随机效用下的渐近最大最小公平分配

中文摘要 AI 辅助

我们研究了在独立同分布随机效用下，不可分割商品的最大最小公平分配的渐近行为。对于$N$个智能体和$K$个商品，其效用${\mU_{i,j}}$独立地取自共同分布$F$，我们通过分布分位数推导了在平衡情形$K=N$（以及$K=LN$扩展情形）下最大最小值的渐近刻画。随后，我们通过将由此产生的总福利与最优总福利进行比较，研究了最大最小公平性的效率影响。对于尾部足够轻的分布，我们证明了随着市场规模的增大，相对效率损失收敛于零，这意味着对于一大类分布，在大型随机实例中最大最小公平性带来的福利损失可忽略不计。

英文摘要

We investigate the asymptotic behavior of max-min fair allocations for indivisible goods under i.i.d. random utilities. For $N$ agents and $K$ goods with utilities ${\mU_{i,j}}$ drawn independently from a common distribution $F$, we derive asymptotic characterizations of the max-min value in the balanced case $K=N$ (and in $K=LN$ extensions) via distributional quantiles. We then study the efficiency impact of max-min fairness by comparing the resulting total welfare with the optimal sum welfare. For distributions with sufficiently light tails, we prove that the relative efficiency loss converges to zero as the market grows, implying that max-min fairness incurs negligible welfare loss in large random instances for a broad class of distributions.

发表机构

• Faculty of Engineering, Bar-Ilan University（巴伊兰大学工程学院）

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