Hospitals are typically optimized to operate near capacity, and there are serious concerns that our healthcare system is not prepared for the next pandemic. Stockpiles of different supplies, e.g., personal protective equipments (PPE) and medical equipment, need to be maintained in order to be able to respond to any future pandemics. Large outbreaks occur with a low probability, and such stockpiles require big investments. Further, hospitals often have mutual sharing agreements, which makes the problem of stockpiling decisions a natural game-theoretical problem. In this paper, we formalize hospital stockpiling as a game-theoretical problem. We use the notion of pairwise Nash stability as a solution concept for this problem, and characterize its structure. We show that stable strategies can lead to high unsatisfied demands in some scenarios, and stockpiles might not be maintained at all nodes. We also show that stable strategies and the social optimum can be computed efficiently.