In a non-atomic network game, there is a continuum of selfish users, independently choosing routes from the origins to thedestinations of their trips. In the static version of the game, each link of the network is endowed by a continuous increasing costfunction of the total flow of agents on the link. It is well known that a Nash equilibrium when no user can decrease own route costby unilaterally changing their route exists and is generally less efficient than the system optimum. In the last decade, a considerableamount of literature was devoted to designing Stackelberg routing in order to reduce the cost of anarchy, i.e. the ratio between thetotal cost of Nash and the system optimum cost, see e.g. Harks (2011) for overview. Apart from the non-atomic users theStackelberg routing assumes an atomic player - a leader - that can unilaterally and consciously influence the cost for thenon-atomic users – the followers - by partially controlling their route choice for a fraction of users. It is usually assumed that thecontrolled users perceive the same cost on each link as the uncontrolled ones.In our paper, we assume that there are two types of agents, which have different cost functions. As an example of such situation,one could consider a continuum of cars and a fleet of trucks. Indeed, the truck speed is normally lower than that of the cars and isless influenced by the congestion. Moreover, the truck route choice may be controlled by a common agency that pursues a strategyof minimising the total cost forof truck (private agency) or the total cost for all vehicles (governmental agency). The costs of usingthe routes areis route specific and vehicle type specific and isare given as specified as a linear functions of the total number ofusers on the route. Each car is atomic and ignores the impact of his decision on congestion. On the contrary, the coordinator of thefleet may take into account the total congestion cost of the trucks and of the cars. We consider several market situations:Stackelberg equilibrium with trucks controlled by the private agency(Stackelberg), the social system optimum, the second-bestoptimum with trucks controlled by the governmental agency, as well as the benchmark (Nash) with no coordination at all. Despitethe simple formulation, all scenarios beside the Nash lead to non-convex minimisation problems. Each of these problems alwayshas a non-interior solution although interior solutions may exist too.Without the coordination, the trucks and the cars choose their routes according to the deterministic user equilibrium. In the socialoptimum, the total cost for cars and trucks is minimised, and it is almost always possible to obtain a non-interior solution withlower total cost than in the user equilibrium.In the network consisting of two identical (i.e. with similar cost functions) parallel routes the trucks cannot benefit from thecoordination and the Stackelberg equilibrium coincides with Nash equilibrium. However, if there are more trucks (car equivalents)than cars and if the car cost function is steeper than the truck cost function, then the governmental agency can improve the socialwelfare compared to the user equilibrium scenario. In this case, moving a truck from the route that accumulates all cars increasethe total cost for trucks but decreases the total cost for the whole collection of vehicles.If the two routes are not identical then the coordination of trucks may actually worsen the situation by reducing the cost for trucksbut increasing the cost for cars and the total cost compared to the Nash equilibrium. On the other side, the governmental agencycontrolling the trucks may decrease the total cost to a value which is lower than the total cost in the Nash equilibrium at the sametime increasing the cost for trucks.In the Stackelberg game, the fleet of trucks is coordinated in order to minimise their total cost. We show that there is always anon-interior solution. However, in the case of identical routes neither cars nor trucks benefit from the coordination of the truck fleetsince equilibrium and optimum coincides. SAY WHAT COULD HAPPEN IN A NON SYMETRIC Finally, in the second bestscenario we envisage, we assume that the trucks choose routes so that the total cost over all vehicles (cars and trucks) is minimised.In the symmetric case, but with different cost functions for trucks and for cars, we have shown that no benefit can be obtained bythe coordination if the number of cars exceeds the number of trucks. However, with more trucks than cars, there is always apossibility to improve the social welfare compared to the user equilibrium scenario. PROVIDE SOME HINT WHY HEREWe finally examine the coordination game, when two fleets of trucks are competingcompeting for what. And the so what?Add the reference that I asked you to add.
Berlin, Germany, 2012.