  We reconsider the well-studied Selfish Routing game with affine latency
functions. The Price of Anarchy for this class of games takes maximum value
4/3; this maximum is attained already for a simple network of two parallel
links, known as Pigou's network. We improve upon the value 4/3 by means of
Coordination Mechanisms.
  We increase the latency functions of the edges in the network, i.e., if
$\ell_e(x)$ is the latency function of an edge $e$, we replace it by
$\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an
adversary fixes a demand rate as input. The engineered Price of Anarchy of the
mechanism is defined as the worst-case ratio of the Nash social cost in the
modified network over the optimal social cost in the original network.
Formally, if $\CM(r)$ denotes the cost of the worst Nash flow in the modified
network for rate $r$ and $\Copt(r)$ denotes the cost of the optimal flow in the
original network for the same rate then [\ePoA = \max_{r \ge 0}
\frac{\CM(r)}{\Copt(r)}.]
  We first exhibit a simple coordination mechanism that achieves for any
network of parallel links an engineered Price of Anarchy strictly less than
4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25.
Then, for the case of two parallel links, we describe an optimal mechanism; its
engineered Price of Anarchy lies between 1.191 and 1.192.
