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We consider the (viscosity) solution $u^\varepsilon$ of the elliptic equation$\varepsilon^2\Delta_p^G u= u$ in a domain (not necessarily bounded),satisfying $u=1$ on its boundary. Here, $\Delta_p^G$ is the {\it game-theoreticor normalized $p$-laplacian}. We derive asymptotic formulas for $\varepsilon\to0^+$ involving the values of $u^\varepsilon$, in the spirit of Varadhan's work\cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizingthat obtained in \cite{MS-AM} for $p=q=2$. As in a related parabolic problem,investigated in \cite{BM}, we link the relevant asymptotic behavior to thegeometry of the domain.

Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian