Integrating a Pareto-Distributed Scale into the Mixed Logit Model: A Mathematical Concept

Abstract

A generalized multinomial logit (G-MNL) model is proposed to alleviate the four challenges inherent to the conditional logit model, including (1) simultaneous unidentifiability, (2) the immediacy of decision-making, (3) the homogeneity of preferences in unobservable variables, and (4) the independence of irrelevant alternatives. However, the G-MNL model has some restrictions that are caused by the assumed logit scale of the lognormal distribution used in the G-MNL model. We propose a mixed logit with integrated Pareto-distributed scale (MIXL-iPS) model to address the restriction of the G-MNL model by introducing a logit scale in accordance with the Pareto distribution type I with an expected value of 1. We have clarified the mathematical properties and examined the distributional properties of the novel MIXL-iPS model. The results suggest that the MIXL-iPS model is a model in which the instability in the estimation of the G-MNL model is modified. Moreover, the apparent preference parameter was confirmed to have a skewed distribution in general in the MIXL-iPS model. In addition, we confirm that in the MIXL-iPS model, bounded rationality is reasonably well represented, as many individuals have below-average choice consistency.