We study the static traffic assignment problem under a Stackelberg routing strategy on a few example networks. We mainly use affine cost functions, with no capacity constraints. However our methods and results can be generalised to more diverse cost functions (as long as the resulting objective functions are convex).
The standard traffic assignment problem consists on finding the stationary traffic flow on all the links on a road network, for a given demand vector that encodes the number of vehicles per unit time that need to travel between origin-destination node pairs.
We consider the following leader-follower scenario: A network manager (acting as the leader) has the freedom to route a certain proportion of the vehicles on the network, whilst the remaining vehicles (the followers) follow a selfish routing strategy.
The network manager attempts to minimise a cost functional, for example, the total network cost. The selfish vehicles’ behaviour leads them to minimise the Beckmann functional for modified cost functions that incorporate the flow of the altruistic vehicles into the free flow cost.
We express the problem as a bilevel convex optimisation problem . We obtain numerical solutions and show the reduction of total system cost as the proportion of centrally routed vehicles increases.
We discuss several approaches to solving the optimisation problem, and compare the problem to solving the traffic assignment problem with tolls.