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Choice probability generating functions


  • Fosgerau, Mogens
  • McFadden, Daniel
  • Bierlaire, Michel


This paper establishes that every random utility discrete choice model (RUM) has a representation that can be characterized by a choice-probability generating function (CPGF) with specific properties, and that every function with these specific properties is consistent with a RUM. The choice probabilities from the RUM are obtained from the gradient of the CPGF. Mixtures of RUM are characterized by logarithmic mixtures of their associated CPGF. The paper relates CPGF to multivariate extreme value distributions, and reviews and extends methods for constructing generating functions for applications. The choice probabilities of any ARUM may be approximated by a cross-nested logit model. The results for ARUM are extended to competing risk survival models.

Suggested Citation

  • Fosgerau, Mogens & McFadden, Daniel & Bierlaire, Michel, 2010. "Choice probability generating functions," MPRA Paper 24214, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:24214

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    References listed on IDEAS

    1. P O Lindberg & E A Eriksson & L-G Mattsson, 1995. "Invariance of achieved utility in random utility models," Environment and Planning A, Pion Ltd, London, vol. 27(1), pages 121-142, January.
    2. Ruud H. Koning & Geert Ridder, 2003. "Discrete choice and stochastic utility maximization," Econometrics Journal, Royal Economic Society, vol. 6(1), pages 1-27, June.
    3. de Palma, Andre & Kilani, Karim, 2007. "Invariance of conditional maximum utility," Journal of Economic Theory, Elsevier, vol. 132(1), pages 137-146, January.
    4. Bierlaire, M. & Bolduc, D. & McFadden, D., 2008. "The estimation of generalized extreme value models from choice-based samples," Transportation Research Part B: Methodological, Elsevier, vol. 42(4), pages 381-394, May.
    5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    6. H C W L Williams, 1977. "On the Formation of Travel Demand Models and Economic Evaluation Measures of User Benefit," Environment and Planning A, , vol. 9(3), pages 285-344, March.
    7. McFadden, Daniel, 1974. "The measurement of urban travel demand," Journal of Public Economics, Elsevier, vol. 3(4), pages 303-328, November.
    8. Daniel McFadden & Kenneth Train, 2000. "Mixed MNL models for discrete response," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 15(5), pages 447-470.
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    Cited by:

    1. Fosgerau, Mogens & de Palma, André, 2015. "Demand systems for market shares," MPRA Paper 62106, University Library of Munich, Germany.
    2. Mattsson, Lars-Göran & Weibull, Jörgen W. & Lindberg, Per Olov, 2014. "Extreme values, invariance and choice probabilities," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 81-95.
    3. Daniel L. McFadden & Mogens Fosgerau, 2012. "A theory of the perturbed consumer with general budgets," NBER Working Papers 17953, National Bureau of Economic Research, Inc.
    4. Fosgerau, Mogens & Frejinger, Emma & Karlstrom, Anders, 2013. "A link based network route choice model with unrestricted choice set," Transportation Research Part B: Methodological, Elsevier, vol. 56(C), pages 70-80.
    5. Pereira, Pedro & Ribeiro, Tiago & Vareda, João, 2013. "Delineating markets for bundles with consumer level data: The case of triple-play," International Journal of Industrial Organization, Elsevier, vol. 31(6), pages 760-773.
    6. Mogens Fosgerau & André De Palma, 2016. "Generalized entropy models," Working Papers hal-01291347, HAL.
    7. Mogens Fosgerau, 2014. "Nonparametric approaches to describing heterogeneity," Chapters,in: Handbook of Choice Modelling, chapter 11, pages 257-267 Edward Elgar Publishing.
    8. Joao Macieira & Pedro Pereira & Joao Vareda, 2013. "Bundling Incentives in Markets with Product Complementarities: The Case of Triple-Play," Working Papers 13-15, NET Institute.

    More about this item


    Discrete choice; random utility; mixture models; duration models; logit; generalised extreme value; multivariate extreme value;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions

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