International Journal of Theoretical Physics

, Volume 49, Issue 12, pp 2971–2990 | Cite as

Quantum Experimental Data in Psychology and Economics

Article

Abstract

We prove a theorem which shows that a collection of experimental data of probabilistic weights related to decisions with respect to situations and their disjunction cannot be modeled within a classical probabilistic weight structure in case the experimental data contain the effect referred to as the ‘disjunction effect’ in psychology. We identify different experimental situations in psychology, more specifically in concept theory and in decision theory, and in economics (namely situations where Savage’s Sure-Thing Principle is violated) where the disjunction effect appears and we point out the common nature of the effect. We analyze how our theorem constitutes a no-go theorem for classical probabilistic weight structures for common experimental data when the disjunction effect is affecting the values of these data. We put forward a simple geometric criterion that reveals the non classicality of the considered probabilistic weights and we illustrate our geometrical criterion by means of experimentally measured membership weights of items with respect to pairs of concepts and their disjunctions. The violation of the classical probabilistic weight structure is very analogous to the violation of the well-known Bell inequalities studied in quantum mechanics. The no-go theorem we prove in the present article with respect to the collection of experimental data we consider has a status analogous to the well known no-go theorems for hidden variable theories in quantum mechanics with respect to experimental data obtained in quantum laboratories. Our analysis puts forward a strong argument in favor of the validity of using the quantum formalism for modeling the considered psychological experimental data as considered in this paper.

