Approximate-EFX Allocations with Ordinal and Limited Cardinal Information

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.

💡 Research Summary

This paper investigates the problem of computing approximately envy‑free up to any good (α‑EFX) allocations in the classic discrete fair division setting where n agents have additive valuations over m indivisible goods, but the algorithm does not have full access to the cardinal values. Instead, it is given each agent’s ordinal ranking of the goods and is allowed to ask a limited number of value‑queries (each query returns the exact value of a specific good for a specific agent). The authors ask two central questions: (i) what is the best achievable α‑EFX guarantee using only ordinal information, and (ii) how many queries per agent are necessary and sufficient to obtain a constant‑α EFX approximation.

Ordinal‑only algorithms.
In Section 2 the authors prove a tight impossibility result: any algorithm that relies solely on the agents’ rankings can guarantee at most α = 1/(m − n). This bound is essentially tight and shows that without any cardinal data the approximation deteriorates dramatically as the number of goods exceeds the number of agents.

Query‑enhanced ordinal algorithms.
The bulk of the paper (Section 3) studies algorithms that may ask a small number of value queries. The main qualitative theorem (Informal Theorem 2) states that a polylogarithmic number of queries in m per agent is both necessary and sufficient to achieve a constant‑α EFX allocation. Concretely, the authors present an algorithm that (a) queries each agent for the exact values of their top n − 1 goods, (b) performs O(log m) binary‑search rounds to bucket the remaining goods into value intervals, and (c) constructs a “virtual” valuation function that approximates the true one. The virtual valuations are then fed to any existing constant‑EFX algorithm (which assumes full information) as a black box. The total query budget per agent is O(n + log² m) = O(log m) when n ≤ log² m. A matching lower bound shows that o(log m·log log m) queries cannot suffice for any constant α, establishing that the upper bound is essentially optimal up to a logarithmic factor.

Special case 1: constant number of agents.
When n is a constant, Section 4 gives a refined analysis. With k queries per agent the algorithm can guarantee an Ω(√k·(m − 1)/(2k − 1))‑EFX approximation. For constant k this yields a constant‑α guarantee, and with O(log m) queries the approximation becomes Ω(1/ log m), which matches the general lower bound up to constant factors.

Special case 2: bivalued instances.
Section 5 focuses on bivalued instances, where each agent’s valuation takes only two possible values (which may differ across agents). Leveraging the structural simplicity, the authors design an algorithm that needs only O(log n) queries per agent to achieve a constant‑α EFX allocation (Theorem 5.1). The method combines the Round‑Robin and Match & Freeze procedures from prior work with a small amount of value‑query information to distinguish high‑value from low‑value goods.

Related work and contribution.
The paper situates itself within a rich literature on EFX, α‑EFX approximations, and distortion in social choice. While previous works have assumed full cardinal knowledge, this is the first to systematically explore the trade‑off between query complexity and α‑EFX quality. It also improves upon prior query‑based fair‑division results (e.g., Oh et al. 2021, Bu et al. 2024) by achieving stronger fairness guarantees (EFX rather than EF1) and by providing tight lower bounds.

Conclusion.
The authors demonstrate that ordinal information alone is insufficient for meaningful EFX guarantees, but a modest number of value queries—polylogarithmic in the number of goods—suffices to obtain constant‑α EFX allocations. Moreover, for settings with few agents or bivalued valuations, the query requirement drops dramatically to O(log n). These findings bridge the gap between theoretical fairness notions and practical elicitation constraints, offering concrete guidance for designing fair allocation mechanisms that respect agents’ limited willingness to disclose precise cardinal preferences.