Reconciling Selfish Routing with Social Good.

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. Right after, 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 of (theta ) for which this task is NP-hard for the concepts of (theta )-Used Nash Equilibrium flow and (theta )-Envy Free flow. Finally, we propose strategies which, in a worst-case approach, can be used by a central planner in order to provide good (theta )-flows.