Constant-Factor EFX Exists for Chores

TL;DR

This paper establishes the existence of 4-EFX allocations, providing the first constant-factor approximation of EFX, and shows the existence of ER equilibria by carefully formulating a linear complementarity problem (LCP) that captures all ER equilibria and proves that the classic complementary pivot algorithm applied to this LCP terminates at an ER equilibrium.

Abstract

We study the problem of fair allocation of chores to agents with additive preferences. In the discrete setting, envy-freeness up to any chore (EFX) has emerged as a compelling fairness criterion. However, establishing its (non-)existence or achieving a meaningful approximation remains a major open question. The current best guarantee is the existence of $O(n^2)$-EFX allocations for $n$ agents, obtained through a sophisticated algorithm (Zhou and Wu, 2022). In this paper, we show the existence of $4$-EFX allocations, providing the first constant-factor approximation of EFX. We also investigate the existence of allocations that are both fair and efficient, using Pareto optimality (PO) as our efficiency criterion. For the special case of bivalued instances, we establish the existence of allocations that are both $3$-EFX and PO, thus improving the current best factor of $O(n)$-EFX without any efficiency guarantees. For general additive instances, the existence of allocations that are $α$-EF$k$ and PO has remained open for any constant values of $α$ and $k$, where EF$k$ denotes envy-freeness up to $k$ chores. We provide the first positive result in this direction by showing the existence of allocations that are $2$-EF$2$ and PO. Our results are obtained via a novel economic framework called earning restricted (ER) competitive equilibrium for fractional allocations, which limits agents' earnings from each chore. We show the existence of ER equilibria by formulating it as an linear complementarity problem (LCP) and proving that the classic complementary pivot algorithm on the LCP terminates at an ER equilibrium. We design algorithms that carefully round fractional ER equilibria, and perform bundle swaps and merges to meet the desired fairness and efficiency criteria. We expect that the concept of ER equilibrium will be useful in deriving further results on related problems.

Figures (3)

• Figure 1: Illustrating the difference between an unrestricted CE and an ER equilibrium of the instance from \ref{['ex:er-eq']}. The chore payments $p_j$ are indicated to the right of the chore, and chore specific earnings $q_{ij}$ are indicated above the edges between agent $i$ and chore $j$. Note that in the unrestricted CE, chore $j_1$ pays out $1$ each to agents $a_1$ and $a_2$, whereas in the ER equilibrium the earning restriction of $1$ forces agent $a_2$ to do chores $j_2$ and $j_3$ in the ER equilibrium.
• Figure 2: Illustrating an $(i,\ell)$ chore swap. In the allocation before the swap, the agent $i\in N_H$ with a single high paying chore $j_i$ envies agent $\ell$ the most. The swap transfers the entire bundle of agent $\ell$ to agent $i$, and transfers the single chore $j_i$ to agent $\ell$. Using the bounds on payments and disutilities one can argue that after the $(i,\ell)$ swap, the agents are $3$-EFX.
• Figure 3: Illustrating the re-allocation of chores in $H$.

Theorems & Definitions (140)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Example 1
• Definition 1
• Definition 2
• Proposition 5
• proof