Random USC Functions, Max-Stable Processes and Continuous Choice

The theory of random utility maximization for a finite set of alternatives is generalized to alternatives which are elements of a compact metric space T. We model the random utility of these alternatives ranging over a continuum as a random process {Yt,t.T} with upper semicontinuous (usc) sample paths. The alternatives which achieve the maximum utility levels constitute a random closed, compact set M. We specialize to a model where the random utility is a max-stable process with a.s. usc paths. Further path properties of these processes are derived and explicit formulas are calculated for the hitting and containment functionals of M. The hitting functional corresponds to the choice probabilities.