Ordinal Maximin Guarantees for Group Fair Division

We investigate fairness in the allocation of indivisible items among groups of agents using the notion of maximin share (MMS). While previous work has shown that no nontrivial multiplicative MMS approximation can be guaranteed in this setting for general group sizes, we demonstrate that ordinal relaxations are much more useful. For example, we show that if $n$ agents are distributed equally across $g$ groups, there exists a $1$-out-of-$k$ MMS allocation for $k=O(g\log (n\mathrm{/}g))$, while if all but a constant number of agents are in the same group, we obtain $k=O(\log n\mathrm{/}\log \log n)$. We also establish the tightness of these bounds and provide non-asymptotic results for the case of two groups.