A characterization theorem for random utility variables

In this note we investigate the condition that the distribution of the maximum of a set of random variables does not depend on which variable attains the maximum. This problem arises in random utility theory. When the random variables are independent, the property implies that all the marginal distributions must be Double Exponential (with distribution function exp(?e^?x) in standard form). When dependence is allowed the property characrerizes a much broader class consisting of arbitrary functions of arbitrary homogeneous functions of the variables e^?xi, a result stated without proof in D. J. Strauss (Journal of Mathematical Psychology, 1979, 20, 35–52). These are the distributions such that the maximum has the same distribution (apart from a location shift) as the marginals, provided the marginals are the same.