Stackelberg thresholds on parallel networks with horizontal queues

We study Stackelberg routing games on parallel networks with horizontal queues, in which a coordinator (leader) controls a fraction α of the total flow on the network, and the remaining players (followers) choose their routes selfishly. The objective of the coordinator is to minimize a system-wide cost function, the total travel-time, while anticipating the response of the followers. Nash equilibria of the routing game (with zero control) are known to be inefficient in the sense that the total travel-time is sub-optimal. Increasing the compliance rate α improves the cost of the equilibrium, and we are interested in particular in the Stackelberg threshold, i.e. the minimal compliance rate that achieves a strict improvement. In this work, we derive the optimal Stackelberg cost as a function of the compliance rate α, and obtain, in particular, the expression of the Stackelberg threshold.