Logit discrete choice model: a new distribution-free justification

Abstract

According to decision making theory, if we know the user’s utility ${U_i=U(s_i)}$ of all possible alternatives s _i , then we can uniquely predict the user’s preferences. In practice, we often only know approximate values ${V_i\approx U_i}$ of the user’s utilities. Based on these approximate values, we can only make probabilistic predictions of the user’s preferences. It is empirically known that in many real-life situations, the corresponding probabilities are described by a logit model, in which the probability p _i of selecting the alternative s _i is equal to ${p_i=e^{\beta\cdot V_i}/\sum_{j=1}^n e^{\beta\cdot V_j}}$ . There exist many theoretical explanations of this empirical formula, some of these explanations led to a 2000 Nobel prize. However, it is known that the logit formula is empirically valid even when the assumptions behind the existing justifications do not hold. To cover such empirical situations, it is therefore desirable to provide a new distribution-free justification of the logit formula. Such a justification is provided in this paper.