Approximate-EFX Allocations with Ordinal and Limited Cardinal Information

TL;DR

This work studies discrete fair division of $m$ goods among $n$ agents with additive valuations under limited information, focusing on the $ ext{$alpha$-EFX}$ objective. It develops a spectrum from purely ordinal algorithms, which cannot beat $alpha=1/(m-n)$, to query-enhanced methods like the $ extrho$-EFX-Virtual$ algorithm that achieve constant-$alpha$ with polylogarithmic queries per agent, via constructing virtual valuations learned from limited queries. The paper also delivers specialized results for constant $n$ and for bivalued valuations, notably a $1/2$-EFX guarantee with $O( ext{log } n)$ queries (and $2$ queries per agent achieving $1/n$-EFX) using Partition-and-RoundRobin and Match&Freeze-based schemes. Overall, it maps the tradeoffs between information needed and fairness achievable, providing near-optimal bounds and outlining directions for broader valuation classes and fairness notions.

Abstract

We study a discrete fair division problem where $n$ agents have additive valuation functions over a set of $m$ goods. We focus on the well-known $α$-EFX fairness criterion, according to which the envy of an agent for another agent is bounded multiplicatively by $α$, after the removal of any good from the envied agent's bundle. The vast majority of the literature has studied $α$-EFX allocations under the assumption that full knowledge of the valuation functions of the agents is available. Motivated by the established literature on the distortion in social choice, we instead consider $α$-EFX algorithms that operate under limited information on these functions. In particular, we assume that the algorithm has access to the ordinal preference rankings, and is allowed to make a small number of queries to obtain further access to the underlying values of the agents for the goods. We show (near-optimal) tradeoffs between the values of $α$ and the number of queries required to achieve those, with a particular focus on constant EFX approximations. We also consider two interesting special cases, namely instances with a constant number of agents, or with two possible values, and provide improved positive results.

Theorems & Definitions (24)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Theorem 3.3
• proof