This work explores the relationship between the set of Wardrop equilibria~(WE) of a routing game, the total demand of that game, and the occurrence of Braess's paradox~(BP). The BP formalizes the counter-intuitive fact that for some networks, removing a path from the network decreases congestion at WE. For a single origin-destination routing games with affine cost functions, the first part of this work provides tools for analyzing the evolution of the WE as the demand varies. It characterizes the piece-wise affine nature of this dependence by showing that the set of directions in which the WE can vary in each piece is the solution of a variational inequality problem. In the process we establish various properties of changes in the set of used and minimal-cost paths as demand varies. As a consequence of these characterizations, we derive a procedure to obtain the WE for all demands above a certain threshold. The second part of the paper deals with detecting the presence of BP in a network. We supply a number of sufficient conditions that reveal the presence of BP and that are computationally tractable. We also discuss a different perspective on BP, where we establish that a path causing BP at a particular demand must be strictly beneficial to the network at a lower demand. Several examples throughout this work illustrate and elaborate our findings.

翻译：本文探讨了路由博弈中Wardrop均衡集、博弈总需求与Braess悖论发生之间的关系。Braess悖论形式化描述了反直觉现象：在某些网络中移除一条路径反而会降低Wardrop均衡时的拥塞程度。对于具有仿射成本函数的单原点-终点路由博弈，本文第一部分提供了分析Wardrop均衡随需求变化演化规律的工具。通过证明在每个分段中Wardrop均衡可能变化的方向集合是变分不等式问题的解，刻画了该依赖关系的分段仿射特性。在此过程中，我们建立了随需求变化时所用路径集与最小成本路径集变化的多种性质。基于这些刻画，我们推导出获取所有高于特定阈值需求的Wardrop均衡的计算流程。论文第二部分涉及网络中Braess悖论的检测。我们提供了若干计算可行的充分条件来揭示Braess悖论的存在。同时探讨了关于Braess悖论的新视角：证明在特定需求下引发Braess悖论的路径在较低需求时必然对网络严格有利。全文通过多个算例阐明并深化了我们的发现。