Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations

Abstract

Abstract. Cake cutting is one of the most fundamental settings in fair division and mechanism design without money. In this paper, we consider different levels of three fundamental goals in cake cutting: fairness, Pareto optimality, and strategyproofness. In particular, we present robust versions of envy-freeness and proportionality that are not only stronger than their standard counter-parts but also have less information requirements. We then focus on cake cutting with piecewise constant valuations and present three desirable algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time, robust envy-free, and non-wasteful. It relies on parametric network flows and recent generalizations of the probabilistic serial algorithm. For the subdomain of piecewise uniform valuations, we show that it is also group-strategyproof. Then, we show that there exists an algorithm (MEA) that is polynomial-time, envy-free, proportional, and Pareto optimal. MEA is based on computing a market-based equilibrium via a convex program and relies on the results of Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise uniform valuations. We then present an algorithm CSD and a way to implement it via randomization that satisfies strategyproofness in expectation, robust proportionality, and unanimity for piecewise constant valuations. For the case of two agents, it is robust envy-free, robust proportional, strategyproof, and polynomial-time. Many of our results extend to more general settings in cake cutting that allow for variable claims and initial endowments. We also show a few impossibility results to complement our algorithms. The impossibilities show that the properties satisfied by CCEA and MEA are maximal subsets of properties that can be satisfied by any algorithm for piecewise constant valuation profiles.

Cake Cutting: Envy and Truth

“…A mechanism is non-wasteful if it never allocates a piece to someone who does not value it; a mechanism is position oblivious if it decides the allocation of an arbitrary portion of the cake only based on the agents' valuations on this portion, but not on its relative position in the cake. These results partially answer the open question raised in [Chen et al, 2010]. We also compare our results to the impossibility results in [Aziz and Ye, 2014] and [Brânzei and Miltersen, 2015].…”

“…Note that this is one of the most natural valuation function classes that has received much attention in the cakecutting study [Bei et al, 2012;Kurokawa et al, 2013;Aziz and Ye, 2014]. Specifically, piecewise constant functions enjoy two nice properties: (i) they are concisely representable, and (ii) they can approximate any functions to an arbitrary precision.…”

“…Informally speaking, an allocation is envy-free if all agents are satisfied with their own assigned pieces, compared with the allocation of any other agent. It is well known that an envy-free allocation always exists [Brams and Taylor, 1995], even if only n − 1 cuts are allowed [Su, 1999], i.e., each agent receives a connected piece. The study of computing envyfree allocations has a long history [Brams and Taylor, 1995;Stromquist, 2008], and some great progress has been made recently in deriving an envy-free algorithm for arbitrary number of agents in bounded running steps [Aziz and Mackenzie, 2016a;2016b].…”

“…This question was first addressed in [Chen et al, 2010] in which the authors gave a truthful cake cutting mechanism for piecewise uniform valuations. In their work, they also left the problem of designing truthful mechanism for piecewise constant valuations as an open problem.…”

“…In their work, they also left the problem of designing truthful mechanism for piecewise constant valuations as an open problem. For piecewise constant valuations, it is known that no truthful, proportional (i.e., one gets at least its average share) and Pareto optimal mechanism exists [Schummer, 1996;Aziz and Ye, 2014]. A characterization of truthful mechanisms for two agents was given by Maya and Nisan [2012].…”