This paper presents a variant of standard fictitious play called average strategy fictitious play (ASFP) for large-scale repeated congestion games, where only a weighted running average of all other players' actions is assumed to be available to each player. It reduces the burden of both information gathering and information processing for each player. Compared to joint strategy fictitious play (JSFP) studied in the literature, the updating process of utility functions for each player is avoided. We prove that there exists at least one pure strategy Nash equilibrium for the congestion game under investigation, and the players' actions generated by ASFP with inertia (players' reluctance to change their previous actions) converge to a Nash equilibrium almost surely. The results are applied in road pricing design to achieve socially beneficial trip timing. Simulation results are provided based on the real traffic data for the Singapore case study.