Summary: In a recent paper, the last three authors showed that a game-theoretic \(p\)-harmonic function \(v\) is characterized by an asymptotic mean value property with respect to a kind of mean value \(\nu_p^r[v](x)\) defined variationally on balls \(B_r(x)\). In this paper, in a domain \(\Omega \subset \mathbb{R}^N\), \(N\ge 2\), we consider the operator \(\mu_p^\varepsilon \), acting on continuous functions on \(\overline{\Omega } \), defined by the formula \(\mu_p^\varepsilon [v](x)=\nu^{r_{\varepsilon}(x)}_p[v](x)\), where \(r_\varepsilon (x)=\min [\varepsilon ,\operatorname{dist}(x,\Gamma )]\) and \(\Gamma\) denotes the boundary of \(\Omega\). We first derive various properties of \(\mu^\varepsilon_p\) such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function \(u^\varepsilon \in C(\overline{\Omega })\) satisfying the Dirichlet-type problem: \[ u(x)=\mu_p^\varepsilon [u](x)\text{ for every } x\in \Omega,\quad u=g \text{ on } \Gamma, \] for any given function \(g\in C(\Gamma )\). This result holds, if we assume the existence of a suitable notion of barrier for all points in \(\Gamma \). That \(u^\varepsilon\) is what we call the variational \(p\)-harmonious function with Dirichlet boundary data \(g\), and is obtained by means of a Perron-type method based on a comparison principle. We then show that the family \(\{ u^\varepsilon \}_{\varepsilon >0}\) gives an approximation for the viscosity solution \(u\in C(\overline{\Omega })\) of \[ \Delta_p^G u=0 \text{ in } \Omega, \quad u=g \text{ on } \Gamma, \] where \(\Delta_p^G\) is the so-called game-theoretic (or homogeneous) \(p\)-Laplace operator. In fact, we prove that \(u^\varepsilon\) converges to \(u\), uniformly on \(\overline{\Omega }\) as \(\varepsilon \rightarrow 0\).