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A First Attempt at Ordinal Projective Measurement

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Quantum Interaction (QI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10106))

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Abstract

To our knowledge, all applications of the quantum framework in social sciences are used to model measurements done on a discrete nominal scale. However, especially in cognition, experiments often produce data on an ordinal scale, which implies some internal structure between the possible outcomes. Since there are no ordinal scales in physics, orthodox projection-valued measurement (PVM) lacks the tools and methods to deal with these ordinal scales. Here, we sketch out an attempt to incorporate the ordinal structure of outcomes into the subspaces representing these outcomes. This will also allow us to reduce the dimensionality of the resulting Hilbert spaces, as these often become too high in more complex quantum-like models. To do so, we loosen restrictions placed upon the PVM (and even POVM) framework. We discuss the two major consequences of this generalization: scaling and the loss of repeatability. We also present two applications of this approach, one in game theory and one concerning Likert scales.

The author would like to thank Kirsty Kitto, James Yearsley, Ariane Lambert-Mogiliansky and especially Ismael Martínez-Martínez for the engaging discussions and comments.

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Notes

  1. 1.

    Ordinal scales are discrete scales with a well defined order on the outcomes.

  2. 2.

    The angle \(\widehat{\mathcal {M}_i\mathcal {M}_j}\) between two subspaces \(\mathcal {M}_i\) and \(\mathcal {M}_j\) is classically defined as min\((\widehat{V_iV_j})\), with \(V_i \in \mathcal {M}_i\) and \(V_j \in \mathcal {M}_j\).

References

  1. Aerts, D., de Bianchi, M.S.: Beyond-quantum modeling of question order effects and response replicability in psychological measurements. CoRR abs/1508.03686 (2015). http://arxiv.org/abs/1508.03686

  2. Aliakbarzadeh, M., Kitto, K.: Applying POVM to model non-orthogonality in quantum cognition. In: Atmanspacher, H., Filk, T., Pothos, E. (eds.) Quantum Interaction. LNCS, vol. 9535, pp. 284–293. Springer, Cham (2016)

    Chapter  Google Scholar 

  3. Blanco, M., Engelmann, D., Koch, A., Normann, H.T.: Preferences and beliefs in a sequential social dilemma. Games Econ. Behav. 87, 122–135 (2014)

    Article  MATH  Google Scholar 

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

    Book  Google Scholar 

  5. Busemeyer, J., Pothos, E., Franco, R., Trueblood, J.: A quantum theoretical explanation for probability judgment errors. Physchol. Rev. 118(2), 193–218 (2011)

    Google Scholar 

  6. Camparo, J.: A geometrical approach to the ordinal data of likert scaling and attitude measurements: the density matrix in psychology. J. Math. Psychol. 57(1), 29–42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Denolf, J., Martínez-Martínez, I., Josephy, H., Barque-Duran, A.: A quantum-like model for complementarity of preferences and beliefs in dilemma games. (submitted)

    Google Scholar 

  8. Gelfand, I., Neumark, M.: On the imbedding of normed rings into the ring of operators in hilbert space. Mat. Sbornik 54, 197–217 (1943)

    MathSciNet  MATH  Google Scholar 

  9. Khrennikov, A., Basieva, I., Dzhafarov, E.N., Busemeyer, J.R.: Quantum models for psychological measurements: an unsolved problem. PloS one 9(10), e110909 (2014)

    Article  Google Scholar 

  10. Martínez-Martínez, I., Denolf, J., Barque-Duran, A.: Do preferences and beliefs in dilemma games exhibit complementarity? In: Atmanspacher, H., Filk, T., Pothos, E. (eds.) Quantum Interaction. LNCS, vol. 9535, pp. 142–153. Springer, Cham (2015)

    Chapter  Google Scholar 

  11. Matvejchuk, M., Widdows, D.: Real-orthogonal projections as quantum pseudo-logic. In: Atmanspacher, H., Filk, T., Pothos, E. (eds.) Quantum Interaction. LNCS, vol. 9535, pp. 275–283. Springer, Cham (2015)

    Chapter  Google Scholar 

  12. Moore, D.W.: Measuring new types of question-order effects: additive and subtractive. Pub. Opin. Q. 66(1), 80–91 (2002)

    Article  MathSciNet  Google Scholar 

  13. de Muynck, W.: POVMs: a small but important step beyond standard quantum mechanics. In: Nieuwenhuizen, T., Mehmani, B., Špicka, B., Aghdami, M., Khrennikov, A. (eds.)Beyond the Quantum, pp. 69–79 (2007)

    Google Scholar 

  14. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  15. Pothos, E., Busemeyer, J.: A quantum probability explanation for violations of ‘rational’ decision theory. In: Proceedings of the Royal Society of London B: Biological Sciences, RSPB-2009 (2009)

    Google Scholar 

  16. Spector, P.E.: Summated rating scale construction: An introduction. No. 82, Sage (1992)

    Google Scholar 

  17. Yearsley, J.M.: Advanced tools and concepts for quantum cognition: a tutorial. J. Math. Psychol. Special Issue on Quantum Probability (in press) (2016)

    Google Scholar 

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Authors and Affiliations

  1. Department of Data Analysis, Ghent University, H. Dunantlaan 1, 9000, Ghent, Belgium

    Jacob Denolf

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  1. Jacob Denolf
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Correspondence to Jacob Denolf .

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Editors and Affiliations

  1. San Francisco State University, San Francisco, California, USA

    Jose Acacio de Barros

  2. University of Oxford, Oxford, United Kingdom

    Bob Coecke

  3. City University of London, London, United Kingdom

    Emmanuel Pothos

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Denolf, J. (2017). A First Attempt at Ordinal Projective Measurement. In: de Barros, J., Coecke, B., Pothos, E. (eds) Quantum Interaction. QI 2016. Lecture Notes in Computer Science(), vol 10106. Springer, Cham. https://doi.org/10.1007/978-3-319-52289-0_18

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  • DOI: https://doi.org/10.1007/978-3-319-52289-0_18

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-52289-0

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