# