We present the design and analysis of a multi-level game-theoretic model of hierarchical policy-making, inspired by policy responses to the COVID-19 pandemic. Our model captures the potentially mismatched priorities among a hierarchy of policy-makers (e.g., federal, state, and local governments) with respect to two main cost components that have opposite dependence on the policy strength, such as post-intervention infection rates and the cost of policy implementation. Our model further includes a crucial third factor in decisions: a cost of non-compliance with the policy-maker immediately above in the hierarchy, such as non-compliance of state with federal policies. Our first contribution is a closed-form approximation of a recently published agent-based model to compute the number of infections for any implemented policy. Second, we present a novel equilibrium selection criterion that addresses common issues with equilibrium multiplicity in our setting. Third, we propose a hierarchical algorithm based on best response dynamics for computing an approximate equilibrium of the hierarchical policy-making game consistent with our solution concept. Finally, we present an empirical investigation of equilibrium policy strategies in this game in terms of the extent of free riding as well as fairness in the distribution of costs depending on game parameters such as the degree of centralization and disagreements about policy priorities among the agents.
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