Online Fair Division Against an Oblivious Adversary

2026-09-23

AI Summary

研究在线分配不可分割物品给n个参与者的问题，针对无意识对手，提出一种算法提高随机分配的PROP1性能至Ω(1/ log log(n/δ))。

Abstract

We study the online allocation of indivisible goods among $n$ agents, where each good must be allocated immediately and irrevocably upon arrival. Against an adaptive adversary, Neoh and Teh [2026] proved that no algorithm can guarantee a positive approximation to proportionality up to one good (PROP1) that is independent of the number of goods, and the same holds for proportionality up to $k$ goods (PROP$k$) for any fixed $k$. We instead consider an oblivious adversary, which fixes the input in advance. Choo et al. [2026] showed that the uniformly random allocation returns a $Θ(1/\log(n/δ))$-PROP1 allocation with probability at least $1-δ$. We improve this to $Ω(1/\log\log(n/δ))$; our algorithm does not take $δ$ as input, so the same algorithm achieves this bound for every $δ\in(0,1)$. Moreover, with the same probability, a variant of our algorithm gives every agent almost her bundle, and even without adding any good when no single good is too valuable relative to this share. In contrast, for envy-freeness up to one good (EF1), we show that, for every $α\in(0,1]$, every randomized algorithm has an input on which its probability of returning an $α$-EF1 allocation is at most $e^{-Ω(n)}$. For envy-freeness up to any good (EFX), this probability is at most $1/n!$ with only $n+1$ goods, a bound that is optimal within a factor of $(n+1)/2$. For the maximin share (MMS), this probability is at most $5/6$, however small $α$ is. Allowing more removals gives a positive envy-freeness guarantee: allocating each good to a uniformly random agent among those with positive values achieves, with high probability, an approximation factor arbitrarily close to one for envy-freeness up to logarithmically many goods, and logarithmically many goods are necessary for this rule.

Problem

• online allocation
• indivisible goods
• fair division
• oblivious adversary
• proportionality

Innovation

• Oblivious Adversary
• Proportionality up to one good (PROP1)
• Randomized Allocation
• Envy-freeness
• Maximin Share (MMS)