Truthful Cake Sharing
Xiaohui Bei, Xinhang Lu, Warut Suksompong
TL;DR
We study cake sharing where a shared subset of length at most $\alpha$ is selected from $[0,1]$ for all agents with piecewise uniform utilities. The analysis centers on two welfare criteria, leximin and MNW, under truthfulness constraints and the ability to block access to unrequested cake; we show Leximin is truthful for any $n$ and achieves the optimal egalitarian ratio $\alpha/(n-(n-1)\alpha)$ among truthful, position-oblivious mechanisms, while MNW is truthful only for $n=2$ and not in general. Blocking emerges as essential for nontrivial fairness, as no positive egalitarian ratio is possible without blocking, and MNW exhibits easy-to-exploit subset-report manipulation. The results illuminate the price of truthfulness and the role of blocking in fair public-resource sharing, and point to extensions to more complex utilities and cost structures, including piecewise constant costs and non-uniform costs.
Abstract
The classic cake cutting problem concerns the fair allocation of a heterogeneous resource among interested agents. In this paper, we study a public goods variant of the problem, where instead of competing with one another for the cake, the agents all share the same subset of the cake which must be chosen subject to a length constraint. We focus on the design of truthful and fair mechanisms in the presence of strategic agents who have piecewise uniform utilities over the cake. On the one hand, we show that the leximin solution is truthful and moreover maximizes an egalitarian welfare measure among all truthful and position oblivious mechanisms. On the other hand, we demonstrate that the maximum Nash welfare solution is truthful for two agents but not in general. Our results assume that mechanisms can block each agent from accessing parts that the agent does not claim to desire; we provide an impossibility result when blocking is not allowed.