The price of optimum in Stackelberg games on arbitrary single commodity networks and latency functions

Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [12, 26]. A Leader can decrease the coordination ratio by assigning flow αr on M, and then all Followers assign selfishly the (1 - α)r remaining flow. This is a Stackelberg Scheduling Instance (M,r,α), 0 ≤ α ≤ 1. It was shown [23] that it is weakly NP-hard to compute the optimal Leader's strategy.For any such network M we efficiently compute the minimum portion βM of flow r needed by a Leader to induce M's optimum cost, as well as his optimal strategy.Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling αr flow can induce ≤ 1<over>α times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example.