We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate concepts of proportionality have been considered in the previous work. Among them, proportionality up to the maximin good (PROPm) has been the best approximate notion of proportionality that can be achieved for all instances. In this paper, we introduce the notion of proportionality up to the least valued good on average (PROPavg), which is a stronger notion than PROPm, and show that a PROPavg allocation always exists for all instances and can be computed in polynomial time. %% for all instances. Our results establish PROPavg as a notable non-trivial fairness notion that can be achieved for all instances. Our proof is constructive, and based on a new technique that generalizes the cut-and-choose protocol and uses a recursive technique.

翻译：我们研究的是将一组不可分割物品公平分配给多个智能体的问题，重点关注比例性这一经典公平性概念。由于物品不可分割时，比例分配并不总是存在，因此已有工作中考虑了比例性的近似概念。其中，达到最大最小物品的比例分配（PROPm）是所有实例中都能实现的最佳近似比例性概念。在本文中，我们引入了达到平均最低价值物品的比例性（PROPavg）这一新概念，它比PROPm更强。我们证明了对于所有实例，PROPavg分配总是存在的，并且可以在多项式时间内计算得出。我们的结果确立PROPavg为一个显著的非平凡公平性概念，且对所有实例均可实现。我们的证明是构造性的，基于一种推广了“切与选”协议并采用递归技术的新方法。