OpenAlex ·
Tight Maximin Share Guarantees of EFX and PMMS
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- OpenAlex
- DOI
- 10.48550/arxiv.2608.30267
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Abstract
In the fair division of indivisible goods among autonomous agents, allocations frequently satisfy strong pairwise fairness criteria while failing to guarantee an agent's full maximin share. Maximin share (MMS) is a global benchmark: the value an agent can guarantee by partitioning all goods into $n$ bundles and receiving her least preferred bundle. In contrast, local pairwise fairness notions---most notably envy-freeness up to any good (EFX) and pairwise maximin share (PMMS)---evaluate fairness through bilateral bundle comparisons. We resolve the open question of determining the optimal universal MMS approximation factor guaranteed by either local notion under nonnegative additive valuations, proving that both factors equal exactly $10/17$. For the lower bound, we show that both EFX and PMMS imply a common local residual inequality: the value of any foreign bundle after removing its least valuable item cannot exceed the focal agent's allocated value. We design a continuous piecewise weight function that converts this local constraint into the global $10/17$ MMS guarantee. The same charging argument establishes that $α$-EFX implies $(10α/17)$-MMS. For tightness, we construct an explicit family of complete allocations that simultaneously satisfy PMMS and EFX$_0$, including comparisons involving zero-valued goods, and whose MMS ratios converge to $10/17$. For each fixed number of agents, we also establish exact reciprocal identities connecting fair division to scheduling: the EFX guarantee is the reciprocal of the pure price of anarchy in selfish identical-machine covering, while the PMMS guarantee is the reciprocal of a locality gap for exact repartition of two machines.
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