Nash Flows Over Time


Motivated by the dynamic traffic assignment problem and with the goal in mind to obtain a better understanding of the complex traffic dynamics, we consider a dynamic game in a flow over time model with deterministic queuing in this thesis. The dynamic equilibria, called Nash flows over time, in the base version of this model were introduced by Koch and Skutella in 2009 and they were already studied intensively. But the flow dynamics lack essential features, such as spillback and kinematic waves, and more crucially, they can only handle a single commodity. Hence, Nash flows over time are only a very rough approximation of a real-world traffic scenario. In order to close the gap between large-scale simulation tools, which work well in practice but lack any provable foundation, and the mathematical theory, we extend this base model by several very natural traffic features and showed that, by extending the proof ideas, we can, in most cases, preserve the existence and the structural insights of dynamic equilibria. As a first starting point we introduce time-dependent capacities and time-dependent speed limits in order to represent changes in the road network, such as planned construction work or school zones. This extension changes the original model only slightly and it is no surprise that it is possible to prove that Nash flows over time in time-dependent networks can, again, be constructed by a sequence of static flows, called thin flows, which are very similar to the thin flows with resetting of the base model. Much more challenging is the consideration of multiple commodities in such a dynamic network game. The essential property of a global FIFO principle does not hold for these scenarios, and therefore, it is not possible anymore to extend multi-commodity Nash flows over time step by step. Instead, we have to consider all infinitesimally small players at the same time as the choice of early particles depends not only on all the flow already in the network but may also depend on all future flow. Even though we cannot use any of the techniques of the base model, we are still able to show the existence of dynamic equilibria in this multi-commodity setting with the help of infinite-dimensional variational inequalities. Since this was known prior to the work in this thesis, the main contribution is the structural insight into these Nash flows over time, as we can show that their derivatives have to satisfy a set of conditions similar to the thin flow equations. The major difference to the single-commodity case is that we cannot consider each thin flow isolated anymore, but instead, we have to take into account the flow of the other commodities (the so-called foreign flow), and therefore, we have to consider all flow from the past and the future simultaneously. Unfortunately, this still does not give a clear instruction on how to construct Nash flows over time with multiple commodities, however, for the special case that all commodities share the same origin we show that the problem of constructing a Nash flow over time reduces to the single-commodity case. The same holds true for the other extreme case, that every particle can start at multiple origins but they all share a common destination. Clearly, the extension of the base model to a model with spillback and kinematic waves is one of the main contributions of this thesis. These fundamental traffic features, which are especially relevant in highly congested networks, were a huge challenge for the deterministic queuing model. The key idea is to restrict the total amount of flow on each arc by an arc specific storage capacity and to model the backwards moving gaps between vehicles by a gap flow over time. In order to obtain a well-defined flow over time based on the route choices of the particles we introduce a priority rule on each intersection. The fair allocation condition guarantees that, in the case of spillback, the particles merge according to the outflow capacity of each incoming link. As we only consider a single-commodity we can, again, use the network-wide FIFO principle in order to construct a Nash flow over time in this kinematic wave model. However, most of the proof ideas do not transfer to this extension, and therefore, most of the proofs become much more involved. In addition, we observe that the arrival times of these dynamic equilibria are not unique anymore and that, due to the fact that congestions can block intersections, the price of anarchy is unbounded in the spillback model. Finally, we consider a different type of flows over time, called instantaneous dynamic equilibria (or IDE flow for short). Here, particles do not predict the future evolution, but instead, each of them chooses a route depending on the current network configuration, i.e., the current shortest distance to the destination. As these might change drastically along the way, each player is allowed to adopt his or her route choice on the way. Even though these flows over time do not form a game theoretical Nash equilibrium, they are well motivated by real-world scenarios since traffic users, following a navigation system, might get re-routed as soon as the current traffic conditions change. As we only have to consider the current waiting times it becomes much easier to construct such IDE flows. In fact, we can even handle multi-commodity IDE flows with the same thin flow technique we use for single-commodity Nash flows over time.