The following approach of finding solutions to a partial differential equation Lu=0, proved to be quite versatile:

(i) develop an asymptotic expansion of a suitable family of averaging operators on u; the operators are parametrized by the radius \epsilon of averaging;

(ii) study the related mean value equation by removing higher order terms in the expansion;

(iii) interpret the mean value equation as the dynamic programming principle of a two-player game incorporating deterministic and stochastic components;

(iv) pass to the limit in the radius of averaging \epsilon, in order to recover solutions to Lu=0 from the values of the game process.

In my talk, I will explain this approach in the following contexts: p-Laplacian; non-local geometric p-Laplacian; Robin boundary conditions; and weighted Laplace-Beltrami operator on a manifold. In each case, finding the appropriate averaging principle and the related game, is the key starting point.