On the Equivalence Between SUE and Fixed-Point States of Day-to-Day Assignment Processes with Serially-Correlated Route Choice

We propose a new theoretical framework of deterministic processes of traffic assignment able to include day-to-day correlation of the random terms. According to the prevailing interpretation of random utility models, random terms are regarded as individual specific. Correlation is justified by persistence of habits and unobservables. The framework includes the deterministic sequence of systematic utilities based on a learning filter, the stochastic process of the random terms based on a stationary autoregressive structure with i.i.d. Gumbel or multivariate normal one-day marginals, and the resulting stochastic process of choice which generally is not Markov. The fixed point states of the choice process are the classical logit and probit Stochastic User Equilibrium (SUE). The linkage between flows at any day and transition flows is made explicit, and, by this, a new perspective on the interpretation of SUE is opened. SUE is the condition where, at macro level, the observed route flows do not change across days, while, at micro level, individuals change route. Only if random terms are unchanged no individual changes route. Transition flows at SUE are symmetric, i.e. the number of shifters from path i to path j equals the number of shifters from path j to path i. The insights are illustrated by numerical examples related to a two-link and a five-link network.