Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations

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 MCSD (Mixed Constrained Serial Dictatorship). CCEA is polynomial-time, robust envy-free, and non-wasteful. Then, we show that there exists an algorithm (MEA) that is polynomial-time, envy-free, proportional, and Pareto optimal. Moreover, we show that for piecewise uniform valuations, MEA and CCEA are group-strategyproof and are equivalent to Mechanism 1 of Chen et. al.(2013). We then present an algorithm MCSD and a way to implement it via randomization that satisfies strategyproofness in expectation, robust proportionality, and unanimity for piecewise constant valuations. We also present impossibility results that show that the properties satisfied by CCEA and MEA are maximal subsets of properties that can be satisfied by any algorithm.