We consider allocating indivisible goods with provable fairness guarantees that are satisfied regardless of which bundle of items each agent receives. Symmetrical allocations of this type are known to exist for divisible resources, such as consensus splitting of a cake into parts, each having equal value for all agents, ensuring that in any allocation of the cake slices, no agent would envy another. For indivisible goods, one analogous concept relaxes envy freeness to guarantee the existence of an allocation in which any bundle is worth as much as any other, up to the value of a bounded number of items from the other bundle. Previous work has studied the number of items that need to be removed. In this paper, we improve upon these bounds for the specific setting in which the number of bundles equals the number of agents. Concretely, we develop the theory of symmetrically envy free up to one good, or symEF1, allocations. We prove that a symEF1 allocation exists if the vertices of a related graph can be partitioned (colored) into as many independent sets as there are agents. This sufficient condition always holds for two agents, and for agents that have identical, disjoint, or binary valuations. We further prove conditions under which exponentially-many distinct symEF1 allocations exist. Finally, we perform computational experiments to study the incidence of symEF1 allocations as a function of the number of agents and items when valuations are drawn uniformly at random.

翻译：我们研究在保证公平性的前提下分配不可分割物品的问题，这种公平性无论每个智能体获得哪一组物品都能得到满足。对于可分资源（如将蛋糕共识分割为若干部分，每部分对所有智能体具有相同价值），已知存在此类对称分配方案，从而确保在任何蛋糕切片分配中都不会产生嫉妒。对于不可分割物品，一个类似的概念通过放宽无嫉妒条件，保证存在一种分配方案，使得任意物品组相对于其他物品组的价值差异不超过另一物品组中有界数量的物品价值。先前的研究已探讨了需要移除的物品数量问题。本文针对捆绑数量等于智能体数量的特定场景改进了这些界。具体而言，我们建立了对称性"至多一件物品无嫉妒"（symEF1）分配的理论体系。我们证明：若相关图的顶点可被划分（着色）为与智能体数量相同的独立集，则存在symEF1分配。该充分条件对于两个智能体始终成立，对于具有相同估值、互斥估值或二元估值的智能体也同样成立。我们进一步证明了存在指数级数量不同symEF1分配的条件。最后，我们通过计算实验研究了当估值服从均匀随机分布时，symEF1分配的出现频率与智能体数量和物品数量的函数关系。