Skip to main content
SpringerLink
Log in

Decision theory with prospect interference and entanglement

  • Published:
Theory and Decision Aims and scope Submit manuscript

Abstract

We present a novel variant 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 describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage’s sure-thing principle, known as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analyzed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect, and novel experiments testing the combined interplay between the two effects are suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Allais M. (1953) Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’ecole Américaine. Econometrica 21: 503–546

    Article  Google Scholar 

  • Al-Najjar, N. I., & Weinstein J. (2009). The ambiguity aversion literature: a critical assessment, Kellogg School of Management Working Paper, Northwestern University.

  • Atmanspacher H. (2003) Mind and matter as asymptotically disjoint inequivalent representations with broken time-reversal symmetry. Biosystems 68: 19–30

    Article  Google Scholar 

  • Atmanspacher H., Römer H., Walach H. (2002) Weak quantum theory: complementarity and entanglement in physics and beyond. Foundation of Physics 22: 379–406

    Article  Google Scholar 

  • Bather J. (2000) Decision theory. Wiley, Chichester

    Google Scholar 

  • Barkan R., Danziger S., Ben-Bashat G., Busemeyer J. R. (2005) Framing reference points: The effect of integration and segregation on dynamic inconsistency. Journal of Behaal Decision Making 18: 213–226

    Article  Google Scholar 

  • Baron J. (1998) Judgement misguided: Intuition and error in public decision-making. Oxford University, Oxford

    Google Scholar 

  • Bechara A., Damasio H., Damasio A. (2000) Emotion, decision making and the orbitofrontal cortex. Cerebral Cortex 10: 295–307

    Article  Google Scholar 

  • Beck F., Eccles J. (1992) Quantum aspects of brain activity and the role of consciousness. Proceedings of National Academy of Sciences of USA, 89: 11357–11361

    Article  Google Scholar 

  • Bell J. S. (1964) On the Einstein-Podolsky-Rosen paradox. Physics 1: 195–200

    Google Scholar 

  • Benioff P. A. (1972) Decision procedures in quantum mechanics. Journal of Mathematical Physics 13: 909–915

    Google Scholar 

  • Benjamin S. C., Hayden P. M. (2001) Multi-player quantum games. Physical Review A 64: 030301

    Article  Google Scholar 

  • Berger J. O. (1985) Statistical decision theory and bayesian analysis. Springer, New York

    Google Scholar 

  • Bohr N. (1929) Wirkungsquantum und Naturbeschreibung. Naturwissenschaft 17: 483–486

    Article  Google Scholar 

  • Bohr, N. (1933). Light and life. Nature, 131, 421–423, 457–459.

    Google Scholar 

  • Bohr N. (1937) Kausalität und Komplemetarität. Erkenntnissenscahft 6: 293–303

    Google Scholar 

  • Bohr N. (1961) La Physique Atomique et la Connaissance Humaine. Gontier, Genève

    Google Scholar 

  • Buchanan J. T. (1982) Discrete and dynamic decision analysis. Wiley, Chichester

    Google Scholar 

  • Busemeyer J. R., Wang Z., Townsend J. T. (2006) Quantum dynamics of human decision- making. Journal of Mathematical Psychology 50: 220–241

    Article  Google Scholar 

  • Camerer, C. F., Loewenstein, G., Rabin, R. (eds) (2003) Advances in behavioral economics. Princeton University, Princeton

    Google Scholar 

  • Chalmers D. (1996) The conscious mind. Oxford University, Oxford

    Google Scholar 

  • Charness, G. B., Levin, D., & Karni, E. (2008). On the conjunction fallacy in probability judgement: New experimental evidence. Department of Economics Working Paper, UCSB.

