On the Evolutionary Dynamics and Optimal Control of a Tripartite Game in the Pharmaceutical Procurement Supply Chain with Regulatory Participation

This study involves the construction of a dynamic evolutionary game model involving three key participants, including the Group Purchasing Organization (GPO), medical institutions, and pharmaceutical suppliers, while comprehensively considering critical factors such as benefit compensation, bad debt risk, and fiscal costs. The model characterizes the strategy evolution of each participant under bounded rationality and imitation learning mechanisms. Based on the replicator dynamics equations, the evolutionary trajectories and equilibrium conditions of the three parties’ strategies are systematically derived. The Jacobian matrix is then used to analyze the local stability of eight boundary equilibria and potential internal mixed equilibria. Furthermore, to capture the optimal adjustment process of the compensation mechanism, the GPO’s compensation level is introduced into an optimal control framework. A controlled evolutionary system is formulated, and the dynamic optimal relationship between compensation intensity and system state is described using the Hamilton–Jacobi–Bellman (HJB) equation. Through analytical linearization and numerical simulations, the optimal feedback compensation law and its closed-loop evolutionary trajectory are obtained, allowing for a comparative analysis between the “fixed compensation” and “optimal compensation” scenarios. The results reveal that an appropriately designed dynamic compensation mechanism can significantly enhance system cooperation stability and overall social welfare. This provides a quantitative theoretical foundation and methodological tool for the refined design and dynamic regulation of pharmaceutical group purchasing policies.