  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.
