Quasi-variational inequality formulation of the mixed equilibrium in multiclass routing games

In the modeling of competition on networks it is usually assumed that users either behave following the Wardrop equilibrium or the Nash equilibrium concept. Nevertheless, in several equilibrium situations, for instance in urban traffic flows, intercity freight flows and telecommunication networks, a mixed behavior is observed. This paper presents a time-dependent network model shared by two types of users: group users (Nash players) and individual users (Wardrop players). A group user has a significant impact on the load of the network, whereas an individual user has a negligible impact. Both classes of users choose the paths to ship their jobs so as to minimize their costs, but they apply different optimization criteria. The source of interaction of users is represented by the travel demand, which is assumed to be elastic with respect to the equilibrium solution. Thus, the equilibrium distribution is proved to be equivalent to the solution of an appropriate time-dependent quasi-variational inequality problem. A result on the existence of solutions is discussed as well as a numerical example.