Fixed-Point States of Day-to-Day Assignment Processes with State-Dependent Route Choice

Abstract

Abstract Stochastic User Equilibrium (SUE) has been proposed to overcome the limitations of the perfect knowledge assumption of classical Wardrop’s user equilibrium. Imperfect knowledge of the network and the associated heterogeneity in the perception of route travel times, as it occurs in the absence of Advanced Traveller Information Systems (ATIS), justifies SUE providing strictly positive flows for all routes. SUE is at the same time the steady state of a day-to-day Markov assignment process under the assumption of state independent route choice, i.e. choice independent of the previously chosen route. Still in a context of absence of ATIS, state-dependent route choice represents a more realistic assumption: users exhibit inertia to change. Experimental evidence supports the assumption. The paper investigates the fixed-point states of Markov assignment processes with state-dependent route choice. These fixed points characterise a new concept of network equilibrium, referred to as State-Dependent Stochastic User Equilibrium (SDSUE), where if each user shifts from her current route to her newly chosen route the observed route flows do not change. The existence of SDSUE is guaranteed under usually satisfied conditions. A modified method of successive averages is proposed for computing SDSUE. An example related to the Braess network with logit route choice illustrates the theoretical insights. Numerical evidence is provided of a symmetry property of SDSUE transition flows whereby, at equilibrium, the number of users shifting from route i to route j equals the number of those shifting from route j to route i.