  We propose a new version of the tug-of-war game and a corresponding dynamic
programming principle related to the $p$-Laplacian with $1<p<2$. For this
version, the asymptotic H\"older continuity of solutions can be directly
derived from recent Krylov-Safonov type regularity results in the singular
case. Moreover, existence of a measurable solution can be obtained without
using boundary corrections. We also establish a comparison principle.
