Invariance of Achieved Utility in Random Utility Models

Abstract

The property in some random utility models that the distribution of achieved utility is invariant across alternatives (the invariance property) is noteworthy as it applies to the multinomial logit model as well as to its generalization: the generalized extreme-value (GEV) models. GEV models constitute the most versatile tool yet known for dealing with discrete choice situations with a structure of similarity—that is, statistical dependence—among alternatives. The invariance property is obviously violated in practice for heterogeneous populations. Therefore it has been argued that invariance constitutes a major problem for GEV models. In contrast these authors argue that invariance is a useful theoretical concept precisely by bringing out heterogeneity. Further, multiple segment GEV models are a suitable tool for dealing with heterogeneity—both theoretically and pragmatically. The class of random utility models possessing the invariance property was characterized by Robertson and Strauss; called the RS class here. However, their proof was not complete. An alternative representation of the RS class is suggested based on the notion of additive homogeneity. This new representation enables the authors to prove the RS characterization theorem and to simplify and systematize the proofs of many other results on RS—and specifically GEV—models. Also, in the new representation, the characterization is naturally stated in terms of the choice probabilities, and of the probability distribution of maximum utility. Assuming that the distribution of actual choices is observable, the choice probabilities are particularly empirically meaningful. This motivates a study of the conditions for a choice probability structure to be RS representable. For the binary choice case conditions that are both necessary and sufficient are given.