  We develop a new and systematic method for proving entropic Ricci curvature
lower bounds for Markov chains on discrete sets. Using different methods, such
bounds have recently been obtained in several examples (e.g., 1-dimensional
birth and death chains, product chains, Bernoulli-Laplace models, and random
transposition models). However, a general method to obtain discrete Ricci
bounds had been lacking. Our method covers all of the examples above. In
addition, we obtain new Ricci curvature bounds for zero-range processes on the
complete graph. The method is inspired by recent work of Caputo, Dai Pra and
Posta on discrete functional inequalities.
