Optimal Provision-After-Wait in Healthcare

Abstract

We investigate computational and mechanism design aspects of allocating medical treatments at hospitals of different costs to patients who each value these hospitals differently. The payer wants to ensure that the total cost of all treatments is at most the budget, B. Access to overdemanded hospitals is rationed through waiting times. We first show that optimizing social welfare in equilibrium is NP-hard. But if the number of hospitals is small and the budget can be relaxed to (1 + ɛ)B for arbitrarily small ɛ, the optimum under budget B can be achieved efficiently. Next, we show waiting times emerge endogenously from the dynamics between hospitals and patients and the payer doesn’t have to explicitly enforce them; all it needs to do is enforce the amount of money paid to each hospital, and the dynamics will converge to the desired waiting times in finite time. Going beyond equilibrium solutions, we investigate the optimization problem over a much larger class of mechanisms. With two hospitals and concave preference profiles of the patients, optimal welfare is actually attained by the randomized assignment, which allocates patients at random and avoids waiting times. Finally, we discuss potential policy implications of our results, followup directions, and open problems.