A theoretical analysis of the cross-nested logit model

The emergence of Intelligent Transportation Systems and the associated technologies has increased the need for complex models and algorithms. Namely, real-time information systems, directly influencing transportation demand, must be supported by detailed behavioral models capturing travel and driving decisions. Discrete choice models methodology provides an appropriate framework to capture such behavior. Recently, the Cross-Nested Logit (CNL) model has received quite a bit of attention in the literature to capture decisions such as mode choice, departure time choice and route choice. The CNL model is an extension of the Nested Logit model, providing more flexibility at the cost of some complexity in the model formulation. In this paper, we develop on the general formulation of the Cross-Nested Logit model proposed by Ben-Akiva and Bierlaire and based on the Generalized Extreme Value (GEV) model. We show that it is equivalent to the formulations by Papola, and Wen and Koppelman. We also show that the formulations by Small and Vovsha are special cases of this formulation. We formally prove that the Cross-Nested Logit model is indeed a member of the GEV models family. In doing so, we clearly distinguish between conditions that are necessary to prove consistency with the GEV theory, from normalization conditions. Finally, we propose to estimate the model with non-linear programming algorithms, instead of heuristics proposed in the literature. In order to make it operational, we provide the first derivatives of the log-likelihood function, which are necessary to such optimization procedures.