 

  • Stapp HP (2007b) Whitehead, James, and the ontology of quantum theory. Mind Matter 5(1):83–109

    Google Scholar 

  • Stapp HP (2008) A model of the quantum-classical and mind-brain connections, and of the role of the quantum Zeno effect in the physical implementation of conscious intent. arXiv:0803.1633v1

  • Tonomura A, Endo J, Matsuda T, Kawasaki T, Ezawa H (1989) Demonstration of single-electron buildup of an interference pattern. Am J Phys 57:117–120

    Article  Google Scholar 

  • Trueblood JS, Busemeyer JR (2011) A quantum probability account of order effects in inference. Cogn Sci 35:1518–1552

    Article  Google Scholar 

  • Trueblood JS, Pothos EM, Busemeyer JR (2014) Quantum probability theory as a common framework for reasoning and similarity. Front Psychol 5(322):1–4

    Google Scholar 

  • Tversky A, Kahneman D (1974) Judgment under uncertainty: heuristics and biases. Science 185:1124–1131

    Article  Google Scholar 

  • Tversky A, Kahneman D (1981) The framing of decisions and the psychology of choice. Science 211:453–458

    Article  Google Scholar 

  • Tversky A, Kahneman D (1983) Extensional vs. intuitive reasoning: the conjunction fallacy in probability judgment. Psychol Rev 90:293–315

    Article  Google Scholar 

  • Tversky A, Shafir E (1992) The disjunction effect in choice under uncertainty. Psychol Sci 3:305–309

    Article  Google Scholar 

  • Vaio F (2015) Quantum-like probability interference in judgment and decision: the approach of quantum cognition to modeling classical rationality violations. In: Bonazzi M, Di Simone V (eds) Redesigning worldwide connections. Cambridge Scholars Publishing, Newcastle upon Tyne, pp 142–180

    Google Scholar 

  • Wang Zh, Busemeer JR, Atmanspacher H, Pothos EM (2013) The potential of using quantum theory to build models of cognition. Top Cognit Sci 5(4):1–17

    Google Scholar 

  • Wang Zh, Soloway T, Shiffrin RM, Busemeyer JR (2014) Context effects produced by question orders reveal quantum nature of human judgments. PNAS 111(26):9431–9436

    Article  Google Scholar 

  • Whitehead AN (1929) Process and reality. An essay in cosmology. Macmillan & Co., London [corrected edition: Griffin DR, Sherburne DW (eds) (1978) Free Press, New York]

  • Whitehead AN (1933) Adventures of ideas. Cambridge University Press, London

    Google Scholar 

  • Young Th (1802) On the theory of light and colours (The 1801 Bakerian lecture). Philos Trans R Soc 92:12–48

    Article  Google Scholar 

  • Young Th (1804) Experiments and calculations relative to physical optics (The 1803 Bakerian lecture). Philos Trans R Soc 94:1–16

    Google Scholar 

  • Young Th (1807) A course of lectures on natural philosophy and the mechanical arts, two volumes. Joseph Johnson, London

    Google Scholar 

  • Yukalov VI, Sornette D (2008) Mathematical structure of quantum decision theory. arXiv:0808.0112v3

  • Zeilinger A, Gähler R, Shull CG, Treimer W, Mampe W (1988) Single- and double-slit diffraction of neutrons. Rev Mod Phys 60(4):1067–1073

    Article  Google Scholar 

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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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