We study the problem of designing truthful and fair mechanisms when allocating a mixture of divisible and indivisible goods. We first show that there does not exist an EFM (envy-free for mixed goods) and truthful mechanism in general. This impossibility result holds even if there is only one indivisible good and one divisible good and there are only two agents. Thus, we focus on some more restricted settings. Under the setting where agents have binary valuations on indivisible goods and identical valuations on a single divisible good (e.g., money), we design an EFM and truthful mechanism. When agents have binary valuations over both divisible and indivisible goods, we first show there exist EFM and truthful mechanisms when there are only two agents or when there is a single divisible good. On the other hand, we show that the mechanism maximizing Nash welfare cannot ensure EFM and truthfulness simultaneously.

翻译：我们研究在混合可分割与不可分割商品分配中设计诚实且公平的机制问题。首先证明，一般情况下不存在同时满足EFM（混合商品无嫉妒性）与诚实性的机制。即使仅涉及一件不可分割商品、一件可分割商品且只有两个代理，该不可能性结果依然成立。因此，我们将关注点转向更受限的设置。当代理对不可分割商品具有二元估值且对单一可分割商品（如货币）具有相同估值时，我们设计了一种同时满足EFM与诚实性的机制。当代理对可分割与不可分割商品均具有二元估值时，我们首先证明在两代理或单一可分割商品的情形下存在EFM且诚实的机制。另一方面，我们证明最大化纳什福利的机制无法同时确保EFM与诚实性。