Fair Cake Division Under Monotone Likelihood Ratios

Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy-free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social (utilitarian) and egalitarian welfare. In addition, we show that, under MLRP, the problem of maximizing Nash social welfare admits a fully polynomial-time approximation scheme (FPTAS).
Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, binomial, Poisson, and exponential distributions. Furthermore, it is known that linear translations of any log-concave function satisfy MLRP. Therefore, our results also hold when the value densities of the agents are linear translations of the following (log-concave) distributions: Laplace, gamma, beta, Subbotin, chi-square, Dirichlet, and logistic. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.