Optimal Provision - After - Wait in Healthcare ∗ [ Working paper ] †

We investigate computational and mechanism design aspects of optimal scarce resource allocation, where the primary rationing mechanism is through waiting times. Specifically we consider the problem of allocating medical treatments to a population of patients. Each patient has demand for exactly one unit of treatment, and can choose to be treated in one of k hospitals, H1, . . . ,Hk. Different hospitals have different costs, which are fully paid by a third party —the “payer”— and do not accrue to the patients. The payer has a fixed budget B and can only cover a limited number of treatments in the more expensive hospitals. Access to over-demanded hospitals is rationed through waiting times: each hospital Hi will have waiting time wi. In equilibrium, each patient will choose his most preferred hospital given his intrinsic preferences and the waiting times. The payer thus computes the waiting times and the number of treatments authorized for each hospital, so that in equilibrium the budget constraint is satisfied and the social welfare is maximized. We show that even if the patients’ preferences are known to the payer, the task of optimizing social welfare in equilibrium subject to the budget constraint is NP-hard. We also show that, with constant number of hospitals, if the budget constraint can be relaxed from B to (1+ )B for an arbitrarily small constant , then the original optimum under budget B can be approximated very efficiently. We further study the endogenous emergence of waiting time from the dynamics between hospitals and patients. When the patients arrive uniformly along time and when they have generic types, we show that the payer does not need to explicitly enforce the optimal waiting times. Rather, each hospital’s waiting time simply changes according to the demand there, and the dynamics will always converge to the desired waiting times in finite time. We then investigate the optimization problem over a much larger class of mechanisms that contains the equilibrium ones as special cases. In the setting with two hospitals, we show that under a natural assumption on the patients’ preference profiles, optimal welfare is in fact attained by the randomized assignment mechanism, which allocates patients to the hospitals at random subject to the budget constraint, but avoids waiting times. Finally, we discuss potential policy implications of our results, as well as follow-up directions and open problems.