This work is motivated by a common urban renewal process called Reconstruct and Divide. It involves the demolition of old buildings and the construction of new ones. Original homeowners are compensated with upgraded apartments, while surplus units are sold for profit, so theoretically it is a win-win project for all parties involved. However, many Reconstruct and Divide projects are withheld or delayed due to disagreements over the assignment of new apartments, claiming they are not fair. The goal of this research is to develop algorithms for envy-free assignment of the new apartments, possibly using monetary payments to reduce envy. In contrast to previous works on envy-free assignment, in our setting the envy depends also on the value of the old apartments, as people with more valuable old apartments expect to get more valuable new apartments. This presents two challenges. First, in some cases, no assignment and payment-vector satisfy the common fairness notions of envy-freeness and proportionality. Hence, we focus on minimizing the envy and the disproportionality (the distance between an agent's value and their proportional share). We present a strongly polynomial-time algorithm that, for a given assignment, finds a payment vector that minimizes the maximum pairwise-envy. We also present a strongly polynomial-time algorithm that computes an assignment and payment-vector that together minimize the maximum disproportionality. Second, directly asking the agents for their subjective valuations for their old apartments is infeasible, as it is a dominant strategy for them to report very high values for their old apartments. We introduce a novel method to elicit agents' valuations indirectly. Using this method, we identify conditions under which our Minimum Disproportionality algorithm is risk-averse truthful.

翻译：本研究受一种常见的城市更新过程——“重建与分割”所启发。该过程涉及拆除旧建筑并建造新建筑。原业主可获得升级后的公寓作为补偿，而剩余单元则出售以获取利润，因此理论上这是一个多方共赢的项目。然而，许多“重建与分割”项目因新公寓分配方案存在争议而被搁置或延迟，业主认为分配不公。本研究旨在开发无嫉妒的新公寓分配算法，可能通过货币支付来减少嫉妒。与以往关于无嫉妒分配的研究不同，在我们的设定中，嫉妒还取决于旧公寓的价值，因为拥有更高价值旧公寓的业主期望获得更有价值的新公寓。这带来了两个挑战。首先，在某些情况下，不存在任何分配方案和支付向量能够同时满足无嫉妒性和比例性这两种常见的公平性概念。因此，我们专注于最小化嫉妒和比例失调（即个体价值与其比例份额之间的距离）。我们提出了一种强多项式时间算法，对于给定的分配方案，该算法能找到最小化最大成对嫉妒的支付向量。我们还提出了一种强多项式时间算法，用于计算能够共同最小化最大比例失调的分配方案与支付向量。其次，直接询问个体对其旧公寓的主观估值是不可行的，因为对他们而言，报告极高的旧公寓价值是占优策略。我们引入了一种间接获取个体估值的新方法。利用该方法，我们确定了在何种条件下我们的最小比例失调算法具有风险规避真实性。