Improving Approximation Guarantees for Maximin Share

We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor 3n/2 \rfloor$ [HSSH21]. We show that $1$-out-of-$\lceil 4n/3\rceil$ MMS allocations always exist. In the multiplicative approximation, the goal is to show the existence of $\alpha$-MMS allocations (for the largest possible $\alpha < 1$) which guarantees each agent at least $\alpha$ times her MMS value. A series of works in the last decade led to the state-of-the-art factor of $\alpha = \frac{3}{4} + \frac{3}{3836}$ [AG23]. We introduce a general framework of $(\alpha, \beta, \gamma)$-MMS that guarantees $\alpha$ fraction of agents $\beta$ times their MMS values and the remaining $(1-\alpha)$ fraction of agents $\gamma$ times their MMS values. The $(\alpha, \beta, \gamma)$-MMS captures both ordinal and multiplicative approximations as its special cases. We show that $(2(1 -\beta)/\beta, \beta, 3/4)$-MMS allocations always exist. Furthermore, since we can choose the $2(1-\beta)/\beta$ fraction of agents arbitrarily in our algorithm, this implies (using $\beta=\sqrt{3}/2$) the existence of a randomized allocation that gives each agent at least 3/4 times her MMS value (ex-post) and at least $(17\sqrt{3} - 24)/4\sqrt{3} > 0.785$ times her MMS value in expectation (ex-ante).