Online MMS Allocation for Chores

TL;DR

The paper resolves a central question in online fair division of chores by showing a tight $n-\varepsilon$-MMS impossibility for any fixed $n$ and $\varepsilon>0$, while simultaneously presenting a practical online algorithm that achieves $\min\{n, O(k), O(\log D)\}$-MMS, where $k$ is the maximum number of disutility types per agent and $D$ the max per-agent ratio of disutilities. The proposed method builds on a generalized round-robin via per-type pressure and a greedy, type-aware allocation rule, augmented by a value-rounding step and a novel Stacking Game discrepancy framework that reduces the online-MMS analysis to a structured adversarial game. The paper also proves a tight bound in the personalized bi-valued case, attaining $(2+\sqrt{3})$-MMS, and discusses how the results extend to broader settings via the discrepancy machinery. These contributions clarify the limits of online MMS allocations and offer practical guarantees under realistic conditions (small $k$ or small $D$), with a rigorous connection between online allocation and discrepancy minimization through the Stacking Game.

Abstract

We study the problem of fair division of indivisible chores among $n$ agents in an online setting, where items arrive sequentially and must be allocated irrevocably upon arrival. The goal is to produce an $α$-MMS allocation at the end. Several recent works have investigated this model, but have only succeeded in obtaining non-trivial algorithms under restrictive assumptions, such as the two-agent bi-valued special case (Wang and Wei, 2025), or by assuming knowledge of the total disutility of each agent (Zhou, Bai, and Wu, 2023). For the general case, the trivial $n$-MMS guarantee remains the best known, while the strongest lower bound is still only $2$. We close this gap on the negative side by proving that for any fixed $n$ and $\varepsilon$, no algorithm can guarantee an $(n - \varepsilon)$-MMS allocation. Notably, this lower bound holds precisely for every $n$, without hiding constants in big-$O$ notation, thereby exactly matching the trivial upper bound. Despite this strong impossibility result, we also present positive results. We provide an online algorithm that applies in the general case, guaranteeing a $\min\{n, O(k), O(\log D)\}$-MMS allocation, where $k$ is the maximum number of distinct disutilities across all agents and $D$ is the maximum ratio between the largest and smallest disutilities for any agent. This bound is reasonable across a broad range of scenarios and, for example, implies that we can achieve an $O(1)$-MMS allocation whenever $k$ is constant. Moreover, to optimize the constant in the important personalized bi-valued case, we show that if each agent has at most two distinct disutilities, our algorithm guarantees a $(2 + \sqrt{3}) \approx 3.7$-MMS allocation.

Figures (5)

• Figure 1: An example operation of the stacking game with $k=1$.
• Figure 2: An example of mapping from pressures to subintervals when $n=3$ and $k=2$.
• Figure 3: An example showing $\tilde{F}^{t+1}(y) \ge F^{t+1}(y)$ with contiguous interval $\tilde{A}\cup\tilde{B}$.
• Figure 4: An example of $(a, b, A, B)$ and $f^t$, where $x$ is to the left of $B$.
• Figure 5: An example for the upgrading interval and the downgrading interval.

Theorems & Definitions (31)

• Definition 1: Minmax share (MMS)
• Definition 2: MMS for type-$\u$ items
• Lemma 1
• proof
• Theorem 1
• Definition 3: Stacking game
• Example 2: Running example for Stacking game
• Lemma 2
• proof
• Proposition 1