The existence of allocations that are fair and efficient, simultaneously, is a central inquiry in fair division literature. A prominent result in discrete fair division shows that the complementary desiderata of fairness and efficiency can be achieved together when allocating indivisible items with nonnegative values; specifically, for indivisible goods and among agents with additive valuations, there always exists an allocation that is both envy-free up to one item (EF1) and Pareto efficient (PO). While a recent breakthrough extends the EF1 and PO guarantee to indivisible chores (items with negative values), the question remains open for indivisible mixed manna, i.e., for indivisible items whose values can be positive, negative, or zero. The current work makes notable progress in resolving this central question. For indivisible mixed manna and additive valuations, we establish the existence of allocations that are PO and introspectively envy-free up to one item (IEF1). In an IEF1 allocation, each agent can eliminate its envy towards all the other agents by either adding an item or removing an item from its own bundle. The notion of IEF1 coincides with EF1 for indivisible chores, and hence, our result generalizes the aforementioned existence guarantee for chores. Our techniques can be adopted to obtain an alternative proof for the existence of EF1 and PO allocations of indivisible goods. Hence, along with the result for mixed manna, we provide a unified approach for establishing the EF1 and PO guarantee for indivisible goods and indivisible chores. We also utilize our result for indivisible items to develop a distinct proof of the noted EF and PO guarantee for divisible mixed manna. Our work highlights an interesting application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division and develops multiple, novel structural insights and algorithmic ideas.

翻译：公平与效率兼备的分配方案是否存在，是公平分配研究领域的核心问题。离散公平分配领域的一个重要结果表明，在分配具有非负价值的不可分割物品时，公平与效率这两个互补的诉求可以同时实现；具体而言，对于不可分割物品且代理人具有可加估值的情形，总存在一个分配方案，既是"除一物品外无嫉妒"（EF1）的，又是帕累托有效（PO）的。尽管近期的一项突破性研究将EF1与PO的保证扩展到了不可分割杂务（具有负价值的物品），但对于不可分割混合资源（即价值可为正、负或零的不可分割物品），该问题仍然悬而未决。本研究在这一核心问题的解决上取得了显著进展。针对不可分割混合资源及可加估值，我们证明了存在同时满足PO和"内省性除一物品外无嫉妒"（IEF1）的分配方案。在一个IEF1分配中，每个代理人可以通过向自己的资源束添加一个物品或从其资源束中移除一个物品，来消除对所有其他代理人的嫉妒。IEF1的概念对于不可分割杂务与EF1一致，因此，我们的结果推广了前述关于杂务的存在性保证。我们的技术可被采纳，从而为不可分割物品的EF1与PO分配的存在性提供另一种证明。因此，结合混合资源的结果，我们为不可分割物品和不可分割杂务的EF1与PO保证提供了一个统一的方法。我们还利用关于不可分割物品的结果，为可分割混合资源中著名的EF与PO保证开发了一个独特的证明。我们的工作凸显了Knaster-Kuratowski-Mazurkiewicz（KKM）定理在离散公平分配中的一个有趣应用，并发展出多种新颖的结构性见解和算法思想。