A Projected Tug-of-War Game for the Regularized $p$-Laplacian

We give a tug-of-war interpretation of the regularized $p$-Laplacian $\operatorname{div}\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $\Omega \subset \mathbb{R}^n$, $p\ge 2$.

The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $\Delta_p w=0$ in $\mathbb{R}^n+1$.

Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\mathbb{R}^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$.