  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.
