The authors study the properties of the functions that satisfy the following identity \[ u_{\epsilon}(x)=\frac{\alpha}{2}\,\left\{ \sup_{\bar{B}_{\epsilon}(x)} u_{\epsilon}+ \inf_{\bar{B}_{\epsilon}(x)} u_{\epsilon}\right\} +\beta\, \frac{1}{|B_{\epsilon}(x)|}\int_{B_{\epsilon}(x) }u_{\epsilon} \,dy \] where \(\epsilon>0\) and \(\alpha\) and \( \beta\) are non negative coefficients such that \(\alpha+\beta=1\). They show that these functions are uniquely determined by their boundary values, approximate \(p\)-harmonic functions for \(2\leq p<\infty\) (for a suitable choice of \(p\) which depends on \(\alpha\) and \(\beta\)) and satisfy the strong comparison principle.
Moreover, their relation with the theory of tug-of-war games with noise is analyzed.