  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.
