Stackelberg thresholds in network routing games or the value of altruism

Noncooperative network routing games are a natural model of userstrying to selfishly route flow through a network in order to minimize their own delays. It is well known that the solution resulting from this selfish routing (called the Nash equilibrium) can have social cost strictly higher than the cost of the optimum solution. One way to improve the quality of the resulting solution is to centrally control a fraction of the flow. A natural problem for the network administrator then is to route the centrally controlled flow in such a way that the overall cost of the solution is minimized after the remaining fraction has routed itself selfishl. This problem falls in the class of well-studied Stackelberg routing games. We consider the scenario where the network administrator wants the final solution to be (strictly) better than the Nash equilibrium. In other words, she wants to control enough flow such that the cost of the resulting solution is strictly less than the cost of the Nash equilibrium. We call the minimum fraction of users that must be centrally routed to improve the quality of the resulting solution the Stackelberg threshold. We give a closed form expression for the Stackelberg threshold for parallel links networks with linear latency functions. The expression is in terms of Nash equilibrium flows and optimum flows. It turns out that the Stackelberg threshold is the minimum of Nash flows on links which have more optimum flow than Nash flow. Using our approach to characterize the Stackelberg thresholds, we are able to give a simpler proof of an earlier result which finds the minimum fraction required to be centrally controlled to induce an optimum solution.