Cake Cutting: Envy and Truth

Abstract

We study envy-free cake cutting with strategic agents, where each agent may manipulate his private information in order to receive a better allocation. We focus on piecewise constant utility functions and consider two scenarios: the general setting without any restriction on the allocations and the restricted setting where each agent has to receive a connected piece. We show that no deterministic truthful envy-free mechanism exists in the connected piece scenario, and the same impossibility result for the general setting with some additional mild assumptions on the allocations. Finally, we study a large market model where the economy is replicated and demonstrate that truth-telling converges to a Nash equilibrium.

Truthful Fair Division without Free Disposal

“…As with the connected piece assumption, Bei et al [2017] showed a similar negative result for position obliviousness with the free disposal assumption but using the larger class of piecewise constant valuations. For piecewise uniform valuations, Chen et al's mechanism is truthful, envy-free, and position oblivious under the free disposal assumption.…”

“…Since this holds for any y > x, if we take some z > y (obviously z > x also), we find that no allocation works when both agents report [0, z], a contradiction. Bei et al [2017] showed that a similar impossibility result holds even with the free disposal assumption, but using the larger class of piecewise constant valuations. For the class of valuations that we consider in Theorem 4, there exists a simple truthful and envy-free mechanism that always returns a connected allocation assuming free disposal.…”

“…For valuation functions beyond piecewise uniform, most results are negative. For example, for piecewise constant valuations, Aziz and Ye [2014] showed that there is no truthful and robust proportional mechanism, two works [Menon and Larson, 2017;Bei et al, 2017] showed that there is no proportional mechanism that allocates connected pieces or is non-wasteful, and Bei et al [2017] also showed that there is no truthful mechanism that satisfies position obliviousness. In the Robertson-Webb query model, Kurokawa et al [2013] showed that there is no truthful envy-free mechanism with bounded queries, while Brânzei and Miltersen [2015] proved that any deterministic truthful mechanism for two agents must be a dictatorship.…”

“…Phase 3: Assume that both agents are at point x of the cake. (It is possible that the two agents meet while both considered by Bei et al [2017]: Our definition only requires that the agents get the same value if the indicator function of their density functions remain the same, whereas Bei et al's definition also requires the pieces to be allocated in "equivalent" ways.…”

Fair Cake-Cutting in Practice

“…This is the standard notion of strategy-proofness in mechanism design. For a weaker notion see[7,8] 4. Formally, for any ϵ ∈ [0, 1], the procedure p is ϵ -strategy-proof if for every agent i, every collection of valuations functions(v i , v −i ), and every other valuation function v ′ i , v i (p i (v i , v −i )) ≥ v i (p i (v ′ i , v −i )) − ϵ .…”