  • Cohen M., Tallon J. M. (2000) Décision dans le risque et l’incertain: l’apport des modèles non additifs. Revue d’Economie Politique 110: 631–681

    Google Scholar 

  • Croson R. T. A. (1999) The disjunction effect and reason-based choice in games. Organizational Behavior and Human Decision Processes 80: 118–133

    Article  Google Scholar 

  • Dickhaut J., McCabe K., Nagode J.C., Rustichini A., Smith K., Pardo J.V. (2003) The impact of the certainty context on the process of choice. Proceedings of National Academy of Sciences of USA 100: 3536–3541

    Article  Google Scholar 

  • Dieudonné J. (2006) Foundations of modern analysis. Hesperides, London

    Google Scholar 

  • Dirac P. A. M. (1958) The principles of quantum mechanics. Clarendon, Oxford

    Google Scholar 

  • Du J., Li H., Xu X., Shi M., Wu J., Zhou X., Han R. (2002) Experimental realization of quantum games on a quantum computer. Physical review letters 88: 137902

    Article  Google Scholar 

  • Du J., Xu X., Li H., Zhou X., Han R. (2001) Entanglement playing a dominating role in quantum games. Physics Letters A 289: 9–15

    Article  Google Scholar 

  • Einstein A., Podolsky B., Rosen N. (1935) Can quantum-mechanical description of physical reality be considered complete?. Physical Review 47: 777–780

    Article  Google Scholar 

  • Eisert J., Wilkens M. (2000) Quantum games. Journal of Modern Optics 47: 2543–2556

    Google Scholar 

  • Epstein L. G. (1999) A definition of uncertainty aversion. The Review of Economic Studies 66: 579–608

    Article  Google Scholar 

  • Feller W. (1970) Introduction to probability theory and its applications. Wiley, New York

    Google Scholar 

  • Fox C., Rogers B., Tversky A. (1996) Option traders exhibit subadditive decision weights. Journal of Risk and Uncertainty 13: 5–17

    Article  Google Scholar 

  • Frederick S., Loewenstein G., O’Donoghue T. (2002) Time discounting and time preference: A critical review. Journal of Economic Literature 40: 351–401

    Article  Google Scholar 

  • French S., Insua D. R. (2000) Statistical decision theory. Arnold, London

    Google Scholar 

  • Frölich H. (1968) Bose condensation of strongly excited longitudinal electric modes. Physics Letters A 26: 402–403

    Article  Google Scholar 

  • Gilboa I. (1987) Expected utility with purely subjective non-additive probabilities. Journal of Mathematical Economics 16: 65–88

    Article  Google Scholar 

  • Gilboa I., Schmeidler D. (1989) Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18: 141–153

    Article  Google Scholar 

  • Goldenberg L., Vaidman L., Wiesner S. (1999) Quantum gambling. Physical Review Letters 82: 3356–3359

    Article  Google Scholar 

  • Hagan S., Hameroff S. R., Tuszynski J. A. (2002) Quantum computation in brain microtubules: Decoherence and biological feasibility. Physical Review E 65: 061901

    Article  Google Scholar 

  • Hastings N. A., Mello J. M. (1978) Decision networks. Wiley, Chichester

    Google Scholar 

  • Iqbal A., Toor A. H. (2001) Evolutionally stable strategies in quantum games. Physics Letters A 280: 249–256

    Article  Google Scholar 

  • Johnson N. F. (2001) Playing a quantum game with a corrupted source. Physical Review A 63: 020302

    Article  Google Scholar 

  • Kaplan S., Garrick B. J. (1981) On the quantitative definition of risk. Risk Analysis 1: 11–27

    Article  Google Scholar 

  • Kühberger A., Komunska D., Perner J. (2001) The disjunction effect: Does it exist for two-step gambles?. Organizational Behavior and Human Decision Processes 85: 250–264

    Article  Google Scholar 

  • Lambdin C., Burdsal C. (2007) The disjunction effect reexamined: Relevant methodological issues and the fallacy of unspecified percentage comparisons. Organizational Behavior and Human Decision Processes 103: 268–276

    Article  Google Scholar 

  • Lee C. F., Johnson N. F. (2003) Efficiency and formalism of quantum games. Physical Review A 67: 022311

    Article  Google Scholar 

  • Li C. F., Zhang Y. S., Huang Y. F., Guo G. C. (2001) Quantum strategies of quantum measurements. Physics Letters A 280: 257–260

    Article  Google Scholar 

  • Li S., Taplin J. E., Zhang Y. (2007) The equate-to-differentiate way of seeing the prisoner’s dilemma. Information Sciences 177: 1395–1412

