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Mathematical Structure Of Quantum Decision Theory

Author

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  • VYACHESLAV I. YUKALOV

    (Department of Management, Technology and Economics, ETH Zürich, Zürich CH-8032, Switzerland;
    Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia)

  • DIDIER SORNETTE

    (Department of Management, Technology and Economics, ETH Zürich, Zürich CH-8032, Switzerland;
    Swiss Finance Institute, c/o University of Geneva, CH 1211 Geneva 4, Switzerland)

Abstract

One of the most complex systems is the human brain whose formalized functioning is characterized by decision theory. We present a "Quantum Decision Theory" of decision-making, based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to explain a variety of interesting fallacies and anomalies that have been reported to particularize the decision-making of real human beings. The theory describes entangled decision-making, non-commutativity of subsequent decisions, and intention interference of composite prospects. We demonstrate how the violation of the Savage's sure-thing principle (disjunction effect) can be explained as a result of the interference of intentions, when making decisions under uncertainty. The conjunction fallacy is also explained by the presence of the interference terms. We demonstrate that all known anomalies and paradoxes, documented in the context of classical decision theory, are reducible to just a few mathematical archetypes, all of which allow the finding of straightforward explanations in the frame of the developed quantum approach.

Suggested Citation

  • Vyacheslav I. Yukalov & Didier Sornette, 2010. "Mathematical Structure Of Quantum Decision Theory," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 13(05), pages 659-698.
    • Handle: RePEc:wsi:acsxxx:v:13:y:2010:i:05:n:s0219525910002803
    DOI: 10.1142/S0219525910002803
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    References listed on IDEAS

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    1. Al-Najjar, Nabil I. & Weinstein, Jonathan, 2009. "The Ambiguity Aversion Literature: A Critical Assessment," Economics and Philosophy, Cambridge University Press, vol. 25(3), pages 249-284, November.
    2. Al-Najjar, Nabil I. & Weinstein, Jonathan, 2009. "Rejoinder: The “Ambiguity Aversion Literature: A Critical Assessment”," Economics and Philosophy, Cambridge University Press, vol. 25(3), pages 357-369, November.
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    Citations

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    Cited by:

    1. Thomas Boyer-Kassem & Sébastien Duchêne & Eric Guerci, 2016. "Quantum-like models cannot account for the conjunction fallacy," Theory and Decision, Springer, vol. 81(4), pages 479-510, November.
    2. Godfrey Cadogan, 2012. "Representation theory for risk on markowitz-tversky-kahneman topology," Economics Bulletin, AccessEcon, vol. 32(4), pages 1-34.
    3. Eichberger, Jürgen & Pirner, Hans Jürgen, 2018. "Decision theory with a state of mind represented by an element of a Hilbert space: The Ellsberg paradox," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 131-141.
    4. Huang, Zhiming & Yang, Lin & Jiang, Wen, 2019. "Uncertainty measurement with belief entropy on the interference effect in the quantum-like Bayesian Networks," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 417-428.
    5. Ismaël Rafaï & Sébastien Duchêne & Eric Guerci & Irina Basieva & Andrei Khrennikov, 2022. "The triple-store experiment: a first simultaneous test of classical and quantum probabilities in choice over menus," Theory and Decision, Springer, vol. 92(2), pages 387-406, March.
    6. Fellnhofer, Katharina & Sornette, Didier, 2022. "Embracing The Intuitive-Analytical Paradox? How Intuitive And Analytical Decision-Making Drive Paradoxes In Simple And Complex Environments," OSF Preprints evjd6, Center for Open Science.
    7. Maroussia Favre & Didier Sornette, 2015. "A Generic Model of Dyadic Social Relationships," PLOS ONE, Public Library of Science, vol. 10(3), pages 1-16, March.
    8. Maroussia Favre & Amrei Wittwer & Hans Rudolf Heinimann & Vyacheslav I Yukalov & Didier Sornette, 2016. "Quantum Decision Theory in Simple Risky Choices," PLOS ONE, Public Library of Science, vol. 11(12), pages 1-29, December.
    9. Yukalov, V.I. & Yukalova, E.P. & Sornette, D., 2022. "Role of collective information in networks of quantum operating agents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 598(C).
    10. Ashtiani, Mehrdad & Azgomi, Mohammad Abdollahi, 2015. "A survey of quantum-like approaches to decision making and cognition," Mathematical Social Sciences, Elsevier, vol. 75(C), pages 49-80.
    11. Eichberger, Jürgen & Pirner, Hans Jürgen, 2017. "Decision Theory with a Hilbert Space as Possibility Space," Working Papers 0637, University of Heidelberg, Department of Economics.
    12. Ferro, Giuseppe M. & Kovalenko, Tatyana & Sornette, Didier, 2021. "Quantum decision theory augments rank-dependent expected utility and Cumulative Prospect Theory," Journal of Economic Psychology, Elsevier, vol. 86(C).
    13. Luke Snow & Shashwat Jain & Vikram Krishnamurthy, 2022. "Lyapunov based Stochastic Stability of Human-Machine Interaction: A Quantum Decision System Approach," Papers 2204.00059, arXiv.org.

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    More about this item

    Keywords

    Decision theory; utility theory; paradoxes in decision-making; action interference and entanglement; action prospects; D03; D81; D83; D84;
    All these keywords.

    JEL classification:

    • D03 - Microeconomics - - General - - - Behavioral Microeconomics: Underlying Principles
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness
    • D84 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Expectations; Speculations

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