In the fair division of items among interested agents, envy-freeness is possibly the most favoured and widely studied formalisation of fairness. For indivisible items, envy-free allocations may not exist in trivial cases, and hence research and practice focus on relaxations, particularly envy-freeness up to one item (EF1). A significant reason for the popularity of EF1 allocations is its simple fact of existence. It is known that EF1 allocations exist for two agents with arbitrary valuations; agents with doubly-monotone valuations; agents with Boolean valuations; and identical agents with negative Boolean valuations. We consider two new but natural classes of valuations, and partly extend results on the existence of EF1 allocations to these valuations. Firstly, we consider trilean valuations - an extension of Boolean valuations - when the value of any subset is 0, $a$, or $b$ for any integers $a$ and $b$. Secondly, we define separable single-peaked valuations, when the set of items is partitioned into types. For each type, an agent's value is a single-peaked function of the number of items of the type. The value for a set of items is the sum of values for the different types. We prove EF1 existence for identical trilean valuations for any number of agents, and for separable single-peaked valuations for three agents. For both classes of valuations, we also show that EFX allocations do not exist.

翻译：在物品公平分配问题中，无嫉妒性可能是最受青睐且被广泛研究的公平性形式化准则。对于不可分割物品，即使在简单情形下无嫉妒分配也可能不存在，因此研究与实践主要关注其松弛条件，特别是“至多一件物品的无嫉妒性”（EF1）。EF1分配广受欢迎的一个重要原因在于其存在性的简单事实。已知EF1分配存在于以下情形：具有任意估值函数的两个智能体；具有双单调估值函数的智能体；具有布尔估值函数的智能体；以及具有负布尔估值函数的相同智能体。我们考虑两类新颖但自然的估值函数，并将EF1分配存在性的结果部分推广至这些估值函数。首先，我们研究三态估值函数——布尔估值函数的扩展——其中任意子集的价值为0、$a$或$b$（$a$和$b$为任意整数）。其次，我们定义可分离单峰估值函数，此时物品集合被划分为若干类型。对于每种类型，智能体对该类型物品的价值是物品数量的单峰函数。对一组物品的总价值是各类型价值的求和。我们证明了对于任意数量智能体的相同三态估值函数，以及三个智能体的可分离单峰估值函数，EF1分配总是存在。对于这两类估值函数，我们还证明了EFX分配并不存在。