Abstract: Infinity Laplacian has received a lot of attractions in the past two decades not only because it is nonlinear, non-variational and highly degenerate but also it has many applications in optimal mass transportation, image processing differential games etc. In this talk, we focus on the biased infinity Laplacian arising from the biased tug-of-war game. We are interested in the eigenvalue problem of this operator. Due to the nonlinear structure, we first give the appropriate characterization of the eigenvalue based on the maximum principle. During this process, the Lipschitz estimate and Harnack inequality are established. By compactness criterion, we obtain the existence of the principal eigenvalue and the corresponding positive eigenfunctions. As an application, we give the decay estimate of the related evolutionary equation.