Keywords

Bell inequalities Polytopes Disjunction effect 

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References

  1. 1.
    Bruza, P.D., Gabora, L. (eds.): Special Issue: Quantum Cognition. J. Math. Psychol. 53, 303–452 (2009) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bruza, P.D., Lawless, W., van Rijsbergen, C.J., Sofge, D. (eds.): Proceedings of the AAAI Spring Symposium on Quantum Interaction, March 26–28, 2007, Stanford University, SS-07-08. AAAI Press, Menlo Park (2007) Google Scholar
  3. 3.
    Bruza, P.D., Lawless, W., van Rijsbergen, C.J., Sofge, D., Coecke, B., Clark, S. (eds.): Proceedings of the Second Quantum Interaction Symposium, University of Oxford, March 26–28, 2008. College Publications, London (2008) Google Scholar
  4. 4.
    Bruza, P.D., Sofge, D., Lawless, W., van Rijsbergen, C.J., Klusch, M. (eds.): Proceedings of the Third International Symposium, QI 2009, Saarbrücken, Germany, March 25–27, 2009. Lecture Notes in Computer Science, vol. 5494. Springer, Berlin, Heidelberg (2009) Google Scholar
  5. 5.
    Aerts, D., Aerts, S.: Applications of quantum statistics in psychological studies of decision processes. Found. Sci. 1, 85–97 (1994). Reprinted in: B.C. van Fraassen (ed.), Topics in the Foundation of Statistics, Springer, Dordrecht MathSciNetGoogle Scholar
  6. 6.
    Aerts, D., Broekaert, J., Smets, S.: The liar paradox in a quantum mechanical perspective. Found. Sci. 4, 115–132 (1999) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Aerts, D., Broekaert, J., Smets, S.: A quantum structure description of the liar paradox. Int. J. Theor. Phys. 38, 3231–3239 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Aerts, D., Aerts, S., Broekaert, J., Gabora, L.: The violation of Bell inequalities in the macroworld. Found. Phys. 30, 1387–1414 (2000) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gabora, L., Aerts, D.: Contextualizing concepts using a mathematical generalization of the quantum formalism. J. Exp. Theor. Artif. Intell. 14, 327–358 (2002) MATHCrossRefGoogle Scholar
  10. 10.
    Aerts, D., Czachor, M.: Quantum aspects of semantic analysis and symbolic artificial intelligence. J. Phys. A 37, L123–L132 (2004) MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Aerts, D., Gabora, L.: A theory of concepts and their combinations I: the structure of the sets of contexts and properties. Kybernetes 34, 167–191 (2005) MATHCrossRefGoogle Scholar
  12. 12.
    Aerts, D., Gabora, L.: A theory of concepts and their combinations II: a Hilbert space representation. Kybernetes 34, 192–221 (2005) MATHCrossRefGoogle Scholar
  13. 13.
    Broekaert, J., Aerts, D., D’Hooghe, B.: The generalised Liar Paradox: a quantum model and interpretation. Found. Sci. 11, 399–418 (2006) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Aerts, D.: General quantum modeling of combining concepts: a quantum field model in Fock space. Archive address and link: http://arxiv.org/abs/0705.1740 (2007)
  15. 15.
    Aerts, D.: Quantum structure in cognition. J. Math. Psychol. 53, 314–348 (2009) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Aerts, D.: Quantum particles as conceptual entities: a possible explanatory framework for quantum theory. Found. Sci. 14, 361–411 (2009) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Aerts, D., Aerts, S., Gabora, L.: Experimental evidence for quantum structure in cognition. In: Bruza, P.D., Sofge, D., Lawless, W., van Rijsbergen, C.J., Klusch, M. (eds.) Proceedings of QI 2009-Third International Symposium on Quantum Interaction. Lecture Notes in Computer Science, vol. 5494, pp. 59–70. Springer, Berlin, Heidelberg (2009) Google Scholar
  18. 18.
    Aerts, D., D’Hooghe, B.: Classical logical versus quantum conceptual thought: examples in economics, decision theory and concept theory. In: Bruza, P.D., Sofge, D., Lawless, W., van Rijsbergen, C.J., Klusch, M. (eds.) Proceedings of QI 2009-Third International Symposium on Quantum Interaction. Lecture Notes in Computer Science, vol. 5494, pp. 128–142 Springer, Berlin, Heidelberg (2009) Google Scholar
  19. 19.
    Aerts, D.: Quantum interference and superposition in cognition: Development of a theory for the disjunction of concepts. In: Aerts, D., D’Hooghe, B., Note, N. (eds.) Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. World Scientific, Singapore (2010) Google Scholar
  20. 20.
    Aerts, D.: Interpreting quantum particles as conceptual entities. Int. J. Theor. Phys. (2010). doi:10.1007/s10773-010-0440-0
  21. 21.
    Aerts, D., Broekaert, J., Gabora, L.: A case for applying an abstracted quantum formalism to cognition. New Ideas Psychol. (2010). doi:10.1016/j.newideapsych.2010.06.002
  22. 22.
