Extreme values, invariance and choice probabilities

Since the pioneering work of McFadden (1974), discrete choice random-utility models have become work horses in many areas in transportation analysis and economics. In these models, the random variables enter additively or multiplicatively and the noise distributions take a particular parametric form. We show that the same qualitative results, with closed-form choice probabilities, can be obtained for a wide class of distributions without such specifications. This class generalizes the statistically independent distributions where any two c.d.f.:s are powers of each others to a class that allows for statistical dependence, in a way analogous to how the independent distributions in the MNL models were generalized into the subclass of MEV distributions that generates the GEV choice models. We show that this generalization is sufficient, and under statistical independence also necessary, for the following invariance property: all conditional random variables, when conditioning upon a certain alternative having been chosen, are identically distributed. While some of these results have been published earlier, we place them in a general unified framework that allows us to extend several of the results and to provide proofs that are simpler, more direct and transparent. Well-known results are obtained as special cases, and we characterize the Gumbel, Fréchet and Weibull distributions.