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Besides the topological structure, there are additional information, i.e., node attributes, on top of the plain graphs. Usually, these systems can be well modeled by attributed graphs, where nodes represent component actors, a set of attributes describe users' portraits and edges indicate their connections. An elusive question associated with attributed graphs is to study how clusters with common internal properties form and evolve in real-world networked systems with great individual diversity, which leads to the so-called problem of attributed graph clustering (AGC). In this paper, we comprehended AGC naturally as a dynamic cluster formation game (DCFG), where each node's feasible action set can be constrained by every cluster in a discrete-time dynamical system. Specifically, we carried out a deep research on a special case of finite dynamic games, named dynamic social game (DSG), the convergence of the finite Nash equilibrium sequence in a DSG was also proved strictly. By carefully defining the feasible action set and the utility function associated with each node, the proposed DCFG can be well related to a DSG; and we showed that a balanced solution of AGC could be found by solving a finite set of coupled static Nash equilibrium problems in the related DCFG. We, finally, proposed a self-learning algorithm, which can start from any arbitrary initial cluster configuration, and, finally, find the corresponding balanced solution of AGC, where all nodes and clusters are satisfied with the final cluster configuration. Extensive experiments were applied on real-world social networks to demonstrate both effectiveness and scalability of the proposed approach by comparing with the state-of-the-art graph clustering methods in the literature.