Optimal transport on a finite graph provides a discrete analogue of Wasserstein geometry in which probability distributions are points of a finite-dimensional simplex. In this introductory talk, I will explain how this geometry leads naturally to Hamilton–Jacobi equations, mean field game systems, and master equations on the probability simplex. A central issue is that the boundary of the simplex corresponds to vanishing probabilities, where the underlying transport geometry may become singular. I will describe a quantitative preservation-of-positivity estimate which prevents solutions from reaching the boundary in finite time and leads to a classical well-posedness theory for the master equation on the open simplex without imposing boundary conditions. I will also discuss how the associated continuous-time Markov-chain game can be recovered from the PDE structure and how this finite-state theory may provide a structure-preserving approximation of continuous-state mean field games.