We derive a non-cooperative and cooperative strategies and state trajectories for a finite-horizon multistage game of renewable resource extraction with asymmetric players. Assuming transferable utility we extend the subgame perfect core concept introduced for extensive-form games to the class of $n$-person multistage games and specify an algorithm for choosing a unique payoff distribution procedure from the core in a two-player game. This quasi proportional payment schedule satisfies several good properties and could be applied to implement a cooperative solution based on the maximization of the relative benefit from cooperation (or the value of cooperation). We provide a numerical example to demonstrate the properties of the obtained solutions and the algorithm implementation.