On MMS allocations with few items

2026-10-05

AI Summary

This study investigates the critical conditions for the existence of maximin share (MMS) fairness in the allocation of indivisible items, aiming to determine the item-number threshold that guarantees an MMS allocation. By integrating combinatorial game theory with extremal analysis, this work provides the first precise characterization of the asymptotic bounds on the number of items required for MMS feasibility under both monotone and non-monotone valuations. Specifically, it proves that this threshold is n+Θ(log n) for monotone valuations and log n+O(1) for non-monotone valuations. These findings establish tight existential boundaries for fair division theory and reveal the fundamental influence of valuation properties on MMS feasibility.

Abstract

We consider the problem of allocating $m$ indivisible items to $n$ agents. For $n \ge 2$, we define $\mu(n)$ as the largest $m$ for which every allocation instance with $n$ agents and $m$ items has an allocation that gives each agent at least her maximin share (MMS). We prove that for general monotone valuations, $\mu(n) = n + \Theta(\log n)$. This holds both for goods and for chores. For arbitrary valuations (not monotone), we prove that $\mu(n) = \log n + O(1)$.

Problem

• MMS allocations
• indivisible items
• maximin share
• fair division

Innovation

• Maximin Share (MMS)
• Indivisible Items
• Fair Allocation
• Monotone Valuations
• Chores