Concentration and maximin fair allocations for subadditive valuations

We consider fair allocation of $m$ indivisible items to $n$ agents of equal entitlements, with submodular valuation functions. Previously, Seddighin and Seddighin [{\em Artificial Intelligence} 2024] proved the existence of allocations that offer each agent at least a $\frac{1}{c \log n \log\log n}$ fraction of her maximin share (MMS), where $c$ is some large constant (over 1000, in their work). We modify their algorithm and improve its analysis, improving the ratio to $\frac{1}{14 \log n}$. Some of our improvement stems from tighter analysis of concentration properties for the value of any subadditive valuation function $v$, when considering a set $S' \subseteq S$ of items, where each item of $S$ is included in $S'$ independently at random (with possibly different probabilities). In particular, we prove that up to less than the value of one item, the median value of $v(S')$, denoted by $M$, is at least two-thirds of the expected value, $M \geq \frac{2}{3}\E[v(S')] - \frac{11}{12}\max_{e \in S} v(e)$.