Caching popular files in the storage of edge networks, namely edge caching, is a promising approach for service providers (SPs) to reduce redundant backhaul transmission to edge nodes (ENs). In this paper, an edge network with one SP, a large number of ENs, and mobile users with time-dependent requests is considered. A convergent and scalable Stackelberg game for edge caching is designed. Specifically, the game is decomposed into two types of sub-games, a storage allocation game (SAG) and a number of user allocation games (UAGs). A Stackelberg game-based alternating direction method of multipliers (Stackelberg game-based ADMM) is proposed to solve either the SAG or each UAG in a distributed manner. The convergence speed and the optimum of the entire game are linearly (or sublinearly) related to the network size, which indicates that this framework can potentially cope with large-scale caching problems. It is also seen in the simulation results that this framework requires fewer backhaul resources than existing approaches.