Flows over time enable a mathematical modeling of traffic that changes as time progresses. In order to evaluate these dynamic flows from a game theoretical perspective we consider the price of anarchy (PoA). In this paper we study the impact of spillback effects on the PoA, which turn out to be substantial. It is known that, in general, the PoA is unbounded in the spillback setting. We extend this by showing that it is still unbounded even when considering networks with unit edge capacities and that the Braess ratio can be arbitrarily large. In contrast to that, we show that on a fixed network the PoA as a function of the flow amount is bounded by a constant and also upper bound the PoA for the set of networks where the outflow capacities satisfy certain constraints depending on the quickest flow. This upper bound only depends on the worst spillback factor of the Nash flows over time of the given network. It therefore provides a way to quantify the impact of spillback to the quality of the dynamic equilibria. In addition, we show the surprising fact that the introduction of spillback behavior can actually speed up dynamic equilibria in some networks.

中文翻译：

---

随着时间的流逝，溢出对无政府状态价格的影响

随着时间的流逝，流量可以对随时间变化的流量进行数学建模。为了从博弈论的角度评估这些动态流，我们考虑无政府状态（PoA）的价格。在本文中，我们研究了溢出效应对 PoA 的影响，结果证明这种影响是巨大的。众所周知，一般来说，PoA 在溢出设置中是无界的。我们通过证明它仍然是无界的，即使考虑具有单位边缘容量的网络并且 Braess 比率可以任意大，我们对此进行了扩展。与此相反，我们表明在固定网络上，作为流量的函数的 PoA 受一个常数的限制，并且也是网络集的 PoA 上限，其中流出容量满足某些约束取决于最快的流量。该上限仅取决于给定网络随时间变化的纳什流的最坏溢出因子。因此，它提供了一种量化溢出对动态平衡质量的影响的方法。此外，我们展示了一个令人惊讶的事实，即溢出行为的引入实际上可以加速某些网络中的动态平衡。