Based on the situation of promoting the effective development of hierarchical healthcare services through hospital associations, this paper studies the contractual mechanism of cooperation between the government and hospitals in coordinating the allocation of high-quality medical resources as well as the mechanism of the operations of hospital associations. By developing a four-stage game-theoretic queueing model, this paper analyzes the dynamic interaction relationships among patients, multiple community hospitals, and tertiary hospitals in a hospital association, and government. Next, this paper presents the equilibrium decisions in terms of patients’ arrival rate, community hospitals’ capacity planning, the tertiary hospital’s resource sinking rate, and the government’s subsidy price, and each community hospital’s profit-sharing price. It finds that, in the absence of resource sinking, the more community hospitals there are, the more conducive they are to reducing the congestion of tertiary hospitals and the healthcare system and improving the welfare of patients. Furthermore, the medium number of community hospitals is more conducive to the improvement of social welfare. By contrast, the fewer the community hospitals, the more favorable the hospital association is to promoting the implementation of the resource-sinking program and improve the effectiveness of the system. When the community hospital’s capacity cost per unit is higher, the maximum number of members in the government-formed consortium should be the smaller. Numerical experiments show that in the process of compensating a tertiary hospital, when the concerned degree of the increase in the rate of patients visiting community hospitals is relatively low or high, the government could save subsidy funds. Specifically, in the case of limited subsidy budgets, this is particularly beneficial for the improvement of social welfare. This paper proposes the contract mechanisms for coordinating the effective allocation of high-quality medical resources in a hospital association. This provides theoretical decision support for the government to establish a scientific hierarchical healthcare delivery system. At the same time, this paper has a high application reference value for improving the overall efficiency of healthcare services.