The classical mean value property is a fundamental characterization of harmonic functions and illustrates the deep connection between the Laplacian and Brownian motion. Building on this interplay between probability and analysis, Peres, Schramm, Sheffield, and Wilson introduced in their influential paper (Journal of the AMS, 22(1), 167–210, 2009) a probabilistic game known as Tug-of-War, which provides a game-theoretic framework for approximating solutions to the infinity-Laplacian. This framework has since been generalized in various directions to handle a wider class of nonlinear PDEs, notably the p-Laplacian. A key tool in this context is the asymptotic mean value property, which serves as a nonlinear analogue of the classical mean value formula and forms the basis of a semidiscrete dynamic programming principle at scale \(\varepsilon \) . The solutions to this principle are shown to converge to the unique viscosity solution of the corresponding Dirichlet problem. The scope of this course is to present an overview of: the classical mean value formulas for the Laplacian; the asymptotic mean value properties for the infinity-Laplacian and the p-Laplacian and their connections to viscosity solutions.