Thinned Quantile Shares are Universally Feasible
For researchers in fair division, this provides a new universally feasible benchmark that is ordinal, self-maximizing, and interpretable, removing the reliance on an unproven conjecture.
The paper introduces thinned quantile shares, a refinement of quantile shares, and proves that for a universal constant c>0, the c-thinned e^{-c}-quantile share is unconditionally universally feasible for fair division of indivisible goods, which is best possible. This is the first nontrivial share besides Feige's residual maximin share to achieve universal feasibility.
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 Erdős 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.