We study the problem of allocating indivisible goods among n agents in a fair
manner. For this problem, maximin share (MMS) is a well-studied solution
concept which provides a fairness threshold. Specifically, maximin share is
defined as the minimum utility that an agent can guarantee for herself when
asked to partition the set of goods into n bundles such that the remaining
(n-1) agents pick their bundles adversarially. An allocation is deemed to be
fair if every agent gets a bundle whose valuation is at least her maximin
share.
Even though maximin shares provide a natural benchmark for fairness, it has
its own drawbacks and, in particular, it is not sufficient to rule out
unsatisfactory allocations. Motivated by these considerations, in this work we
define a stronger notion of fairness, called groupwise maximin share guarantee
(GMMS). In GMMS, we require that the maximin share guarantee is achieved not
just with respect to the grand bundle, but also among all the subgroups of
agents. Hence, this solution concept strengthens MMS and provides an ex-post
fairness guarantee. We show that in specific settings, GMMS allocations always
exist. We also establish the existence of approximate GMMS allocations under
additive valuations, and develop a polynomial-time algorithm to find such
allocations. Moreover, we establish a scale of fairness wherein we show that
GMMS implies approximate envy freeness.
Finally, we empirically demonstrate the existence of GMMS allocations in a
large set of randomly generated instances. For the same set of instances, we
additionally show that our algorithm achieves an approximation factor better
than the established, worst-case bound.Comment: 19 page
