🤖 AI Summary
This paper studies mechanism design for fair and efficient allocation of divisible resources—such as cake-cutting and homogeneous divisible goods—under strategic agents. Addressing the challenge of reconciling incentive compatibility with fairness, it first establishes that the Maximum Nash Welfare (MNW) mechanism achieves an incentive ratio of 2 for cake-cutting even without free disposal. It then introduces a performance interpolation framework between MNW and Probabilistic Arbitration (PA), proving tight incentive ratio bounds of 2 and [e^{1/e}, e], respectively. Furthermore, it constructs the first envy-free cake-cutting mechanism for two agents attaining the optimal incentive ratio of 4/3. Finally, it demonstrates that connectivity constraints on allocated pieces imply a lower bound of Θ(n) on the incentive ratio. Collectively, these results unify and extend the theoretical frontiers of fair division and mechanism design.
📝 Abstract
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 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. (EC'13), 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 optimal incentive ratio achievable by envy-free cake cutting mechanisms. We first give an envy-free mechanism for two agents with an incentive ratio of $4 / 3$. Then, we show that any envy-free cake cutting mechanism with the connected pieces constraint has an incentive ratio of $Theta(n)$.
