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Here, $\\Delta_p^G$ is the game-theoretic $p$-laplacian, a $1$-homogeneous version of the standard $p$-laplacian. Also, we consider the (viscosity) solution $u^\\varepsilon$ of the nonlinear elliptic equation $\\varepsilon^2\\Delta_p^G u^\\varepsilon= u^\\varepsilon$ in $\\Omega$, satisfying $u^\\varepsilon=1$ on its boundary. 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