Per Olov Lindberg: Choice Probabilities in Random Utility Models

Random Utility (RU) Models have reached a wide applicability in many areas of applied Economics. In all application areas, the formulas for the choice probabilities play a central role. For the formulas to be useful, the probabilities further need to sum up to 1. In this note, we give necessary and sufficient conditions for the validity of the classical choice probability formula (i.e. that the choice probabilities are the integrals of the partial derivatives of the cdf of the random term along the diagonal), and that the probabilities sum up to 1. Further, it is proved sufficient that the gradient of said cdf is continuous and piecewise differentiable with locally bounded gradient, along the diagonal. Along the way we prove many intermediate results relating different derivatives of the cdf. Finally the results are applied to derive formulas for the distribution of achieved utility conditioned on a given alternative being chosen.