Utility independence versus IIA property in independent probit models

Abstract

In the field of conjoint choice analysis consumer preferences are commonly estimated via multinomial logit (MNL) or multinomial probit (MNP/IP) models. The basic assumption of both the MNL model and the independent probit (IP) model is independence of utilities between alternatives, also referred to as IID assumption. Since the MNL model in addition exhibits the Independence of Irrelevant Alternatives (IIA) property, it is often directly concluded that the IP model likewise exhibits the IIA property, thereby relating IID to IIA. However, this is not true considering Luce’s definition of the IIA property. The objective of this paper is twofold: First, we clarify that IID does not imply IIA in the specific context of the IP model. Second, we work out the sources for the non-proportional shifting of the ratio of choice shares that distinguishes the IP model from the MNL model. Accordingly, we first theoretically review the true property of the IP model and subsequently illustrate with a conjoint example that the IP model does not exhibit the IIA property. In addition, we provide a case-by-case analysis to explain the non-proportional shifting of choice shares in the substitution pattern of the IP model and work out the principles for that non-proportional shifting mechanism.