On Lower Bounds for Maximin Share Guarantees

TL;DR

This paper investigates the existence of maximin share (MMS) allocations in fair division, establishing new bounds on the number of agents and items for guaranteed MMS allocations, and narrowing the gap between known upper and lower bounds.

Contribution

It proves new bounds showing MMS allocations exist for larger classes of instances, significantly improving previous results and closing the gap between known bounds.

Findings

MMS allocations exist for any instance with n + c items when n is sufficiently large.

For n ≠ 3, all instances with n + 6 goods have an MMS allocation.

Bounds depend on exponential functions of c and factorials, refining previous limits.

Abstract

We study the problem of fairly allocating a set of indivisible items to a set of agents with additive valuations. Recently, Feige et al. (WINE'21) proved that a maximin share (MMS) allocation exists for all instances with $n$ agents and no more than $n + 5$ items. Moreover, they proved that an MMS allocation is not guaranteed to exist for instances with $3$ agents and at least $9$ items, or $n \ge 4$ agents and at least $3n + 3$ items. In this work, we shrink the gap between these upper and lower bounds for guaranteed existence of MMS allocations. We prove that for any integer $c > 0$, there exists a number of agents $n_c$ such that an MMS allocation exists for any instance with $n \ge n_c$ agents and at most $n + c$ items, where $n_c \le \lfloor 0.6597^c \cdot c!\rfloor$ for allocation of goods and $n_c \le \lfloor 0.7838^c \cdot c!\rfloor$ for chores. Furthermore, we show that for $n \neq 3$ agents, all instances with $n + 6$ goods have an MMS allocation.