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A classical solution to a partial differential equation is a suitably smooth function that satisfies an equation pointwise in a domain. However, many equations that appear in applications admit no such solutions and therefore the notion of solution needs to be extended. One such extension is achieved by integration by parts in the theory of distributional weak solutions. Another class of extended solutions is the viscosity solutions defined by generalized pointwise derivatives. If both viscosity and weak solutions can be meaningfully defined, it is natural to ask whether they coincide. This dissertation studies the equivalence of solutions to different equations related to the p-Laplacian and stochastic tug-of-war games.