Almost-MMS allocations with small agent redundancy

For a fixed $ε\ge 0$, let $ρ_{ε, n}$ be the maximum value of $ρ_n$ such that, in every allocation instance with $m$ indivisible goods and $(1 + ε)n$ agents, there is a subset of $n$ agents with the following property. The allocation instance restricted to only this subset of $n$ agents has an allocation that gives each agent at least $ρ_n$-fraction of her $n$-maximin share (MMS). When $ε> 0$ we refer to $εn$ as the agent redundancy. The case $ε= 0$ is the standard setting without agent redundancy, and we use $ρ_{0,n}$ and $ρ_n$ interchangeably. For allocation instances with additive valuations, % agents, it is known that $\frac{7}{9} \le ρ_n < 1-Ω(\frac{1}{\log^2 n})$ for every $n \ge 2$ (with stronger lower bounds for small $n$). It is an open question whether $ρ_n$ tends to~1 as $n$ grows. We show that with agent redundancy, the answer to this question is positive. That is, for every $ε> 0$, $ρ_{ε, n}$ tends to~1 as $n$ grows. A straightforward corollary of this result is that for every $c > 0$, if the number of items satisfies $m \le (2 - c)n$, then the answer is positive also for $ρ_n$ (with no agent redundancy). Adapting our techniques to a related setting, we also improve a recent result of Ezra and Garbuz, and show that without agent redundancy, there always exists an allocation that gives $n - O(\log^2 n)$ agents their full MMS.