Online Fair Division Meets Reordering Buffers

2026-07-01

AI Summary

This study addresses fair allocation of indivisible goods in an online setting where items arrive sequentially and must be assigned immediately, with the goal of achieving envy-freeness (EF) or its relaxation EF1. The work introduces, for the first time, a reordering buffer to enable dynamic fairness guarantees under personalized $k$-value instances and general additive valuations. It proposes a buffer-based mechanism whose size scales linearly with both $k$ and the number of agents, ensuring EF1 at every step and EF at almost all time steps. For settings where each agent’s valuation takes at most $k$ distinct values, the authors combine combinatorial constructions of envy-free matchings with buffer scheduling strategies. The paper also establishes impossibility results, showing that smaller buffers cannot achieve comparable guarantees, and reveals that performance under general additive valuations depends critically on the maximum ratio between values of the same sign.

Abstract

We study the online fair division of indivisible mixed manna among agents with additive valuation functions. Under the standard online model, at each time step an indivisible item arrives; each agent may assign it a positive, negative, or zero value, and it must be irrevocably allocated, before the arrival of the next item. At the same time, we also wish to maintain some fairness guarantee, and in this work we focus on envy-freeness (EF) and one of its most prominent relaxations, envy-freeness up to one item (EF1). Given the strong negative and the scarce positive results for this problem without additional assumptions, we augment our algorithms with buffers that can store and rearrange a limited number of items. This setting interpolates naturally between the fully online case (no buffer) and the fully offline case (a buffer large enough to hold all items). We show that algorithms equipped with reasonably sized buffers can achieve strong guarantees for personalized $k$-value instances, i.e., instances in which each agent assigns at most $k$ distinct values to items. In particular, we construct allocations that are EF1 at every time step and EF at most time steps, using a buffer of size linear in $k$ and in the number of agents. Our approach relies on novel combinatorial arguments and on constructing a sequence of envy-free matchings that allocates most items. Finally, we extend our results to general additive valuation functions, with a dependence on the largest per-agent ratio between two values of the same sign, and we also identify limitations of our approach via impossibility results on the use of buffers with smaller size.

Problem

• online fair division
• indivisible mixed manna
• envy-freeness
• EF1
• reordering buffers

Innovation

• online fair division
• reordering buffers
• envy-freeness
• EF1
• additive valuations