超越功利社会福利的先知不等式

中文摘要 AI 辅助

在经典的独立同分布先知不等式问题中，单个物品被分配给按顺序到达的 $n$ 个代理人之一，其价值从已知分布中独立抽取。当代理人到达时，其价值被揭示，算法必须立即分配物品或继续。通常的目标是功利社会福利：接受者的期望价值。该目标的保证可扩展到将 $m$ 个不可分割物品分配给具有独立同分布加性价值的顺序到达代理人。然而，功利社会福利忽略了期望效用在代理人之间的分配方式。受公平分配领域丰富文献的启发，我们转而通过其广义 $p$-均值福利来评估在线规则，该福利包括 $p=1$ 时的功利福利、$p=0$ 时的纳什福利（效用的几何均值）以及 $p\to-\infty$ 时的平等福利（最小效用）。当物品数量很大时，我们证明这个多物品公平分配问题恰好被一个由代理人期望效用的 $p$-均值评估的单物品先知问题所刻画。我们刻画了这个单物品问题：每个在线规则都被一个分位数阈值规则弱帕累托支配，并且最优的平等规则使代理人的期望效用相等。我们证明对于每个 $n$，在线最优值至少是先知的平等福利的 $\Gamma\approx0.7059$ 倍；由广义均值的单调性，相同的保证对每个 $p\le1$ 成立。此外，对于平等福利，当 $n\to\infty$ 时，最优比率收敛到 $\Gamma$。因此，渐近地，优化平等福利而非功利福利相对于经典的功利比率 $0.7451$ 仅损失约四个百分点。最后，当 $m=n$ 时，对于每个 $p\le0$，最坏情况竞争比率在 $n\to\infty$ 时收敛到零，表明大物品假设是必要的。

英文摘要

In the classical i.i.d. prophet-inequality problem, a single item is allocated to one of $n$ agents who arrive sequentially, with values drawn independently from a known distribution. When an agent arrives, their value is revealed, and the algorithm must immediately allocate the item or continue. The usual objective is utilitarian welfare: the expected value of the recipient. Guarantees for this objective extend to allocating $m$ indivisible items to sequentially arriving agents with i.i.d.\ additive values. Utilitarian welfare, however, ignores how expected utility is distributed across agents. Motivated by a rich literature in fair division, we instead evaluate an online rule by its generalized $p$-mean welfare, which includes utilitarian welfare at $p=1$, Nash welfare (the geometric mean of utilities) at $p=0$, and egalitarian welfare (the minimum utility) as $p\to-\infty$. When the number of items is large, we show that this many-item fair-division problem is captured exactly by a single-item prophet problem evaluated by the $p$-mean of agents' expected utilities. We characterize this single-item problem: every online rule is weakly Pareto dominated by a quantile-threshold rule, and an optimal egalitarian rule equalizes agents' expected utilities. We prove that for every $n$, the online optimum is at least $Γ\approx0.7059$ times the prophet's egalitarian welfare; by monotonicity of generalized means, the same guarantee holds for every $p\le1$. Further, for egalitarian welfare, the optimal ratio converges to $Γ$ as $n\to\infty$. Thus, asymptotically, optimizing egalitarian rather than utilitarian welfare costs only about four percentage points relative to the classical utilitarian ratio of $0.7451$. Finally, when $m=n$, the worst-case competitive ratio converges to zero as $n\to\infty$ for every $p\le0$, showing that the large-item assumption is necessary.

发表机构

• Google Research（谷歌研究院）
• Purdue University（普渡大学）

机构由 AI 辅助整理，请以论文原文为准。