Thinned Quantile Shares are Universally Feasible

TL;DR

This paper introduces thinned quantile shares, a new class of fair division benchmarks, proving their universal feasibility unconditionally for certain parameters, thus improving upon previous results and removing conditional assumptions.

Contribution

It defines the c-thinned quantile share, proves its universal feasibility for some constant c, and refines prior results by removing the rainbow EMC assumption.

Findings

Existence of a universal constant c > 0 for which the c-thinned e^{-c}-quantile share is universally feasible.

The c-thinned q-quantile share can be infeasible for q > e^{-c}.

Prior to this, only Feige's residual maximin share was known to be universally feasible.

Abstract

Quantile shares, introduced by Babichenko, Feldman, Holzman, and Narayan [STOC 2024], offer an ordinal, self-maximizing, and interpretable benchmark for fair division of indivisible goods, but their universal feasibility is known only conditional on the rainbow Erd\H{o}s matching conjecture (EMC). Specifically, Babichenko et al. showed that assuming the rainbow EMC in the near-perfect matching regime, the $(1/2e)$-quantile share is universally feasible. In contrast, a simple argument shows that the $q$-quantile share can be infeasible for any $q > 1/e$. We introduce a one-parameter refinement of quantile shares, the $c$-thinned quantile share, obtained by thinning the inclusion probability in the random benchmark bundle by a factor of $c$ for a fixed constant $c\in(0,1]$. Our main result is that there exists a universal constant $c >0$ for which the $c$-thinned $e^{-c}$-quantile share is unconditionally universally feasible; this is best possible in the sense that for any $c \in (0,1]$, the $c$-thinned $q$-quantile share can be infeasible for any $q > e^{-c}$. Prior to this work, the only nontrivial share known to be universally feasible was Feige's residual maximin share. The thinning viewpoint also lets us remove the factor-two loss in the conditional result for the original quantile share: assuming the rainbow EMC, the $(1/e)$-quantile share is universally feasible.