Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Expected utility theory with non-commutative probability theory

  • 295 Accesses

  • 2 Citations

Abstract

In this paper, we extend von Neumann and Morgenstern’s expected utility approach to a non-commutative probability theory. We introduce a new representation of the decision maker’s set of events which extends the canonical representation. We reformulate von Neumann and Morgenstern’s approach to modeling decision maker behavior by non-commutative probability theory. We introduce a set of preference axioms similar to von Neumann and Morgenstern’s axioms, and show that they lead to a generalization of the expected utility theorem. Our generalization allows for decision makers to make an intuitive distinction between representations of a set of events. We find that this methodology enables several paradoxes and inconsistencies in traditional expected utility theory (e.g., Allais paradox, etc.) to be solved or better understood.

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

References

  1. Aerts D, Broekaert J, Czachor M, D’Hooghe B (2011) A quantum-conceptual explanation of violations of expected utility in economics. In: Quantum interaction. Lecture Notes in Computer Science, vol 7052 Springer, Berlin, pp 192–198

  2. Allais M (1953) Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine. Econometrica 21(4): 503–546

  3. Busemeyer JR, Wang Z, Townsend JT (2006) Quantum dynamics of human decision-making. J Math Psychol 50(3):220–241 (Jean-Claude Falmagne: Part II)

  4. Danilov VI, Lambert-Mogiliansky A (2005, Dec) Non-classical measurement theory: a framework for behavioral sciences. Levine’s working paper archive, David K. Levine

  5. Danilov VI, Lambert-Mogiliansky A (2010) Expected utility theory under non-classical uncertainty. Theory Decis 68(1): 25–47

  6. Doran R, American Mathematical Society (1994) C*-algebras: 1943–1993: a fifty year celebration: AMS special session commemorating the first fifty years of c*-algebra theory, 13–14 January, 1993, San Antonio. Contemporary mathematics, American Mathematical Society

  7. Eilenberg S (1941) Ordered topological spaces. Am J Math 63: 39–45

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

  9. Goodman F, de la Harpe P, Jones V (1989) Coxeter graphs and towers of algebras. Mathematical Sciences Research Institute publications. Springer, Berlin

  10. Gyntelberg J, Hansen F (2005) Expected utility theory with “small worlds”. Discussion Papers 04-20, University of Copenhagen. Department of Economics

  11. Gyntelberg J, Hansen F (2009) Subjective expected utility theory with “small worlds”. Discussion Papers 09-26, University of Copenhagen. Department of Economics

  12. Hansen F (2005) A general theory of decision making. Aust J Math Anal Appl 2: 1–13

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

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

  15. Kahneman D, Tversky A (1984) Choices, values and frames. Am Psychol 39: 341–350

  16. Khrennikov A (2007) Quantum-like probabilistic models outside physics. ArXiv Physics e-prints

  17. La Mura P (2009) Projective expected utility. J Math Psychol 53(5):408–414, Special Issue: Quantum Cognition

  18. Lambertini L (2000) Quantum mechanics and mathematical economics are isomorphic. John von Neumann between physics and economics. Working papers, Dipartimento Scienze Economiche, Universita’ di Bologna

  19. Lambert-Mogiliansky A, Zamir S, Zwirn H (2009) Type indeterminacy: a model of the kt(kahneman-tversky)-man. J Math Psychol 53: 249–361

  20. Loomes G, Sugden R (1982) Regret theory: an alternative theory of rational choice under uncertainty. Econ J 92: 805–824

  21. Machina MJ (1982) Expected utility without the independence axiom. Econometrica 50: 277–323

  22. Quiggin J (1982) A theory of anticipated utility. J Econ Behav Organ 3: 323–343

  23. Savage LJ (1972) The foundations of statistics. Wiley, New York (1954, 2nd revised edn)

  24. Von Neumann J (1932) Mathematische Grundlagen der Quantenmechanic. Springer, Berlin

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

  26. Yukalov VI, Sornette D (2010) Decision theory with prospect interference and entanglement. Theory Decis 70: 283–328 doi:10.1007/s11238-010-9202-y

Download references

Author information

Correspondence to Dino Borie.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borie, D. Expected utility theory with non-commutative probability theory. J Econ Interact Coord 8, 295–315 (2013). https://doi.org/10.1007/s11403-012-0098-1

Download citation

Keywords

  • Expected utility
  • Decision theory
  • Non-expected utility
  • Quantum decision theory

JEL Classification

  • C60
  • D03
  • D81