Resource Allocation with Budgets: Optimal Stable Allocations and Optimal Lotteries∗

We introduce the resource allocation problem where a planner needs to purchase differ-ent resources from providers of different qualities and costs, and the planner allocates the re-sources/providers to consumers with different preferences. The planner has a budget that limitshow much he can spend on the resources. He wants to maximize the social welfare generatedfrom the consumers, while keeping his total expenditure for purchasing the resources within hisbudget. Previous studies have either focused on the resource acquisition part, with one buyerand many strategic sellers, or the resource allocation part, with one seller and many strategicbuyers. This is for the first time both providers and consumers are included in the picture.The consumers do not pay for the resources and will act to maximize their individual utilities.Thus the planner must use proper rationing tools to make sure that they will stick to theproviders allocated to them. Two widely existing rationing tools are studied in this paper:waiting times and lotteries (and the combination of them).We characterize (sometimes partially) the structures of optimal allocation schemes whendifferent rationing tools are used, and we identify conditions under which lotteries can do betterand under which waiting times can do better. We also settle the computation complexity forcomputing/approximating them. For resource allocation with waiting times, we show that theoptimal solution is NP-hard to find, and we construct an FPTAS for it. For resource allocationwith lotteries, we show that for a large class of the problem the optimal solution has a surprisinglysimple structure, and can be solved by a linear program.Following our results, neither waiting time nor lottery is absolutely better than the otherin terms of generating social welfare. A planner should choose an appropriate tool based onthe conditions that we identify. Our results then let the planner computes/approximates thecorresponding optimal allocations efficiently. Indeed, our results are the first systematic studyof both rationing tools when resource acquisition and resource allocation occur together (whichis often the case in real life), and we provide useful approaches for future study on this moregeneral and realistic model.