  Selfish routing is a central problem in algorithmic game theory, with one of
the principal applications being that of routing in road networks. Inspired by
the emergence of routing technologies and autonomous driving, we revisit
selfish routing and consider three possible outcomes of it: (i)
$\theta$-Positive Nash Equilibrium flow, where every path that has non-zero
flow on all of its edges has cost no greater than $\theta$ times the cost of
any other path, (ii) $\theta$-Used Nash Equilibrium flow, where every used path
that appears in the path flow decomposition has cost no greater than $\theta$
times the cost of any other path, and (iii) $\theta$-Envy Free flow, where
every path that appears in the path flow decomposition has cost no greater than
$\theta$ times the cost of any other path in the path flow decomposition. We
first examine the relations of these outcomes among each other and then measure
their possible impact on the network's performance. Afterwards, we examine the
computational complexity of finding such flows of minimum social cost and give
a range for $\theta$ for which this task is easy and a range for $\theta$ for
which this task is NP-hard. Finally, we propose deterministic strategies which,
in a worst case approach, can be used by a central planner in order to provide
good such flows, and further introduce a natural idea for randomly routing
players after giving them specific guarantees about their costs in the
randomized routing, as a tool for the central planner to implement a desired
flow.
