Summary: In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the \(h\) degree homogeneous \(p\)-Laplacian \[ \begin{cases} \begin{aligned} &|Du|^{h-1} \Delta_p^N u+ \lambda a(x)|u|^{h-1}u = 0, \quad \text{in }\Omega,\\ &u = 0, \quad \text{on } \partial\Omega. \end{aligned} \end{cases} \] Here \(a(x)\) is a positive continuous bounded function in the closure of \(\Omega\subset \mathbb{R}^n\) (\(n\geq 2\)), \(h>1\), \(2< p<\infty\), and \(\Delta_p^N u = \frac{1}{p}|Du|^{2-p} \text{div} (|Du|^{p-2}Du)\) is the normalized version of the \(p\)-Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue \(\lambda_\Omega\), which is positive and has a corresponding positive eigenfunction for \(p>n\). The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter \(\lambda<\lambda_\Omega\), we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.