Allocating indivisible goods is a ubiquitous task in fair division. We study additive welfarist rules, an important class of rules which choose an allocation that maximizes the sum of some function of the agents' utilities. Prior work has shown that the maximum Nash welfare (MNW) rule is the unique additive welfarist rule that guarantees envy-freeness up to one good (EF1). We strengthen this result by showing that MNW remains the only additive welfarist rule that ensures EF1 for identical-good instances, two-value instances, as well as normalized instances with three or more agents. On the other hand, if the agents' utilities are integers, we demonstrate that several other rules offer the EF1 guarantee, and provide characterizations of these rules for various classes of instances.

翻译：在公平分配中，分配不可分割物品是一项普遍存在的任务。我们研究加法福利主义规则，这是一类重要的规则，其选择能够最大化代理人效用函数之和的分配方案。先前的研究表明，最大纳什福利（MNW）规则是唯一能保证至多一件物品的无嫉妒性（EF1）的加法福利主义规则。我们通过证明MNW在以下情况下仍然是唯一能确保EF1的加法福利主义规则来强化这一结论：同质物品实例、双值实例，以及具有三个或更多代理人的归一化实例。另一方面，如果代理人的效用为整数，我们证明其他几种规则也能提供EF1保证，并针对各类实例对这些规则进行了刻画。