In fair division of indivisible goods, l-out-of-d maximin share (MMS) is the value that an agent can guarantee by partitioning the goods into d bundles and choosing the l least preferred bundles. Most existing works aim to guarantee to all agents a constant fraction of their one-out-of-n MMS. But this guarantee is sensitive to small perturbation in agents' cardinal valuations. We consider a more robust approximation notion, which depends only on the agents' ordinal rankings of bundles. We prove the existence of l-out-of-(l+1/2)n MMS allocations of goods for any integer l >= 1, and develop a polynomial-time algorithm that achieves this guarantee when l = 1.

中文翻译：

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商品的序数最大份额近似

在不可分割的商品的公平划分中，l-out-of-d maximin share (MMS) 是代理可以通过将商品分成 d 束并选择 l 个最不喜欢的束来保证的价值。大多数现有工作旨在向所有代理保证其 n 分之一 MMS 的恒定比例。但这种保证对代理人基本估值的小扰动很敏感。我们考虑了一个更稳健的近似概念，它仅取决于代理对束的序数排名。我们证明对于任何整数 l >= 1 存在 l-out-of-(l+1/2)n 商品的 MMS 分配，并开发了一个多项式时间算法，当 l = 1 时实现这一保证。