We give a self-contained and short proof for the existence, uniqueness and measurability of the $p$-harmonious functions in the case $2\le p < \infty$. The proofs only use elementary analytic tools. As a consequence, we obtain existence, uniqueness and measurability of value functions for the tug-of-war game with noise.
"On the existence and uniqueness of $p$-harmonious functions." Differential Integral Equations 27 (3/4) 201 - 216, March/April 2014.