Fair allocations with subadditive and XOS valuations
Uriel Feige, Vadim Grinberg
TL;DR
This work advances fair division for subadditive and XOS valuations under arbitrary entitlements by deriving ex-ante MES guarantees and strong ex-post APS allocations. It leverages a configuration LP framework with rounding to obtain $\tfrac{1}{2}$-MES for subadditive and $1-\tfrac{1}{e}$-MES for XOS ex-ante, and introduces a novel extended bidding-game strategy that achieves $$(1 - o(1))\frac{\log\log m}{\log m}$$-APS for subadditive valuations, plus a $\tfrac{1}{6}$-APS result for XOS under arbitrary entitlements. For equal entitlements, the paper provides a $\tfrac{4}{17}$-APS allocation for XOS, improving prior MMS-based bounds and illustrating the nuanced gap between APS and MMS benchmarks. The results establish the first APS-type guarantees for subadditive and XOS valuations with arbitrary entitlements and reveal a separation between the plain and extended bidding-game mechanisms, with practical implications for mechanism design in fair allocation systems.
Abstract
We consider the problem of fair allocation of $m$ indivisible goods to $n$ agents with either subadditive or XOS valuations, in the arbitrary entitlement case. As fairness notions, we consider the anyprice share (APS) ex-post, and the maximum expectation share (MES) ex-ante. We observe that there are randomized allocations that ex-ante are at least $\frac{1}{2}$-MES in the subadditive case and $(1-\frac{1}{e})$-MES in the XOS case. Our more difficult results concern ex-post guarantees. We show that $(1 - o(1))\frac{\log\log m}{\log m}$-APS allocations exist in the subadditive case, and $\frac{1}{6}$-APS allocations exist in the XOS case. For the special case of equal entitlements, we show $\frac{4}{17}$-APS allocations for XOS. Our results are the first for subadditive and XOS valuations in the arbitrary entitlement case, and also improve over the previous best results for the equal entitlement case.