Mixed behavior network equilibria and quasi-variational inequalities

In the modeling of competition on networks it is usually assumed that users either behave following the Wardropian user equilibrium or the system optimum 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-based model shared by two types of users: generalized Nash players and user equilibrium players. Generalized Nash players have a significant impact on the load of the network, whereas user equilibrium players have a negligible impact. Both classes of players choose the paths to send their flows so as to minimize their own costs, but they apply different optimization criteria. Players interact via some implicit balance constraints which depend on the equilibrium solution. Thus, the equilibrium distribution is proved to be equivalent to the solution of a time-dependent quasi-variational inequality problem. Results on existence of solutions are discussed as well as a numerical example.