    Aerts, D., D’Hooghe, B.: A quantum-conceptual explanation of violations of expected utility in economics. In: Aerts, D., D’Hooghe, B., Note, N. (eds.) Worldviews, Science and Us: Bridging Knowledge and Its Implications for Our Perspectives of the World. World Scientific, Singapore (2010) Google Scholar
  23. 23.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Chapter IV.1,2. Springer, Berlin. (1932) MATHGoogle Scholar
  24. 24.
    Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935) MATHCrossRefADSGoogle Scholar
  25. 25.
    Bohr, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48, 696–702 (1935) MATHCrossRefADSGoogle Scholar
  26. 26.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) MATHMathSciNetGoogle Scholar
  27. 27.
    Jauch, J., Piron, C.: Can hidden variables be excluded from quantum mechanics? Helv. Phys. Acta 36, 827–837 (1963) MATHMathSciNetGoogle Scholar
  28. 28.
    Bell, J.S.: On the Einstein Podolsky Rosen Paradox. Physics 1, 195–200 (1964) Google Scholar
  29. 29.
    Bohm, D., Bub, J.: A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory. Rev. Mod. Phys. 38, 453–469 (1966) MATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966) MATHCrossRefADSGoogle Scholar
  31. 31.
    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967) MATHMathSciNetGoogle Scholar
  32. 32.
    Jauch, J.M., Piron, C.: Hidden variables revisited. Rev. Mod. Phys. 40, 228–229 (1968) CrossRefADSGoogle Scholar
  33. 33.
    Gudder, S.P.: Hidden variables in quantum mechanics reconsidered. Rev. Mod. Phys. 40, 229–231 (1968) CrossRefADSGoogle Scholar
  34. 34.
    Bohm, D., Bub, J.: On hidden variables-A reply to comments by Jauch and Piron and by Gudder. Rev. Mod. Phys. 40, 235–236 (1968) CrossRefADSGoogle Scholar
  35. 35.
    Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969) CrossRefADSGoogle Scholar
  36. 36.
    Gudder, S.P.: On hidden variable theories. J. Math. Phys. 11, 431–436 (1970) MATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Clauser, J.F., Horne, M.A.: Experimental consequences of objective local theories. Phys. Rev. D 10, 526–535 (1974) CrossRefADSGoogle Scholar
  38. 38.
    Accardi, L., Fedullo, A.: On the statistical meaning of complex numbers in quantum mechanics. Lett. Nuovo Cimento 34, 161–172 (1982) CrossRefMathSciNetGoogle Scholar
  39. 39.
    Aspect, A., Grangier, P., Roger, G.: Experimental realization of Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: a new violation of Bell’s Inequalities. Phys. Rev. Lett. 49, 91–94 (1982) CrossRefADSGoogle Scholar
  40. 40.
    Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell’s Inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807 (1982) CrossRefMathSciNetADSGoogle Scholar
  41. 41.
    Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea Publishing, New York (1956) MATHGoogle Scholar
  42. 42.
    Accardi, L.: The probabilistic roots of the quantum mechanical paradoxes. In: Diner, S., Fargue, D., Lochak, G., Selleri, F. (eds.) The Wave-Particle Dualism: A Tribute to Louis de Broglie on his 90th Birthday, pp. 297–330. Springer, Dordrecht (1984) Google Scholar
  43. 43.
    Aerts, D.: A possible explanation for the probabilities of quantum mechanics and a macroscopical situation that violates Bell inequalities. In: Mittelstaedt, P., Stachow, E.W. (eds.): Recent Developments in Quantum Logic, Grundlagen der Exacten Naturwissenschaften, Wissenschaftverlag, vol. 6, pp. 235–251. Bibliographisches Institut, Mannheim (1985) Google Scholar
  44. 44.
    Aerts, D.: A possible explanation for the probabilities of quantum mechanics. J. Math. Phys. 27, 202–210 (1986) CrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Aerts, D.: The origin of the non-classical character of the quantum probability model. In: Blanquiere, A., Diner, S., Lochak, G. (eds.) Information, Complexity, and Control in Quantum Physics, pp. 77–100. Springer, Wien-New York (1987) Google Scholar
  46. 46.
    Redhead, M.: Incompleteness, Nonlocality and Realism. Clarendon Press, Oxford (1987) MATHGoogle Scholar
  47. 47.
    Pitowsky, I.: Quantum Probability, Quantum Logic. Lecture Notes in Physics, vol. 321. Springer, Heidelberg (1989) MATHGoogle Scholar
  48. 48.
    Hampton, J.A.: Overextension of conjunctive concepts: Evidence for a unitary model for concept typicality and class inclusion. J. Exp. Psychol. Learn. Mem. Cogn. 14, 12–32 (1988) CrossRefGoogle Scholar
  49. 49.
    Hampton, J.A.: Inheritance of attributes in natural concept conjunctions. Mem. Cogn. 15, 55–71 (1987) Google Scholar
  50. 50.
    Hampton, J.A.: The combination of prototype concepts. In: Schwanenugel, P. (ed.) The Psychology of Word Meanings. Erlbaum, Hillsdale (1991) Google Scholar
