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 there are flourishing positive results against strategic agents for homogeneous divisible items, very few of them are known to hold in cake cutting. We show that the Maximum Nash Welfare (MNW) mechanism, which provides desirable fairness and efficiency guarantees and achieves an \emph{incentive ratio} of $2$ for homogeneous divisible items, also has an incentive ratio of $2$ in cake cutting. 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. Moreover, we show that, for cake cutting, the Partial Allocation (PA) mechanism proposed by Cole et al. \cite{DBLP:conf/sigecom/ColeGG13}, 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 $\Theta(n)$.

翻译：我们研究在 $n$ 个智能体之间以公平且高效的方式分配可分割资源的问题。当存在策略性智能体时，额外的激励保证也必不可少，而设计公平高效机制的问题变得难以处理。尽管在均匀可分割物品方面针对策略性智能体的积极成果层出不穷，但在蛋糕切割中已知成立的结论却寥寥无几。我们证明最大纳什福利（MNW）机制在均匀可分割物品中能提供理想的公平性与效率保证，并实现激励比为 $2$，同时该机制在蛋糕切割中也具有激励比 $2$。值得注意的是，即使不采用自由处置假设（该假设在真实蛋糕切割机制设计中难以避免），该结果依然成立。此外，我们证明对于蛋糕切割，Cole 等人提出的部分分配（PA）机制（该机制对于均匀可分割物品是真实的且具有 $1/e$ 的 MNW 近似比）其激励比位于 $[e^{1 / e}, e]$ 区间内；当允许随机化时，可将其转化为期望意义下的真实机制。针对 MNW 与 PA 机制在激励比与纳什福利之间的权衡，我们为蛋糕切割和均匀可分割物品建立了二者之间的插值关系。最后，我们研究在连续片段约束下具有低激励比的公平机制的存在性。我们证明任何满足连续片段约束的无嫉妒蛋糕切割机制，其激励比为 $\Theta(n)$。