Invariance of conditional maximum utility

The consumer benefit in a discrete choice model is often measured by maximum utility. We characterize the conditional (on the chosen alternative) and the unconditional distribution of maximum utility. We show that among a wide class of distributions (independent with convex supports) of error terms, the Type I extreme-value distribution is the unique distribution which ensures that all the conditional distributions of maximum utility coincide. Moreover, we show that for i.i.d. (with convex support) error terms, the invariance of conditional expected maximum utility characterizes the multinomial logit model.