Theorem 14, p. 14 — an allocation meeting Theorem 8's three conditions gives every agent at least MMS_i(n, M)/(2 − |P|/n)

Let $N=[n]$ be a set of $n$ agents with additive valuations and $M$ a set of $m$ goods. Let $X=\langle X_1,\dots,X_n\rangle$ be a partial allocation with pool of unallocated goods $P=M\setminus\bigcup_i X_i$ and $k=|P|$, satisfying the three conditions of Theorem 8:

1. $X$ is EFX;
2. $v_i(X_i)\ge v_i(P)$ for all agents $i$;
3. $|P|$ is less than the number of sources (unenvied agents) of the envy graph $G_X$.

Then for every agent $i\in N$,

The guarantee interpolates between the factor $\tfrac12$ for an empty pool and the factor $\bigl(1+\tfrac1n\bigr)^{-1}$ when $|P|=n-1$: the more goods the algorithm leaves to charity, the closer every agent gets to her full maximin share.

Formalization Note The paper's Theorem 14 asserts that there exists an allocation satisfying Theorem 8's conditions, a ½-Nash-social-welfare guarantee, and this bound. Existence of such an allocation is Theorem 8 (the goal of mission I of this series); its proof of the third bullet uses only the three conditions, so the goal is posed as the implication for every such allocation, and combining it with Theorem 8 gives the printed bullet. The Nash-social-welfare bullet is not posed: it rests on Lemma 12, which initialises the algorithm with the allocation of [12], a black box not proved in the paper. Condition 3 forces $n\ge 1$ and $k<n$, so the denominator $2-k/n$ (computed in $\mathbb{R}$) exceeds 1 and $\mathrm{MMS}_i(n,M)$ uses $n\ge 1$ bundles. Agents are Fin n, 0-based.