Exponential Absolute Minimizing extension and biased infinity Laplacian

We study the variational structure of the biased infinity Laplacian by introducing a notion of the $\beta$\textit{-Exponential Absolute Minimizing Extension} ($\beta$--AM) on arbitrary length space, which absolutely minimizing the exponential slope $$L^{\beta}_u (E) := \beta \sup_{x,y \in E} \frac{u(y) - e^{-\beta |x-y|} u(x)}{1- e^{-\beta |x-y|}}.$$We also define the corresponding Exponential McShane-Whitney-type extension, and $\beta$-biased convexity, which equivalently characterize $\beta$-AM and may be of independent interest. These generalize the classical Absolute Minimizing Lipschitz Extension as a special case when $\beta = 0$. In Euclidean space with Euclidean norm, this corresponds to the Aronsson equation with Hamiltonian $$H(u, \nabla u) = |\nabla u| + \beta u,$$ equivalently viscosity solutions of $\Delta_{\infty}^{\beta} u = 0$. We show that $\beta$-AM arises as the continuum value of a biased tug-of-war game. Analogous to the unbiased case, we derive various properties of this extension. As an application, we further show that the linear blow-up property holds for biased infinity harmonic functions.