Randomness in Transportation Utility Models The Triangular Distribution may be a better choice than the Normal and Gumbel

Discrete Choice Random Utility Models (RUMs) analyze, predict and model human choices and have been successfully used in modeling driving decisions. A random term is included in RUMs to resolve similar alternatives and model the unpredictable factors leading a person to make a choice among a discrete set of options. There is no agreed and scientifically proven best way of modelling this term despite having been applied in many diverse fields and little attention has been paid to the underlying phenomena that justify one choice over another. In this paper, we examine theoretically and in a particular taxi traffic simulation in Singapore, the most common choices for the random term, logit and probit, and also contrast them with simple alternatives such as the uniform or triangular distributions. We provide a Unified Random Graph Model (URGM) to explore randomness in RUMs, identify the differences between distribution choices, introduce a novel distribution distance metric, and explore the effect such choices have in a RUM. We contend that the theoretical and practical advantages of the Gaussian and the Gumbel distribution (GaGs) have been overrated in the traffic simulation literature, even without any categorical theoretical or empirical justification for one over the other (we show the Gaussian approximates the Gumbel 81-94% depending on the metric). We provide evidence in favor of the triangular distribution and show that it can approximate the GaGs in the 90s% without some of their demerits. These results are corroborated by our theoretical and experimental analysis, were the Triangular outperforms GaGs and has several theoretical advantages. The analysis is in many ways comparable to the activation function debate in the deep learning literature.