Undiscounted Recursive Path Choice Models: Convergence Properties and Algorithms

Abstract

Traffic flow predictions are central to a wealth of problems in transportation. Path choice models can be used for this purpose, and in state-of-the-art models—so-called recursive path choice (RPC) models—the choice of a path is formulated as a sequential arc choice process using undiscounted Markov decision process (MDP) with an absorbing state. The MDP has a utility maximization objective with unknown parameters that are estimated based on data. The estimation and prediction using RPC models require repeatedly solving value functions that are solutions to the Bellman equation. Although there are several examples of successful applications of RPC models in the literature, the convergence of the value iteration method has not been studied. We aim to address this gap. For the two closed-form models in the literature—recursive logit (RL) and nested recursive logit (NRL)—we study the convergence properties of the value iteration method. In the case of the RL model, we show that the operator associated with the Bellman equation is a contraction under certain assumptions on the parameter values. On the contrary, the operator in the NRL case is not a contraction. Focusing on the latter, we study two algorithms designed to improve upon the basic value iteration method. Extensive numerical results based on two real data sets show that the least squares approach we propose outperforms two value iteration methods.