How Hard Is For Agents To Learn The User Equilibrium? Characterizing Traffic Networks By Means Of Entropy

Traffic assignment is an important stage in the task of modeling a transportation system. Several methods for solving the traffic assignment problem (TAP) were proposed, mostly based on iterative procedures. However, little was done in the direction of analyzing the difficulty of such procedures. For instance, why is it that some networks require orders of magnitude more iterations than others? What matters in this task? Clearly, the topology of the network can only give hints up to a certain level; the assignment task is fundamentally tied to how the demand is distributed (among other characteristics of the problem). This means that methods to estimate the complexity of a network (e.g. those based on centrality measures) can only help up to a certain level. The motivation for this work is to, a priori, estimate how hard will the effort underlying solving the TAP be (i.e. without doing the actual assignment). It arose from the fact that after performing assignment in several different networks, we noted that finding the solution for the problem was much easier for some networks than for others. Specifically, the more complex the network, the more difficulty it is to setup the parameters underlying the procedure for solving the TAP. In this work, we propose a new measure of how coupled routes in a network are, based on an estimation of the demand distribution. Our approach involves three main steps: (i) sampling the universe of all possible assignments, (ii) creating a model that gives the incentive a road user has for changing routes, as well as the asymptotic distribution of preference for routes, and (iii) computing the entropy of this distribution. This approach is experimentally validated using several networks of different natures. We then solve the TAP by letting road users use reinforcement learning to learn the user equilibrium. With this, we are able to make important relationships between the entropy values and how hard are the learning tasks. Results suggest that it is possible to use both the entropy values as well as the asymptotic distribution of preferences of the road users to gain important information that guides the traffic expert. For instance, the higher the entropy, the higher the indication that more than one route is perceived as preferable and, hence, the more difficult the learning task.