    Article  Google Scholar 

  • Lindgren B. W. (1971) Elements of decision theory. Macmillan, New York

    Google Scholar 

  • Lockwood M. (1989) Mind, brain and the quantum. Basil Blackwell, Oxford

    Google Scholar 

  • Machina M. J. (2008) Non-expected utility theory. In: Durlauf S. N., Blume L. E. (eds) The new palgrave dictionary of economics. Macmillan, Basingstoke

    Google Scholar 

  • Marshall K. T., Oliver R. M. (1995) Decision making and forecasting. McGraw-Hill, New York

    Google Scholar 

  • Mendelson E. (1965) Introduction to mathematical logic. Van Nostrand, Princeton

    Google Scholar 

  • Meyer D. (1999) Quantum strategies. Physical Review Letters 82: 1052–1055

    Article  Google Scholar 

  • Montesano A. (2008) Effects of uncertainty aversion on the call option market. Theory and Decision 65: 97–123

    Article  Google Scholar 

  • Morier D. M., Borgida E. (1984) The conjunction fallacy: A task-specific phenomenon?. Personality Social Psychology Bulletin 10: 243–253

    Article  Google Scholar 

  • Penrose R. (1989) The emperor’s new mind. Oxford University, Oxford

    Google Scholar 

  • Pessa E., Vitiello G. (2003) Quantum noise, entanglement and chaos in the quantum field theory of mind-brain states. Mind and Matter 1: 59–79

    Google Scholar 

  • Primas H. (2003) Time-entanglement between mind and matter. Mind and Matter 1: 81–119

    Google Scholar 

  • Quiggin J. (1982) A theory of anticipated utility. Journal of Economic Behavior and Organization 3: 323–343

    Article  Google Scholar 

  • Raiffa H., Schlaifer R. (2000) Applied statistical decision theory. Wiley, New York

    Google Scholar 

  • Rivett P. (1980) Model building for decision analysis. Wiley, Chichester

    Google Scholar 

  • Rottenstreich Y., Tversky A. (1997) Unpacking, repacking and anchoring: Advances in support theory. Psychological Review 104: 406–415

    Article  Google Scholar 

  • Safra Z., Segal U. (2008) Calibration results for non-expected utility theories. Econometrica 76: 1143–1166

    Article  Google Scholar 

  • Satinover J. (2001) The quantum brain. Wiley, New York

    Google Scholar 

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

    Google Scholar 

  • Schmeidler D. (1989) Subjective probability and expected utility without additivity. Econometrica 57: 571–587

    Article  Google Scholar 

  • Shafir E. (1994) Uncertainty and the difficulty of thinking through disjunctions. Cognition 50: 403–430

    Article  Google Scholar 

  • Shafir E., Simonson I., Tversky A. (1993) Reason-based choice. Cognition 49: 11–36

    Article  Google Scholar 

  • Shafir E., Tversky A. (1992) Thinking through uncertainty: Nonconsequential reasoning and choice. Cognitive Psychology 24: 449–474

    Article  Google Scholar 

  • Shafir E. B., Smith E. E., Osherson D. N. (1990) Typicality and reasoning fallacies. Memory and Cognition 18: 229–239

    Article  Google Scholar 

  • Sheremeta, R. M., & Zhang, J. (2009). Can groups solve the problem of over-building in contests? Department of Economics Working Paper, McMaster University.

  • Shor P. (1997) Polynomial-type algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal of Scientific and Statistical Computing 26: 1484–1494

    Google Scholar 

  • Simon H. A. (1955) A behavioral model of rational choice. Quarterly Journal of Economics 69: 99–118

    Article  Google Scholar 

  • Sornette D., Davis A.B., Ide K., Vixie K.R., Pisarenko V., Kamm J.R. (2007) Algorithm for model validation: Theory and applications. Proceedings of National Academy of Sciences of USA 104: 6562–6567

    Article  Google Scholar 

  • Stapp H. P. (1993) Mind, matter, and quantum mechanics. Springer, Berlin

    Google Scholar 

  • Stapp H. P. (1999) Attention, intention, and will in quantum physics. Journal of Consciousness Studies 6: 143–164

    Google Scholar 

  • Stuart C. I. J., Takahashi Y., Umezawa H. (1978) On the stability and non-local properties of memory. Journal of Theoretical Biology 71: 605–618

