<p>In a generalized model of a stochastic common property differential game problem, we present a more flexible framework that extends the model to handle natural growth, price, and cost functions across a wider range of parameters. Unlike previous studies in the literature, we do not prioritize parsimony. Instead, we consider nonlinear resource stock dynamics and non-quadratic payoff functions, focusing on monopoly, duopoly, and oligopoly games with a feedback information structure. The choice of model parameters significantly influences the nature and analytical behavior of the solutions to the nonlinear Hamilton–Jacobi–Bellman PDEs (ODEs), consequently affecting Nash equilibrium strategies for finite- (infinite-) horizon planning. Due to the lack of closed-form solutions, we employ a collocation method based on the fractional-order shifted Chelyshkov polynomials to solve the nonlinear HJB PDEs (ODEs) in the generalized stochastic common property differential game problem with finite- (infinite-) horizons. Numerical results demonstrate that different parameter choices in the approximate solutions based on the fractional-order shifted Chelyshkov polynomials greatly affect the performance of the collocation method. Furthermore, the results show that the firms’ optimal extraction plans are strongly influenced by the natural growth, price, and cost functions with respect to different parameters.</p>