The incentive guarantees behind Nash welfare in divisible resources allocation

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. [1], 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 $\Theta \left(n\right)$.