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arXiv · 2609.33588

On best of both worlds allocations with subadditive valuations

Uriel Feige
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Abstract

We consider allocation of indivisible goods to agents with equal entitlements and subadditive valuations. As an ex-post fairness notion we consider the maximin share (MMS), and as an ex-ante fairness notion we consider the maximum expectation share (MES), which is always at least as large as the MMS, and sometimes much larger. We present a simple transformation that for every $0 < ρ\le 1$, given any algorithm that produces $ρ$-MMS allocations, transforms it into a randomized allocation algorithm that offers $ρ$-MMS ex-post simultaneously with $η$-MES ex-ante. We prove several new properties of MES, and use them to show that $η\ge \min[\fracρ{2 + ρ}, \frac{1}{4}]$. We also present cases in which the transformation results in a higher value of $η$. Applying our transformation to currently known allocation algorithms shows for subadditive valuations the existence of randomized allocations that are simultaneously $Ω(\frac{1}{\log\log n})$-MES ex-ante and $Ω(\frac{1}{\log\log n})$-MMS ex-post, and for XOS valuations the existence of randomized allocations that are simultaneously $\frac{4}{27}$-MES ex-ante and $\frac{4}{17}$-MMS ex-post.

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Uriel Feige. 2026-09-27. On best of both worlds allocations with subadditive valuations. https://arxiv.org/abs/2609.33588

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