We study the problem of allocating divisible resources among $n$ agents, hopefully in a fair and efficient manner. With the presence of strategic agents, additional incentive guarantees are also necessary, and the problem of designing fair and efficient mechanisms becomes much less tractable. While the maximum Nash welfare (MNW) mechanism has been proven to be prominent by providing desirable fairness and efficiency guarantees as well as other intuitive properties, no incentive property is known for it. We show a surprising result that, when agents have piecewise constant value density functions, the incentive ratio of the MNW mechanism is $2$ for cake cutting, where the incentive ratio of a mechanism is defined as the ratio between the largest possible utility that an agent can gain by manipulation and his utility in honest behavior. Remarkably, this result holds even without the free disposal assumption, which is hard to get rid of in the design of truthful cake cutting mechanisms. We also show that the MNW mechanism is group strategyproof when agents have piecewise uniform value density functions. Moreover, we show that, for cake cutting, the Partial Allocation (PA) mechanism proposed by Cole et al., which is truthful and $1/e$-MNW for homogeneous divisible items, has an incentive ratio between $[e^{1 / e}, e]$ and when randomization is allowed, can be turned to be truthful in expectation. Given two alternatives for a trade-off between incentive ratio and Nash welfare provided by the MNW and PA mechanisms, we establish an interpolation between them for both cake cutting and homogeneous divisible items. Finally, we study the existence of fair mechanisms with a low incentive ratio in the connected pieces setting. We show that any envy-free cake cutting mechanism with the connected pieces constraint has an incentive ratio of at least $\Omega(n)$.

翻译：我们研究在$n$个智能体间公平高效地分配可分割资源的问题。当存在策略性智能体时，额外的激励保障是必要的，此时设计公平高效的机制变得更为棘手。尽管最大化纳什福利（MNW）机制通过提供理想的公平性和效率保障及其他直观性质被证明具有显著优势，但其激励性质尚不明确。本文揭示了一个令人惊讶的结果：当智能体拥有分段常数价值密度函数时，MNW机制在 cake cutting 问题上的激励比率为$2$，其中机制激励比率定义为智能体通过操纵可获得的最大效用与其诚实行为的效用之比。值得注意的是，该结论即使在无自由处置假设条件下依然成立，而这一假设在诚实 cake cutting 机制设计中难以规避。我们还证明了当智能体拥有分段均匀价值密度函数时，MNW机制具有群体策略证明性。此外，本研究证明：对于蛋糕切割问题，Cole等人提出的部分分配（PA）机制（该机制对同质可分割物品具有诚实性且达到$1/e$-MNW）的激励比率介于$[e^{1/e}, e]$之间，且允许随机化时可转化为期望诚实性。针对MNW与PA机制在激励比率与纳什福利之间的权衡取舍，我们在 cake cutting 与同质可分割物品场景下建立了二者间的插值关系。最后，我们探讨了在连续段约束下低激励比率公平机制的存在性，证明任何满足连续段约束的无嫉妒蛋糕切割机制，其激励比率至少为$\Omega(n)$。