Understanding EFX Allocations: Counting and Variants
Tzeh Yuan Neoh, Nicholas Teh
TL;DR
This work investigates the minimum number of EFX allocations and the complexity of counting them, revealing $\#P$-hardness in general while deriving tight bounds in restricted regimes where the number of goods $m$ is close to the number of agents $n$. It extends the analysis to weighted and monotone-valuation variants, proving polynomial-time computability of $WEFX$ and PO allocations under binary additive valuations, and establishing a $1/4$-WEFX guarantee for $n=2$. The paper also introduces the $EFX+$ variant and analyzes its existence and nonexistence across small instances (e.g., $n=2$ vs $(n,m)=(3,4)$), highlighting incomparability with $EFX$ and the brittle nature of these fairness notions. Together, these results illuminate transition thresholds for the existence of fair allocations, offer algorithmic tools for specific valuation classes, and map out future directions for understanding EFX-type fairness in broader settings.
Abstract
Envy-freeness up to any good (EFX) is a popular and important fairness property in the fair allocation of indivisible goods, of which its existence in general is still an open question. In this work, we investigate the problem of determining the minimum number of EFX allocations for a given instance, arguing that this approach may yield valuable insights into the existence and computation of EFX allocations. We focus on restricted instances where the number of goods slightly exceeds the number of agents, and extend our analysis to weighted EFX (WEFX) and a novel variant of EFX for general monotone valuations, termed EFX+. In doing so, we identify the transition threshold for the existence of allocations satisfying these fairness notions. Notably, we resolve open problems regarding WEFX by proving polynomial-time computability under binary additive valuations, and establishing the first constant-factor approximation for two agents.