The Exact MMS Guarantees of EFX and PMMS

Envy-freeness up to any good (EFX) and pairwise maximin share (PMMS) are standard local fairness criteria for indivisible goods, whereas maximin share (MMS) is a global benchmark. We determine the exact quantitative relationship between these local fairness notions and the global MMS guarantee under nonnegative additive valuations. We show that the optimal universal factor for both notions is $ρ^{\mathrm{EFX}\to\mathrm{MMS}}=ρ^{\mathrm{PMMS}\to\mathrm{MMS}}=\frac{10}{17}$. We prove the lower bound by a combinatorial charging argument. After an initial reduction, both EFX and PMMS imply the same local condition on every foreign bundle from the perspective of a focal agent: deleting its least valuable good leaves value at most the focal bundle. A three-piece concave weight function translates this local condition into the global $10/17$ guarantee. We then construct an explicit family of complete allocations that are simultaneously PMMS and EFX$_0$, whose MMS ratios converge to $10/17$, showing that both constants are tight even in the presence of zero-valued goods. The argument also gives $α$-EFX $\Rightarrow (10α/17)$-MMS. Finally, we establish an exact correspondence between these fair-division guarantees and scheduling equilibria. For every fixed number of agents $n$, the EFX-to-MMS extremal ratio equals the reciprocal of the pure price of anarchy for selfish identical-machine covering. Similarly, the PMMS-to-MMS ratio equals the reciprocal of a locality gap based on exact pairwise machine repartition. These correspondences explain why the constant $10/17$ governs both problems.