I have two utility functions, $u_1(c,s)$ and $u_2(c,s)$. They are derivable. $u_1(c,s)$ refers to Player 1, while $u_2(c,s)$ to Player 2. The strategic leverage of Player 1 is $s$ while the strategic leverage of Player 2 is $c$.
In order to obtain the best response functions and Nash equilibria, I have to solve the system of the following two equations: $$\frac{\partial }{\partial s}u_1(c,s)=0$$ $$\frac{\partial }{\partial c}u_2(c,s)=0$$ From this I obtain both best response functions, i.e. $$s=f_1(c)$$ $$c=f_1(s)$$ If I plot them in a (c,s)-plane, or solve the above system, I obtain two Nash equilibria.
Now, I'm not an expert in game theory, so please apologize if the question may seem naive or denote a lack of knowledge of the theory.
I would like to know if the game converges to these equilibria, or if it converges to only one of them. I do not know if in this game, from where I was standing, it is possible to ask a similar question or it lacks something in the definition of the model. Example, textbook and other are welcome.
Thanks in advance!