Placement of edge computing servers at the edge of the network can reduce task transmission delay. Connecting them into a system can provide services for a wider range. However, due to the mobility of the crowd and mobile devices, the number of tasks offloaded to each edge server may be quite different, which will seriously affect the QoS of the system. To this end, we investigate the QoS improvement of the distributed edge computing system from the game-theoretic perspective and propose a multi-agent state-based learning algorithm. Firstly, by modeling the cost of an edge computing server as the deviation between its execution time and the system average execution time, we formulate the QoS improvement of the system as a state-based game where each agent competes to maximize its own utility. Then, we propose a multi-agent state-based learning algorithm to obtain the pure Nash equilibrium strategy of each agent. Finally, compared with the existing approaches, the experiments show that the proposed algorithm can improve the QoS of the distributed edge computing system.