In a network composite congestion game, two types of players (nonatomic player of weight zero and atomic splittable player with positive weight) have different strategic behaviors. Statistic properties of equilibrium have been studied recently. This work considers the dynamical aspects, i.e. the evolution of the players' behavior at disequilibium states. Suppose that players adjust their strategies in a selfish and myopic way. The evolution of the strategy profile can be approximated by a continuous-time dynamical system. One shows that several dynamics well-known in the framework of population games (thus with only nonatomic players) such as replicator, BNN, Smith, and two projection dynamics can well be adapted to this more general framework with heterogeneous players. Their asymptotic properties are analysed. Explicitly, one investigates under which condition these dynamics converge to the equilibria.

The main approach of the work is to characterize an equilibrium as a solution to a variational inequality problem (VIP). Then some known "gap functions", whose minimum points coincide with the solutions of the VIP, are natural candidates to serve as Lyapunov function in the corresponding dynamics.

This work is the first to study dynamics in composite congestion games or more generally in games with similar heterogeneous players (which are not rare in economics). Besides, some of the dynamics can be adapted (via a quasi-variational inequality formulation of the equilibrium) to generalized equilibrium problems where the strategy space of a player depends on his opponents' strategies. This setting is more realistic when one takes the real world constraints such as road capacity or production capacity into account.