We present a polynomial-time algorithm that computes an ex-ante envy-free lottery over envy-free up to one item (EF1) deterministic allocations. It has the following advantages over a recently proposed algorithm: it does not rely on the linear programming machinery including separation oracles; it is SD-efficient (both ex-ante and ex-post); and the ex-ante outcome is equivalent to the outcome returned by the well-known probabilistic serial rule. As a result, we answer a question raised by Freeman, Shah, and Vaish (2020) whether the outcome of the probabilistic serial rule can be implemented by ex-post EF1 allocations. In the light of a couple of impossibility results that we prove, our algorithm can be viewed as satisfying a maximal set of properties. Under binary utilities, our algorithm is also ex-ante group-strategyproof and ex-ante Pareto optimal. Finally, we also show that checking whether a given random allocation can be implemented by a lottery over EF1 and Pareto optimal allocations is NP-hard.

中文翻译：

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同时实现事前和事后公平

我们提出了一种多项式时间算法，该算法通过最多一个项目（EF1）确定性分配来计算事前免嫉妒彩票。与最近提出的算法相比，它具有以下优点：它不依赖于包括分离预言机在内的线性规划机制；它是 SD 高效的（事前和事后）；并且事前结果等价于众所周知的概率序列规则返回的结果。因此，我们回答了 Freeman、Shah 和 Vaish（2020）提出的问题，概率序列规则的结果是否可以通过事后 EF1 分配来实现。根据我们证明的几个不可能结果，我们的算法可以被视为满足最大的一组属性。在二进制实用程序下，我们的算法也是事前组策略证明和事前帕累托最优的。最后，我们还表明，检查给定的随机分配是否可以通过 EF1 和帕累托最优分配的抽签来实现是 NP 难的。