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Contextuality and the Weak Axiom in the Theory of Choice

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Quantum Interaction (QI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9535))

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Abstract

Recent work on the logical structure of non-locality has constructed scenarios where observations of multi-partite systems cannot be adequately described by compositions of non-signaling subsystems. In this paper we apply these frameworks to economics. First we construct a empirical model of choice, where choices are understood as observable outcomes in a certain sense. An analysis of contextuality within this framework allows us to characterize which scenarios allow for the possible construction of an adequate global choice rule. In essence, we mathematically characterize when it makes sense to consider the choices of a group as composed of individual choices. We then map out the logical space of some relevant empirical principles, relating properties of these contextual choice scenarios to no-signalling theories and to the weak axiom of revealed preference.

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Notes

  1. 1.

    In other literature, C(U) is referred to as the support of the measurement context U under the model C.

  2. 2.

    The Greenberger-Horne-Zeilinger states [10].

  3. 3.

    There are many alternative explanations. See also [9] on the possibility that the measurement itself is changing the agent’s preferences.

  4. 4.

    As noted in [2], the terminology global section arises because these binary empirical models can be given the structure of a presheaf. More details on this approach can be found in [1].

  5. 5.

    The notion of strong contextuality in contextual semantics comes from [1].

  6. 6.

    Obviously, one could also consider a blend of the two views: (i) what is presented in a menu and (ii) what is affordable? We focus on the extreme case for simplicity.

References

  1. Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13(11), 113036 (2011)

    Article  Google Scholar 

  2. Abramsky, S., Hardy, L.: Logical bell inequalities. Phys. Rev. A 85(6), 062114 (2012)

    Article  Google Scholar 

  3. Abramsky, S.: Contextual semantics: from quantum mechanics to logic, databases, constraints, and complexity. Bulletin of EATCS 2(113) (2014)

    Google Scholar 

  4. Abramsky, S., Mansfield, S., Barbosa, R.S.: The cohomology of non-locality and contextuality. In: 2011 Proceedings of Quantum Physics and Logic, Electronic Proceedings in Theoretical Computer Science (EPTCS), pp. 1–14 (2011)

    Google Scholar 

  5. Bao, N., Halpern, N.Y.: Quantum voting and violation of arrow’s impossibility theorem. arXiv preprint arXiv:1501.00458 (2015)

  6. Brandenburger, A., La Mura, P.: Quantum decision theory. arXiv preprint arXiv:1107.0237 (2011)

  7. Brunner, N., Linden, N.: Connection between bell nonlocality and Bayesian game theory. Nature communications 4 (2013)

    Google Scholar 

  8. Busemeyer, J.R., Bruza, P.D.: Quantum Models of Cognition and Decision. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  9. Danilov, V., Lambert-Mogiliansky, A.: Measurable systems and behavioral sciences. Math. Soc. Sci. 55(3), 315–340 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bells theorem. In: Kafatos, M. (ed.) Bells theorem, quantum theory and conceptions of the universe. Fundamental Theories of Physics, vol. 37, pp. 69–72. Springer, Netherlands (1989)

    Chapter  Google Scholar 

  11. Luce, R., Raiffa, H.: Games and Decisions: Introduction and Critical Survey. Dover books on advanced mathematics. Dover books on advanced mathematics. Dover Publications, New York (1957)

    Google Scholar 

  12. Mansfield, S.: The mathematical structure of non-locality and contextuality. D.Phil. thesis, Oxford University (2013)

    Google Scholar 

  13. Mas-Colell, A., Whinston, M.D., Green, J.: Microeconomic Theory. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  14. Mogiliansky, A.L., Zamir, S., Zwirn, H.: Type indeterminacy: A Model for the KT(Kahneman-Tversky)-Man. J. Math. Psychol. 53(5), 349–361 (2009). special Issue: Quantum Cognition

    Article  MATH  Google Scholar 

  15. Pothos, E.M., Busemeyer, J.R.: A quantum probability explanation for violations of ‘rational’ decision theory. In: Proceedings of the Biological sciences / The Royal Society, vol. 276(1665), pp. 2171–2178 (2009)

    Google Scholar 

  16. Saari, D.G.: The profile structure for Luce’s choice axiom. J. Math. Psychol. 49(3), 226–253 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang, Z., Busemeyer, J.R.: A quantum question order model supported by empirical tests of an a priori and precise prediction. Top. Cogn. Sci. 5(4), 689–710 (2013)

    Google Scholar 

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Acknowledgements

The authors thank Samson Abramsky for suggesting investigation of the no-signalling condition. Zeng gratefully acknowledges the support of The Rhodes Trust and Zahn gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG) through SFB 884 “Political Economy of Reforms”.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Oxford, UK

    William Zeng

  2. Department of Economics, University of Mannheim, Mannheim, Germany

    Philipp Zahn

Authors
  1. William Zeng
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  2. Philipp Zahn
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Correspondence to Philipp Zahn .

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Editors and Affiliations

  1. ETH Zurich, Zurich, Switzerland

    Harald Atmanspacher

  2. University of Freiburg, Freiburg, Germany

    Thomas Filk

  3. City University London, London, United Kingdom

    Emmanuel Pothos

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Zeng, W., Zahn, P. (2016). Contextuality and the Weak Axiom in the Theory of Choice. In: Atmanspacher, H., Filk, T., Pothos, E. (eds) Quantum Interaction. QI 2015. Lecture Notes in Computer Science(), vol 9535. Springer, Cham. https://doi.org/10.1007/978-3-319-28675-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-28675-4_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28674-7

  • Online ISBN: 978-3-319-28675-4

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