We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for $n$ agents with piecewise linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for $\epsilon$-envy-freeness for mixed goods ($\epsilon$-EFM), and present an algorithm that finds an $\epsilon$-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and $1/\epsilon$.
翻译:我们研究的是在资源含有可分割和不可分割的商品时的公平分割问题;不能直接将嫉妒自由(EF)和嫉妒自由(EF1)等典型的公平概念直接适用于混合商品环境;在这项工作中,我们提出对混合商品(EFM)实行新的公平概念,即混合商品(EFM)的无嫉妒自由(EFM),这是将EF和EF1直接概括到混合商品环境;我们证明EFM总是对任何数目的代理人进行分配;我们还提出有效的算法,以计算两种代理人和美元代理人的EFM分配,对可移动商品进行笔直线估值;最后,我们放宽对无嫉妒要求的要求,而不是要求混合商品(Esilon$-EFM)免税的美元,而提出一种算法,在制剂数量、不可分割货物数量和1美元/美元/美元中,在时间多盘分配美元-EFM。