With very few exceptions, research in fair division has mostly focused on deterministic allocations. Deviating from this trend, we define and study the novel notion of interim envy-freeness (iEF) for lotteries over allocations, which aims to serve as a sweet spot between the too stringent notion of ex-post envy-freeness and the very weak notion of ex-ante envy-freeness. Our new fairness notion is a natural generalization of envy-freeness to random allocations in the sense that a deterministic envy-free allocation is iEF (when viewed as a degenerate lottery). It is also certainly meaningful as it allows for a richer solution space, which includes solutions that are provably better than envy-freeness according to several criteria. Our analysis relates iEF to other fairness notions as well, and reveals tradeoffs between iEF and efficiency. Even though several of our results apply to general fair division problems, we are particularly interested in instances with equal numbers of agents and items where allocations are perfect matchings of the items to the agents. Envy-freeness can be trivially decided and (when it can be achieved, it) implies full efficiency in this setting. Although computing iEF allocations in matching allocation instances is considerably more challenging, we show how to compute them in polynomial time, while also maximizing several efficiency objectives. Our algorithms use the ellipsoid method for linear programming and efficient solutions to a novel variant of the bipartite matching problem as a separation oracle. We also extend the interim envy-freeness notion by introducing payments to or from the agents. We present a series of results on two optimization problems, including a generalization of the classical rent division problem to random allocations using interim envy-freeness as the solution concept.

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过渡期免嫉妒：随机分配的新公平概念

除极少数例外，公平分工的研究主要集中在确定性分配上。偏离这种趋势，我们定义并研究了彩票对分配过度的“羡慕无间期”（iEF）的新颖概念，其目的是在过于严格的事后嫉妒无力概念与非常薄弱的​​概念之间找到一个甜蜜点。前嫉妒的自由。我们的新的公平性概念是将无羡慕性自然化为随机分配，这意味着确定性无羡慕性分配为iEF（当被视为简并彩票时）。这当然也很有意义，因为它提供了更丰富的解决方案空间，其中包括根据一些标准可证明比无羡慕更好的解决方案。我们的分析也将iEF与其他公平性概念相关联，并揭示了iEF与效率之间的权衡。即使我们的一些结果适用于一般公平分配问题，我们对代理和项目数量相等的实例尤其感兴趣，在这些实例中，分配是项目与代理的完美匹配。嫉妒的程度可以轻易决定，并且（在可以实现时，）意味着在这种情况下可以充分发挥效率。尽管在匹配的分配实例中计算iEF分配更具挑战性，但我们展示了如何在多项式时间内计算它们，同时还最大化了多个效率目标。我们的算法使用椭圆体方法进行线性规划，并有效地解决了二分匹配问题的一个新变体，即分离预言。我们还通过向代理人收取费用或从代理人收取费用来扩展临时免除嫉妒的概念。