A General and Operational Representation of Generalised Extreme Value Models
Generalised extreme value models provide an interesting theoretical framework to develop closed-form random utility models. Unfortunately, few instances of this family are to be found as operational models in practice. The form of the model, based on a generating function G which must satisfy specific properties, is rather complicated. Fundamentally, it is not an easy task to translate an intuitive perception of the correlation structure by the modeller into a concrete G function. And even if the modeller succeeds in proposing a new G function, the task of proving that it indeed satisfies the properties is cumbersome. In modelling transportation demand, researchers face the problem that many of the choices they wish to model interact in complex ways. One approach to this problem is to use mixed logit models, exploiting the power of simulation-based estimation, to incorporate the interactions required. An alternative approach, however, which is followed in this paper, is to exploit further the GEV model family originally proposed by McFadden [McFadden, D., 1978. Modelling the choice of residential location. In: Karlquist, A. et al. (Eds.), Spatial Interaction Theory and Residential Location. North-Holland, Amsterdam, pp. 7596]. The main objectives of this paper are (i) to provide a general theoretical foundation, so that the development of new GEV models will be easier in the future, and (ii) to propose an easy way of generating new GEV models without a need for complicated proofs. Our technique requires only a network structure capturing the underlying correlation of the choice situation under consideration. If the network complies with some simple conditions, we show how to build an associated model. We prove that it is indeed a GEV model and, therefore, complies with random utility theory. The multinomial logit, the nested logit and the cross-nested logit models are specific instances of our class of models. So are the recent GenL model, combining choice set generation and choice model and some specialised compound models used in recent transportation work. Probability, expected maximum utility and elasticity formulae for the class of models are provided.