Provision-After-Wait with Common Preferences

Abstract

In this article, we study the Provision-after-Wait problem in healthcare (Braverman, Chen, and Kannan, 2016). In this setting, patients seek a medical procedure that can be performed by different hospitals of different costs. Each patient has a value for each hospital and a budget-constrained government/planner pays for the expenses of the patients. Patients are free to choose hospitals, but the planner controls how much money each hospital gets paid and thus how many patients each hospital can serve (in one budget period, say, one month or one year). Waiting times are used in order to balance the patients’ demand, and the planner’s goal is to find a stable assignment that maximizes the social welfare while keeping the expenses within the budget. It has been shown that the optimal stable assignment is NP-hard to compute, and, beyond this, little is known about the complexity of the Provision-after-Wait problem. We start by showing that this problem is in fact strongly NP-hard, and thus does not have a Fully Polynomial-Time Approximation Scheme (FPTAS). We then focus on the common preference setting, where the patients have the same ranking over the hospitals. Even when the patients perceive the hospitals’ values to them based on the same quality measurement—referred to as proportional preferences , which has been widely studied in resource allocation—the problem is still NP-hard. However, in a more general setting where the patients are ordered according to the differences of their values between consecutive hospitals, we construct an FPTAS for it. To develop our results, we characterize the structure of optimal stable assignments and their social welfare, and we consider a new combinatorial optimization problem that may be of independent interest, the ordered Knapsack problem. Optimal stable assignments are deterministic and ex-post individually rational for patients. The downside is that waiting times are dead-loss to patients and may burn a lot of social welfare. If randomness is allowed, then the planner can use lotteries as a rationing tool: The hope is that they reduce the patients’ waiting times, although they are interim individually rational instead of ex-post. Previous study has only considered lotteries for two hospitals. In our setting, for arbitrary number of hospitals, we characterize the structure of the optimal lottery scheme and conditions under which using lotteries generates better (expected) social welfare than using waiting times.