再探公平分配中的重构问题

中文摘要 AI 辅助

我们重新探讨不可分割物品公平分配中的重构问题，其目标是通过一系列交换将一个公平分配转换为另一个公平分配，并在每一步都保持公平性。我们关注的是直至k件物品的无嫉妒性（EFk）层级。我们证明，对于任意固定的k，具有相同大小向量的两个EF1分配不一定存在中间分配满足EFk的重构路径。即使这两个分配来自标准的EF1方法（如嫉妒环消除算法或最大纳什福利解），这种不可能性依然存在。相反，我们证明由递归平衡选取序列（包括轮转法）产生的具有相同大小向量的分配，总是通过一条保持EF2的路径相连。我们还证明，对于任意固定数量的智能体，判断EF1重构路径是否存在是NP难的。此外，我们通过研究一个也允许转移的更宽松模型来补充这些基于交换的结果，并建立了额外的连通性保证。

英文摘要

We revisit reconfiguration in the fair allocation of indivisible goods, where the goal is to transform one fair allocation into another through a sequence of exchanges while preserving fairness at every step. Our focus is on the hierarchy of envy-freeness up to $k$ goods (EF$k$). We show that for any fixed $k$, two EF1 allocations with the same size vector need not admit a reconfiguration path whose intermediate allocations satisfy EF$k$. This impossibility persists even when the two allocations arise from standard EF1 approaches: the envy cycle elimination algorithm or the maximum Nash welfare solution. In contrast, we prove that allocations with the same size vector produced by recursively balanced picking sequences, including round-robin, are always connected via a path that maintains EF2. We also show that deciding whether an EF1 reconfiguration path exists is NP-hard for any fixed number of agents. Furthermore, we complement these exchange-based results by studying a more permissive model that also allows transfers, establishing additional connectivity guarantees.

发表机构

• Technical University of Munich（慕尼黑工业大学）
• National University of Singapore（新加坡国立大学）

机构由 AI 辅助整理，请以论文原文为准。