Single-or-Sample: Online Fair Allocation for Combinatorial Agents

2026-09-26

AI Summary

This study addresses the challenge of fairly allocating items to combinatorial agents in an online setting, specifically aiming to maximize the maximin share (MMS) without prior knowledge of agent valuations. To this end, it proposes the Single-or-Sample randomized algorithm, which integrates greedy submodular maximization, single-item reduction, and adaptive concentration bound analysis to ensure robustness against adversarial inputs. The work achieves constant-factor MMS approximations under both additive and submodular valuations. Furthermore, it reveals a fundamental trade-off boundary between approximation ratios and success probabilities, while proving the tightness of this trade-off for XOS valuations and providing matching algorithms.

Abstract

We study the problem of fairly allocating $m$ indivisible goods among $n$ agents who arrive online, under the notion of maximin share (MMS) fairness. Fair allocation with online arrivals is notoriously challenging: prior work achieves constant-factor MMS guarantees only when agents'preferences belong to a set of valuation functions known in advance, while no guarantees were known without such prior information. We develop a new randomized online algorithm for additive and submodular valuations, that we call Single-or-Sample, and that achieves a constant-factor approximation to MMS simultaneously for all agents, with constant probability. The algorithm requires no prior knowledge about the agents'valuations, and works against adversarial (oblivious) inputs. We further establish a fundamental tradeoff between approximation and success probability. Specifically, for any $c \ge 1$, no online algorithm can guarantee a $1/c$-approximation to MMS with probability exceeding $1 - 1/c^2$, even for binary additive valuations. This rules out constant MMS with probability asymptotically closer to $1$ than a constant. For XOS, the lower bound is much stronger: no algorithm can achieve even $1/\log\log n$-MMS to all agents with a constant probability. We complement this lower bound with an algorithm for the regime $c\in \Omega(\log n)$, namely $1/c$-MMS to all agents with probability $(1-O(1/c))$, and show that this tradeoff is tight for XOS. Our constant-factor algorithm introduces several new ideas, combining greedy submodular maximization with randomized allocation and single-item reduction. A key technical ingredient is a new approach for analyzing iterative sampling without replacement. We develop concentration bounds that apply to a broad class of adaptive processes with complex dependencies across elements and rounds, which may be of independent interest.

Problem

• Online Fair Allocation
• Indivisible Goods
• Maximin Share
• Combinatorial Agents

Innovation

• Online Fair Allocation
• Maximin Share (MMS)
• Randomized Algorithm
• Submodular Maximization
• Concentration Bounds