Analysis of Discrete Choice Models: A Welfare-Based Framework

Based on the observation that many existing discrete choice models admit a welfare function of utilities whose gradient gives the choice probability vector, we propose a new representation of discrete choice model which we call the welfare-based choice model. The welfare-based choice model is meaningful on its own by providing a new way of constructing choice models. More importantly, it provides great analysis convenience for establishing connections among existing choice models. We prove by using convex analysis theory, that the welfare-based choice model is equivalent to the representative agent choice model and the semi-parametric choice model, establishing the equivalence of the latter two. We show that these three models are all strictly more general than the random utility model, while when there are only two alternatives, those four models are equivalent. In particular, we show that the distinction between the welfare-based choice model and the random utility model lies in the requirement of the higher-order derivatives of the welfare function. We then define a new concept in choice models: substitutability/complementarity between alternatives. We show that the random utility model only allows substitutability between different alternatives; while the welfare-based choice model allows flexible substitutability/complementarity patterns. We argue that such flexibility could be desirable in capturing certain practical choice patterns and expanding the scope of discrete choice models. Examples are given of new choice models proposed under our framework.