Fair Assignment of Indivisible Chores to Asymmetric Agents

TL;DR

This work addresses fair allocation of indivisible chores to agents with asymmetric entitlements, using the weighted maximin share (WMMS) as the fairness benchmark. It introduces a constructive Layered Moving Knife Algorithm that yields a constant-WMMS assignment, proven to be $20$-WMMS, by reducing to a sorted-chores setting and employing an entitlement-divisible relaxation to facilitate analysis. The paper also develops a chore-oblivious analysis that improves small-$n$ bounds (notably $n=3$ and $n=4$) and reports empirical improvements up to $n=10$, indicating practical gains beyond the theoretical worst-case guarantee. Overall, the results significantly advance constant-factor WMMS guarantees for asymmetric chore division, with potential refinements via tighter analysis of the moving-knife procedure.

Abstract

We consider the problem of assigning indivisible chores to agents with different entitlements in the maximin share value (\MMS) context. While constant-\MMS\ allocations/assignments are guaranteed to exist for both goods and chores in the symmetric setting, the situation becomes much more complex when agents have different entitlements. For the allocation of indivisible goods, it has been proven that an $n$-\WMMS\ (weighted \MMS) guarantee is the best one can hope for. For indivisible chores, however, it was recently discovered that an $O(\log n)$-\WMMS\ assignment is guaranteed to exist. In this work, we improve this upper bound to a constant-\WMMS\ guarantee.\footnote{We prove the existence of a 20-\WMMS\ assignment, but we did not attempt to optimize the constant factor. We believe our methods already yield a slightly better bound with a tighter analysis.}

Figures (5)

• Figure 1: Comparison of chore-oblivious bounds and $\log n+1$ values
• Figure 2: The initial state of the layered moving knife algorithm is illustrated in this figure.
• Figure 3: The layered moving knife algorithm is explained in this figure. $a_x$ is the lowest index agent who is in $D$.
• Figure 4: A visualization of the 2D simplex defined by $\textsf{w}_1 + \textsf{w}_2 + \textsf{w}_3 = 1$, where each point represents a triple of entitlements in barycentric coordinates. Triangle vertices correspond to full entitlement for one agent (i.e., $(1,0,0)$, $(0,1,0)$, or $(0,0,1)$), and interior points represent proportional combinations. The triangle is partitioned by which upper bound on $\mathcal{F}(\langle \textsf{w}_1, \textsf{w}_2, \textsf{w}_3 \rangle)$ dominates: green for $\frac{(\textsf{w}_1 + \textsf{w}_3)k}{\textsf{w}_3}$ and blue for $\frac{15\textsf{w}_3}{13\textsf{w}_1}$. On the right is a heatmap of the upper bound across the simplex, with lighter colors indicating larger values.
• Figure 5: Comparison of chore-oblivious bounds.

Theorems & Definitions (19)

• Lemma 4.1: barman2017approximation, restated for our setting
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Theorem 4.5
• proof