With very few exceptions, recent research in fair division has mostly focused on deterministic allocations. Deviating from this trend, we study the fairness notion of interim envy-freeness (iEF) for lotteries over allocations, which serves as a sweet spot between the too stringent notion of ex-post envy-freeness and the very weak notion of ex-ante envy-freeness. iEF 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 study the extension of interim envy-freeness notion when payments to or from the agents are allowed. 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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关于临时免费羡慕分配彩票

除极少数例外，最近在公平分配方面的研究主要集中在确定性分配上。偏离这种趋势，我们研究了彩票对分配的过度嫉妒的临时概念（iEF）的公平性，这是事后嫉妒自由的过于严格概念与事前事态的非常薄弱的​​概念之间的最佳结合点。无羡慕。从确定性的无羡慕分配就是iEF（当被视为简并彩票）的意义上讲，iEF是无羡慕对随机分配的自然概括。这当然也很有意义，因为它提供了更丰富的解决方案空间，其中包括根据多个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的公平分配问题，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。尽管在匹配分配实例中计算iEF分配更具挑战性，但是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。当允许向代理人付款或从代理人付款时，我们还研究了临时性的“无嫉妒”概念的扩展。