  We design and analyze a multi-level game-theoretic model of hierarchical
policy interventions for epidemic control, such as those in response 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 cost components that have opposite dependence
on the policy strength -- post-intervention infection rates and the
socio-economic cost of policy implementation. Additionally, our model 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
counties with state-level policies. We propose two novel algorithms for
approximating solutions to such games. The first is based on best response
dynamics (BRD), and exploits the tree structure of the game. The second
combines quadratic integer programming (QIP), which enables us to collapse the
two lowest levels of the game, with best response dynamics. Through extensive
experiments, we show that our QIP-based approach significantly outperforms the
BRD algorithm both in running time and the quality of equilibrium solutions.
Finally, we apply the QIP-based algorithm to experiments based on both
synthetic and real-world data under various parameter configurations and
analyze the resulting (approximate) equilibria to gain insight into the impact
of decentralization on overall welfare (measured as the negative sum of costs)
as well as emergent properties like free-riding and fairness in cost
distribution among policy-makers.
