A closed-form bounded route choice model accounting for heteroscedasticity, overlap, and choice set formation

The Multinomial Logit (MNL) model is widely used in route choice modelling due to its simple closed-form choice probability function. However, MNL assumes that the error terms are independently and identically distributed with infinite support. As a result, it imposes homoscedasticity, meaning that long and short trips share the same error variance, disregards correlations between overlapping routes, and assigns non-zero choice probabilities to all available routes, regardless of their cost. This paper addresses these limitations by developing a closed-form route choice model. We introduce the Bounded q-Product Logit (BqPL) model, which incorporates heteroscedastic error terms with bounded support. The parameter $q$ controls the rate at which error term variance increases with trip cost, and routes that violate cost bounds receive zero choice probabilities, implicitly defining the route choice set. Furthermore, we extend the BqPL model to account for correlations between overlapping routes by integrating path size correction terms within the choice probability function, resulting in the Bounded Path Size q-Product Logit (BPSqPL) model. We illustrate the properties of the BPSqPL model on small-scale networks, contrasting it with a range of existing choice models into which it can collapse. We then present a method to estimate the model parameters and standard errors, using bootstrapping. Finally, we estimate the model using a large-scale bicycle route choice case study, comparing its goodness-of-fit, interpretability, and forecasting ability with relevant collapsing models. We also test the impact of the choice set size on the estimated parameters. The results underscore the importance of addressing the three key limitations of the MNL model and demonstrate the effectiveness of the BPSqPL model in doing so.