  We consider the problem of fairly allocating a set of indivisible goods among
$n$ agents with additive valuations, using the popular fairness notion of
maximin share (MMS). Since MMS allocations do not always exist, a series of
works provided existence and algorithms for approximate MMS allocations. The
Garg-Taki algorithm gives the current best approximation factor of
$(\frac{3}{4} + \frac{1}{12n})$. Most of these results are based on complicated
analyses, especially those providing better than $2/3$ factor. Moreover, since
no tight example is known of the Garg-Taki algorithm, it is unclear if this is
the best factor of this approach. In this paper, we significantly simplify the
analysis of this algorithm and also improve the existence guarantee to a factor
of $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. For small $n$, this
provides a noticeable improvement. Furthermore, we present a tight example of
this algorithm, showing that this may be the best factor one can hope for with
the current techniques.
