  The maximin share (MMS) guarantee is a desirable fairness notion for
allocating indivisible goods. While MMS allocations do not always exist,
several approximation techniques have been developed to ensure that all agents
receive a fraction of their maximin share. We focus on an alternative
approximation notion, based on the population of agents, that seeks to
guarantee MMS for a fraction of agents. We show that no optimal approximation
algorithm can satisfy more than a constant number of agents, and discuss the
existence and computation of MMS for all but one agent and its relation to
approximate MMS guarantees. We then prove the existence of allocations that
guarantee MMS for $\frac{2}{3}$ of agents, and devise a polynomial time
algorithm that achieves this bound for up to nine agents. A key implication of
our result is the existence of allocations that guarantee
$\text{MMS}^{\lceil{3n/2}\rceil}$, i.e., the value that agents receive by
partitioning the goods into $\lceil{\frac{3}{2}n}\rceil$ bundles, improving the
best known guarantee of $\text{MMS}^{2n-2}$. Finally, we provide empirical
experiments using synthetic data.
