We consider the problem of fair allocation of $m$ indivisible items to a group of $n$ agents with subsidy (money). Our work mainly focuses on the allocation of chores but most of our results extend to the allocation of goods as well. We consider the case when agents have (general) additive cost functions. Assuming that the maximum cost of an item to an agent can be compensated by one dollar, we show that a total of $n/4$ dollars of subsidy suffices to ensure a proportional allocation. Moreover, we show that $n/4$ is tight in the sense that there exists an instance with $n$ agents for which every proportional allocation requires a total subsidy of at least $n/4$. We also consider the weighted case and show that a total subsidy of $(n-1)/2$ suffices to ensure a weighted proportional allocation.

翻译：我们研究了在存在补贴（金钱）的情况下，将$m$个不可分割物品公平分配给$n$个智能体的问题。研究主要聚焦于家务分配，但多数结论同样适用于物品分配。考虑智能体具有（一般）可加成本函数的情形。假设每件物品对智能体产生的最大成本可通过一美元补偿，我们证明总额为$n/4$美元的补贴足以确保比例分配。进一步研究表明$n/4$是紧界——存在一个包含$n$个智能体的实例，其中任何比例分配所需总补贴至少为$n/4$。我们还考虑了加权情形，证明总额为$(n-1)/2$的补贴足以确保加权比例分配。