This paper discusses random utility representations of the Luce model (Luce, Individual choice behavior: a theoretical analysis, 1959). Earlier works, such as McFadden (Frontier in econometrics, 1973), Yellott (J Math Psychol 15:109–144, 1977), and Strauss (J Math Psychol 20:35–52, 1979) have discussed random utility representations under the assumption that utilities are additively (or multiplicatively) separable in a deterministic and a random part. Under various conditions, they have established that a separable and independent random utility representation exists if and only if the random terms are type III (type I) extreme value distributed. This paper analyzes independent random utility representations without the separability condition and with an infinite universal set of alternatives. Under these assumptions, it turns out that the most general random utility representation of the Luce model is a utility function that is an arbitrary strictly increasing transformation of a separable utility function (additive or multiplicative) with extreme value distributed random terms.
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In a recent paper by Fosgerau and Bierlaire (2009), it is argued in their abstract that sometimes the multiplicative formulation may be a more plausible than the additive one because decision-makers may evaluate relative differences rather than absolute differences. However, since the utility concept in this context is ordinal the multiplicative and additive formulations are equivalent a priori (that is, before a functional form of the respective deterministic part of the utility functions have been chosen).
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I am grateful for comments by Tony Marley.
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Dagsvik, J.K. What independent random utility representations are equivalent to the IIA assumption?. Theory Decis 80, 495–499 (2016). https://doi.org/10.1007/s11238-014-9479-3
- Independent random utility models
- Independence from irrelevant alternatives
- Non-separable random utility representations