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A stricter canon: general Luce models for arbitrary menu sets

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  • Jose A. Rodrigues-Neto

    (Research School of Economics, Australian National University)

  • Matthew Ryan

    (Department of Economics and Finance, Auckland University of Technology)

  • James Taylor

    (Research School of Economics, Australian National University)

Abstract

The classical Luce model (Luce, 1959) assumes positivity of random choice: each available alternative is chosen with strictly positive probability. The model is characterised by Luce's choice axiom. Ahumada and Ulku (2018) and (independently) Echenique and Saito (2019) define the general Luce model (GLM), which relaxes the positivity assumption, and show that it is characterised by a cyclical independence (CI) axiom. Cerreia-Vioglio et al. (2021) subsequently proved that the choice axiom characterises an important special case of the GLM in which a rational choice function (i.e., one that may be rationalised by a weak order) first selects the acceptable alternatives from the given menu, with any residual indifference resolved randomly in Luce fashion. The choice axiom is thus revealed as a fundamental "canon of probabilistic rationality". This result assumes that choice behaviour is specified for all non-empty, finite menus that can be constructed from a given universe, X, of alternatives. We relax this assumption by allowing choice behaviour to be specified for an arbitrary collection of non-empty, finite menus. In this context, we show that the Cerreia-Vioglio et al. (2021) result obtains when the choice axiom is replaced with a mild strengthening of CI. The latter condition implies the choice axiom, thus providing a "stricter canon".

Suggested Citation

  • Jose A. Rodrigues-Neto & Matthew Ryan & James Taylor, 2024. "A stricter canon: general Luce models for arbitrary menu sets," Working Papers 2024-04, Auckland University of Technology, Department of Economics.
    • Handle: RePEc:aut:wpaper:2024-04
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    References listed on IDEAS

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    1. Ahumada, Alonso & Ülkü, Levent, 2018. "Luce rule with limited consideration," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 52-56.
    2. Chambers,Christopher P. & Echenique,Federico, 2016. "Revealed Preference Theory," Cambridge Books, Cambridge University Press, number 9781107087804, January.
    3. John C. Harsanyi, 1967. "Games with Incomplete Information Played by "Bayesian" Players, I-III Part I. The Basic Model," Management Science, INFORMS, vol. 14(3), pages 159-182, November.
    4. Horan, Sean, 2021. "Stochastic semi-orders," Journal of Economic Theory, Elsevier, vol. 192(C).
    5. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, March.
    6. Cerreia-Vioglio, Simone & Lindberg, Per Olov & Maccheroni, Fabio & Marinacci, Massimo & Rustichini, Aldo, 2021. "A canon of probabilistic rationality," Journal of Economic Theory, Elsevier, vol. 196(C).
    7. Alós-Ferrer, Carlos & Mihm, Maximilian, 2025. "A characterization of the Luce choice rule for an arbitrary collection of menus," Journal of Economic Theory, Elsevier, vol. 223(C).
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