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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 (3/4 + 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 (3/4 + min(1/36, 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. 

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed till now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics.We propose two alternative fairness concepts—called epistemic EFX (EEFX) and minimum EFX value fairness (MXS)---inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts are excellent alternatives to EFX and MMS.