Softmax and McFadden’s Discrete Choice Under Interval (and Other) Uncertainty

  • Bartlomiej Jacek Kubica
  • Laxman Bokati
  • Olga Kosheleva
  • Vladik KreinovichEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)


One of the important parts of deep learning is the use of the softmax formula, that enables us to select one of the alternatives with a probability depending on its expected gain. A similar formula describes human decision making: somewhat surprisingly, when presented with several choices with different expected equivalent monetary gain, we do not just select the alternative with the largest gain; instead, we make a random choice, with probability decreasing with the gain – so that it is possible that we will select second highest and even third highest value. Both formulas assume that we know the exact value of the expected gain for each alternative. In practice, we usually know this gain only with some certainty. For example, often, we only know the lower bound \(\underline{f}\) and the upper bound \(\overline{f}\) on the expected gain, i.e., we only know that the actual gain f is somewhere in the interval \(\left[ \,\underline{f},\overline{f}\right] \). In this paper, we show how to extend softmax and discrete choice formulas to interval uncertainty.


Deep learning Softmax Discrete choice Interval uncertainty 


  1. 1.
    Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Camridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  2. 2.
    Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016)zbMATHGoogle Scholar
  3. 3.
    Hurwicz, L.: Optimality Criteria for Decision Making Under Ignorance, Cowles Commission Discussion Paper, Statistics, no. 370 (1951)Google Scholar
  4. 4.
    Kosheleva, O., Kreinovich, V., Sriboonchitta, S.: Econometric models of probabilistic choice: beyond McFadden’s formulas. In: Kreinovich, V., Sriboonchitta, S., Huynh, V.-N. (eds.) Robustness in Econometrics. SCI, vol. 692, pp. 79–87. Springer, Cham (2017). Scholar
  5. 5.
    Kreinovich, V.: Decision making under interval uncertainty (and beyond). In: Guo, P., Pedrycz, W. (eds.) Human-Centric Decision-Making Models for Social Sciences. SCI, vol. 502, pp. 163–193. Springer, Heidelberg (2014). Scholar
  6. 6.
    Kreinovich, V.: Decision making under interval (and more general) uncertainty: monetary vs. utility approaches. J. Comput. Technol. 22(2), 37–49 (2017)zbMATHGoogle Scholar
  7. 7.
    Kreinovich, V.: From traditional neural networks to deep learning: towards mathematical foundations of empirical successes. In: Shahbazova, S.N., et al. (eds.) Proceedings of the World Conference on Soft Computing, Baku, Azerbaijan, 29–31 May 2018 (2018)Google Scholar
  8. 8.
    Luce, D.: Inividual Choice Behavior: A Theoretical Analysis. Dover, New York (2005)CrossRefGoogle Scholar
  9. 9.
    Luce, R.D., Raiffa, R.: Games and Decisions: Introduction and Critical Survey. Dover, New York (1989)zbMATHGoogle Scholar
  10. 10.
    McFadden, D.: Conditional logit analysis of qualitative choice behavior. In: Zarembka, P. (ed.) Frontiers in Econometrics, pp. 105–142. Academic Press, New York (1974)Google Scholar
  11. 11.
    McFadden, D.: Economic choices. Am. Econ. Rev. 91, 351–378 (2001)CrossRefGoogle Scholar
  12. 12.
    Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction, 2nd edn. MIT Press, Cambridge (2018)zbMATHGoogle Scholar
  13. 13.
    Train, K.: Discrete Choice Methods with Simulation. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of Life SciencesWarsawPoland
  2. 2.University of Texas at El PasoEl PasoUSA

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