Evolutionary game analysis of family–school–community collaboration in child outdoor physical exercise

Abstract

The widespread insufficiency of children’s outdoor physical exercise has become a critical issue affecting children’s physical health development and school physical education quality. The crux lies not in the absence of any single actor, but in the lack of a stable coordination mechanism among families, schools, and communities. This paper constructs a family–school–community tripartite evolutionary game model, positions the government as an exogenous regulatory variable, introduces the concept of community network externalities, and employs replicator dynamics equations and Jacobian matrix methods to analyze equilibrium stability. The model mechanisms are examined against the case of China’s 2024 “Jiaolianti” (Education-Community Coalition) policy jointly launched by seventeen government departments. The system exhibits three evolutionarily stable strategies (ESS)—E1(0,0,0) low-level trap, E5(1,1,0) family–school coordination, and E8(1,1,1) full coordination—forming a tristable structure. Government family subsidy \({D}_{F}\) acts as a mild accelerator, community support \({D}_{C}\) exhibits a threshold-switch effect, and synergy gain \({\gamma}_{C}\) displays a “high-threshold, high-return” profile. Double-parameter joint sensitivity analysis, attraction domain analysis, and free-riding coefficient robustness testing confirm that the core conclusions are robust within reasonable parameter intervals. The Jiaolianti’s institutional design—taking schools as the center, regions as the principal body, and resources as the connective tissue, with institutional arrangements providing safeguards—shows high consistency with the model’s transition pathways from \({E}_{1}\) to \({E}_{5}\) to \({E}_{8}\) . The study provides a mechanistic explanation for collaborative governance of children’s outdoor physical exercise and offers interval-based, mechanism-based policy references for optimizing Jiaolianti implementation.