We consider a dynamic mechanism design problem where an auctioneer sells an indivisible good to groups of buyers in every round, for a total of $T$ rounds. The auctioneer aims to maximize their discounted overall revenue while adhering to a fairness constraint that guarantees a minimum average allocation for each group. We begin by studying the static case ($T=1$) and establish that the optimal mechanism involves two types of subsidization: one that increases the overall probability of allocation to all buyers, and another that favors the groups which otherwise have a lower probability of winning the item. We then extend our results to the dynamic case by characterizing a set of recursive functions that determine the optimal allocation and payments in each round. Notably, our results establish that in the dynamic case, the seller, on the one hand, commits to a participation bonus to incentivize truth-telling, and on the other hand, charges an entry fee for every round. Moreover, the optimal allocation once more involves subsidization, which its extent depends on the difference in future utilities for both the seller and buyers when allocating the item to one group versus the others. Finally, we present an approximation scheme to solve the recursive equations and determine an approximately optimal and fair allocation efficiently.

翻译：我们研究一个动态机制设计问题，其中拍卖者在每一轮向多个买家群体出售一个不可分割的物品，总共进行$T$轮。拍卖者的目标是在遵守公平约束的前提下最大化其折现总收益，该公平约束保证每个群体获得最低的平均分配概率。我们首先研究静态情形（$T=1$），并证明最优机制包含两种类型的补贴：一种提高所有买家的总体分配概率，另一种则倾向于那些原本获胜概率较低的群体。随后，我们将结果推广到动态情形，通过刻画一组递归函数来确定每一轮的最优分配与支付。值得注意的是，我们的结果表明，在动态情形中，卖方一方面承诺提供参与奖励以激励真实报价，另一方面则对每一轮收取入场费。此外，最优分配再次涉及补贴，其程度取决于将物品分配给某一群体相对于其他群体时，卖方与买家未来效用的差异。最后，我们提出一种近似方案来求解递归方程，从而高效地确定近似最优且公平的分配。