Abstract: In this talk we will discuss 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. Different hospitals have different costs, which are fully paid by a third party ---the “payer”--- and do not accrue to the patients. Access to over-demanded hospitals is rationed through waiting times: each hospital H_i will have waiting time w_i. In equilibrium, each patient will choose his most preferred hospital given his intrinsic preferences and the waiting times. Assuming the payer has a fixed budget; it needs to decide how many treatments of each type to procure to optimize the welfare in equilibrium.
We show that the equilibrium waiting times will emerge endogenously. We then 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+\epsilon)B for an arbitrarily small constant \epsilon, then the original optimum under budget B can be approximated very efficiently.
Going beyond equilibrium solutions we investigate the optimization problem over a much larger class of mechanisms containing 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 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.