In this paper, we are interested in the connection between some stochastic games, namely the Tug-of-War Games, and non-local PDEs on graphs. We consider a general formulation of Tug-of-War Games related to many continous PDEs. Using the framework of Partial difference Equation, we transcribe this formulation on graph, and show that it encompasses several PDEs on graphs such as ∞-Laplacian, Game p-Laplacian with and without gradient terms, and Eikonal equation. We then interpret these discrete games as non-local Tug-of-War Games. The proposed framework is illustrated with general interpolation problems on graphs.