Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

What independent random utility representations are equivalent to the IIA assumption?

  • 715 Accesses

  • 1 Citations

Abstract

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.

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

Notes

  1. 1.

    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).

  2. 2.

    The work of Hausdorff (1921a, b) on the moment problem is famous in mathematics and probability theory and recognized as a deep result.

References

  1. Barberà, S., & Pattanaik, P. K. (1986). Falmagne and the rationalizability of stochastic choices in terms of random orderings. Econometrica, 54, 707–715.

  2. Colonius, H. (1984). Stochastische Theorien individuellen Wahlverhaltens. Berlin: Springer.

  3. Dagsvik, J. K. (1994). Discrete and continuous choice, max-stable processes and independence from irrelevant attributes. Econometrica, 62, 1179–1205.

  4. Dagsvik, J. K. (1995). How large is the class of generalized extreme value random utility models? Journal of Mathematical Psychology, 39, 90–98.

  5. Falmagne, J.-C. (1978). A representation theorem for finite random scale systems. Journal of Mathematical Psychology, 18, 52–72.

  6. Feller, W. (1971). An introduction to probability theory and its applications (Vol. II). New York: Wiley.

  7. Fiorini, S. (2004). A short proof of a theorem of Falmagne. Journal of Mathematical Psychology, 48, 80–82.

  8. Fosgerau, M., & Bierlaire, M. (2009). Discrete choice models with multiplicative error terms. Transportation Research Part B, 43, 494–505.

  9. Hausdorff, F. (1921a). Summationsmethoden und Momentfolgen. I. Mathematische Zeitschrift, 9, 74–109.

  10. Hausdorff, F. (1921b). Summationsmethoden und Momentfolgen. II. Mathematische Zeitschrift, 9, 280–299.

  11. Luce, R. D. (1959). Individual choice behavior: A theoretical analysis. New York: Wiley.

  12. Luce, R. D., & Suppes, P. (1965). Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.), Handbook of mathematical psychology (Vol. III). New York: Wiley.

  13. McFadden, D. (1973). Conditional logit analysis of qualitative choice behavior. In P. Zarembka (Ed.), Frontier in Econometrics (pp. 105–142). New York: Academic Press.

  14. Monderer, D. (1992). The stochastic choice problem: A game-theoretic approach. Journal of Mathematical Psychology, 36, 547–554.

  15. Strauss, D. (1979). Some results on random utility models. Journal of Mathematical Psychology, 20, 35–52.

  16. Thurstone, L. L. (1927). A law of comparative judgment. Psyhological Review, 34, 273–286.

  17. Yellott, J. I. (1977). The relationship between Luce’s choice axiom, Thurstone’s theory of comparative judgment, and the double exponential distribution. Journal of Mathematical Psychology, 15, 109–144.

Download references

Acknowledgments

I am grateful for comments by Tony Marley.

Author information

Correspondence to John K. Dagsvik.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Independent random utility models
  • Independence from irrelevant alternatives
  • Non-separable random utility representations

JEL Classification

  • C25