    Article  Google Scholar 

  • Stuart C. I. J., Takahashi Y., Umezawa H. (1979) Mixed system brain dynamics: Neural memory as a macroscopic ordered state. Foundations of Physics 9: 301–327

    Article  Google Scholar 

  • Tegmark M. (2000) Importance of quantum decoherence in brain processes. Physical Review E 61: 4194–4205

    Article  Google Scholar 

  • Tentori K., Bonini N., Osherson D. (2004) The conjunction fallacy: A misunderstanding about conjunction?. Cognitive Science 28: 467–477

    Article  Google Scholar 

  • Tversky A., Kahneman D. (1973) Availability: A heuristic for judging frequency and probability. Cognitive Psychology 5: 207–232

    Article  Google Scholar 

  • Tversky A., Kahneman D. (1980) Judgements of and by representativeness. In: Kahneman D., Slovic P., Tversky A. (eds) Judgements under uncertainty: Heuristics and biases. Cambridge University, New York, pp 84–98

    Google Scholar 

  • Tversky A., Kahneman D. (1983) Extensional versus intuitive reasoning: The conjunction fallacy in probability judgement. Psychological Review 90: 293–315

    Article  Google Scholar 

  • Tversky A., Koehler D. (1994) Support theory: A nonexistential representation of subjective probability. Psychological Review 101: 547–567

    Article  Google Scholar 

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

    Article  Google Scholar 

  • van Enk S. J., Pike R. (2002) Classical rules in quantum games. Physical Review A 66: 024306

    Article  Google Scholar 

  • Vitiello G. (1995) Dissipation and memory capacity in the quantum brain model. International Journal of Modern Physics B 9: 973–989

    Article  Google Scholar 

  • von Neumann J. (1955) Mathematical foundations of quantum mechanics. Princeton University, Princeton

    Google Scholar 

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

    Google Scholar 

  • Wells G. L. (1985) The conjunction error and the representativeness heuristic. Social Cognition 3: 266–279

    Article  Google Scholar 

  • Weirich P. (2001) Decision space. Cambridge University, Cambridge

    Book  Google Scholar 

  • White D. I. (1976) Fundamentals of decision theory. Elsevier, New York

    Google Scholar 

  • Yates J. F., Carlson B. W. (1986) Conjunction errors: Evidence for multiple judgement procedures, including signed summation. Organizational Behavior and Human Decision Processes 37: 230–253

    Article  Google Scholar 

  • Yukalov V. I. (1975) Causality problem in quantum physics. Philosophical Sciences 18: 145–147

    Google Scholar 

  • Yukalov V. I. (2003a) Entanglement measure for composite systems. Physical Review Letters 90: 167905

    Article  Google Scholar 

  • Yukalov V. I. (2003b) Quantifying entanglement production of quantum operations. Physical Review A 68: 022109

    Article  Google Scholar 

  • Yukalov V. I. (2003c) Evolutional entanglement in nonequilibrium processes. Modern Physics Letters B 17: 95–103

    Article  Google Scholar 

  • Yukalov V. I., Sornette D. (2008) Quantum decision theory as quantum theory of measurement. Physics Letters A 372: 6867–6871

    Article  Google Scholar 

  • Yukalov V. I., Sornette D. (2009a) Scheme of thinking quantum systems. Laser Physics Letters 6: 833–839

    Article  Google Scholar 

  • Yukalov V. I., Sornette D. (2009b) Physics of risk and uncertainty in quantum decision making. European Physical Journal B 71: 533–548

    Article  Google Scholar 

  • Zeckhauser R. (2006) Investing in the unknown and unknowable. Capitalism and Society 1: 1–39

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management, Technology and Economics, ETH Zürich, 8032, Zürich, Switzerland

    V. I. Yukalov & D. Sornette

  2. Bogolubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980, Russia

    V. I. Yukalov

  3. Swiss Finance Institute, C/o University of Geneva, 40 blvd. Du Pont d’Arve, 1211, Geneva 4, Switzerland

    D. Sornette

Authors
  1. V. I. Yukalov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Sornette
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Yukalov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yukalov, V.I., Sornette, D. Decision theory with prospect interference and entanglement. Theory Decis 70, 283–328 (2011). https://doi.org/10.1007/s11238-010-9202-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11238-010-9202-y

Keywords

JEL Classification

Search

Navigation