Many models in economics, transportation research and engineering involve probabilistic selection of one alternative or unit from a finite set; the selected alternative being the consumption bundle with the highest utility to a consumer, the production method with the lowest cost for a producer, the firm with the highest productivity in a competitive market, etc. The consumer chooses the alternative with the highest utility, the producer the production method with the lowest cost, the market the firm with the highest productivity, etc. The analyst is interested in the selection probabilities and in the probability distribution for the value of the selected alternative. We say that the random vector associated with the alternatives at hand has the invariance property if the distribution of the value of any given alternative, conditional on that alternative being selected, is the same for all alternatives. We show that this invariance property holds if and only if the marginal distributions of the components of the random vector, representing the alternatives, are positive powers of each other. We illustrate the analytical power of the invariance property by way of examples. We also show how it generalizes the pioneering work of McFadden (1974) on multinomial logit models, in which the random components are assumed to be additive and the random terms Gumbel distributed. We show that the same qualitative results, with closed-form selection probabilities, hold for a wide class of random selection processes without such structural and parametric assumptions.
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