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.
翻译:最大份额(MMS)的保证是分配不可分割货物的可取的公平概念。虽然MMS分配并不总是存在,但已经开发了几种近似技术,以确保所有代理商获得最大份额的一小部分。我们注重基于代理商人数的替代近似概念,力求为一小部分代理商保证MMS。我们表明,任何最佳近似算法都不能满足一个不变的代理商数量,并讨论除一个代理商以外的所有代理商的存在和计算及其与近似MMS保障的关系。我们随后证明,存在着保证MMS $\frac{2 ⁇ 3}的代理商分配款,并设计了一种多边时间算法,实现这一约束的最多九个代理商。我们结果的一个关键含义是,存在着保证美元(text{MMS{lice{3n/2 ⁇ rceil}美元,即通过将货物分成一个代理商获得的价值(美元)[frac]{{3 ⁇ 2}n ⁇ rceil$(rceil$)的保证款包件,从而改进了已知的美元/MIS2n2}实验数据的最佳保证。最后。