  51. 51.
    Storms, G., De Boeck, P., Van Mechelen, I., Geeraerts, D.: Dominance and non-commutativity effects in concept conjunctions: extensional or intensional basis? Mem. Cogn. 21, 752–762 (1993) Google Scholar
  52. 52.
    Hampton, J.A.: Conjunctions of visually-based categories: overextension and compensation. J. Exp. Psychol. Learn. Mem. Cogn. 22, 378–396 (1996) CrossRefGoogle Scholar
  53. 53.
    Hampton, J.A.: Conceptual combination: conjunction and negation of natural concepts. Mem. Cogn. 25, 888–909 (1997) Google Scholar
  54. 54.
    Storms, G., de Boeck, P., Hampton, J.A., van Mechelen, I.: Predicting conjunction typicalities by component typicalities. Psychon. Bull. Rev. 6, 677–684 (1999) Google Scholar
  55. 55.
    Tversky, A., Kahneman, D.: Judgments of and by representativeness. In: Kahneman, D., Slovic, P., Tversky, A. (eds.): Judgment under Uncertainty: Heuristics and Biases. Cambridge University Press, Cambridge (1982) Google Scholar
  56. 56.
    Tversky, A., Kahneman, D.: Extension versus intuitive reasoning: the conjunction fallacy in probability judgment. Psychol. Rev. 90, 293–315 (1983) CrossRefGoogle Scholar
  57. 57.
    Hampton, J.A.: Disjunction of natural concepts. Mem. Cogn. 16, 579–591 (1988) Google Scholar
  58. 58.
    Carlson, B.W., Yates, J.F.: Disjunction errors in qualitative likelihood judgment. Organ. Behav. Hum. Decis. Process. 44, 368–379 (1989) CrossRefGoogle Scholar
  59. 59.
    Tversky, A., Shafir, E.: The disjunction effect in choice under uncertainty. Psychol. Sci. 3, 305–309 (1992) CrossRefGoogle Scholar
  60. 60.
    Bar-Hillel, M., Neter, E.: How alike is it versus how likely is it: a disjunction fallacy in probability judgments. J. Personal. Soc. Psychol. 65, 1119–1131 (1993) CrossRefGoogle Scholar
  61. 61.
    Croson, R.T.A.: The disjunction effect and reason-based choice in games. Organ. Behav. Hum. Decis. Process. 80, 118–133 (1999) CrossRefGoogle Scholar
  62. 62.
    Kühberger, A., Komunska, D., Perner, J.: The disjunction effect: Does it exist for two-step gambles? Organ. Behav. Hum. Decis. Process. 85, 250–264 (2001) CrossRefGoogle Scholar
  63. 63.
    Li, S., Taplin, J.E.: Examining whether there is a disjunction effect in prisoner’s dilemma games. Chin. J. Psychol. 44, 25–46 (2002) Google Scholar
  64. 64.
    van Dijk, E., Zeelenberg, M.: The discounting of ambiguous information in economic decision making. J. Behav. Decis. Mak. 16, 341–352 (2003) CrossRefGoogle Scholar
  65. 65.
    van Dijk, E., Zeelenberg, M.: The dampening effect of uncertainty on positive and negative emotions. J. Behav. Decis. Mak. 19, 171–176 (2006) CrossRefGoogle Scholar
  66. 66.
    Bauer, M.I., Johnson-Laird, P.N.: How diagrams can improve reasoning. Psychol. Sci. 4, 372–378 (2006) CrossRefGoogle Scholar
  67. 67.
    Lambdin, C., Burdsal, C.: The disjunction effect reexamined: relevant methodological issues and the fallacy of unspecified percentage comparisons. Organ. Behav. Hum. Decis. Process. 103, 268–276 (2007) CrossRefGoogle Scholar
  68. 68.
    Bagassi, M., Macchi, L.: The ‘vanishing’ of the disjunction effect by sensible procrastination. Mind & Society 6, 41–52 (2007) CrossRefGoogle Scholar
  69. 69.
    Hristova, E., Grinberg, M.: Disjunction effect in prisoner’s dilemma: Evidences from an eye-tracking study. In: Cogsci 2008, Proceedings, Washington, July 22–26 (2008) Google Scholar
  70. 70.
    Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école Américaine. Econometrica 21, 503–546 (1953) MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Ellsberg, D.: Risk, ambiguity, and the savage axioms. Q. J. Economics 75, 643–669 (1961) CrossRefGoogle Scholar
  72. 72.
    Savage, L.J.: The Foundations of Statistics. Wiley, New York (1954) MATHGoogle Scholar
  73. 73.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944) MATHGoogle Scholar
  74. 74.
    Busemeyer, J.R., Wang, Z., Townsend, J.T.: Quantum dynamics of human decision-making. J. Math. Psychol. 50, 220–241 (2006) MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Pothos, E.M., Busemeyer, J.R.: A quantum probability explanation for violations of ‘rational’ decision theory. Proc. R. Soc. B (2009) Google Scholar
  76. 76.
    Khrennikov, A., Haven, E.: Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts. J. Math. Psychol. 53, 378–388 (2009) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Diederik Aerts
    • 1
  • Bart D’Hooghe
    • 1
  • Emmanuel Haven
    • 2
  1. 1.Leo Apostel Center for Interdisciplinary StudiesBrussels Free UniversityBrusselsBelgium
  2. 2.School of ManagementUniversity of LeicesterLeicesterUK

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