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In practice, both comprehensive hospital (e.g. AAA hospital) and primary hospital (e.g. community hospital) can exist in healthcare system, where the comprehensive hospital can provide a guaranteed service, but the waiting time of patients is relatively long. By contrast, the primary hospital is less congested, but the patients cannot be treated if the illness is found to be severe. Then the trade-off between efficiency (primary hospital) and quality (comprehensive hospital) should be considered. In this paper, we consider a resource allocation problem in a public service system with multi-type service providers and patients. To capture the interactions between the multi-type hospitals, a queueing-game-theoretical model is established. And we obtain the following results. First, the socially optimal reimbursement policy is obtained, and the sensitivities of parameters are examined, which indicate that, somewhat interestingly, both the optimal budget to primary hospital and the maximal social welfare are non-monotone in the effectiveness parameter or the joining probability of patients. Second, by comparing the socially optimal strategy with individually equilibrium strategy, we find that individual behaviour of patients does not necessarily lead to systems more congested than what is socially desirable. Third, we demonstrate the robustness of our model by extending it to a three-level system and a system with multiple parallel primary hospitals.

Efficiency-quality trade-off in allocating resource to public healthcare systems