Bicriteria Nash flows over time

Traffic congestion imposes a huge economic loss to the economy. As such, there has been a huge effort to understand congestion using theoretical models. The dynamic model that gained most attention for modelling traffic is the deterministic fluid queuing model, already introduced by Vickrey. A common drawback of most of the models is the simplified assumption that road network users only aim for minimizing their arrival time. However, in traffic networks in particular, users are not always that single-minded. In this paper we extend the state-of-the-art game theoretic traffic models with a multi-criteria objective function. We assume that users try to minimize costs subject to arriving at the sink before a given deadline. Here, costs could be thought of as an intrinsic preference a user has regarding the different route choices and queuing dynamics only play a role for the arrival time of a user.

We determine the existence and the structure of Nash flows over time and fully characterize the price of anarchy for this model, which measures the ratio of the quality of the Nash flow and the optimal flow. We evaluate the quality both with respect to the throughput for a given deadline and the makespan for a given amount of flow. We prove the following three results. (i) In series-parallel graphs, both prices of anarchy are unbounded. (ii) In parallel path graphs the throughput-PoA is at most 2, or at most e/(e-1) if all transit times are 0. Both bounds are tight. (iii) In parallel path graphs the makespan-PoA is at most e/(e-1), independent of transit times, and this is tight. All our upper bounds are also valid for dynamic equilibria in the deterministic fluid queuing model.