  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.
