Ordinal Maximin Guarantees for Group Fair Division
Pasin Manurangsi, Warut Suksompong
TL;DR
The paper investigates fair division of indivisible items among groups through an ordinal relaxation of MMS, defining p_MMS as the smallest p for which a 1-out-of-p MMS allocation exists for all agents. It proves asymptotically tight bounds that separate balanced and unbalanced group settings, showing p_MMS = Theta(g log(n/g)) in the balanced case and giving two regimes for general cases with corresponding upper and matching lower bounds. The authors provide efficient randomized algorithms achieving these upper bounds and use covering designs to derive lower bounds, including non-asymptotic results for the two-group case. The results demonstrate that ordinal MMS guarantees can be substantially stronger than cardinal MMS guarantees in group settings and offer concrete guidance for fair allocation across diverse group structures.
Abstract
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 $k = O(g\log(n/g))$, while if all but a constant number of agents are in the same group, we obtain $k = O(\log n/\log \log n)$. We also establish the tightness of these bounds and provide non-asymptotic results for the case of two groups.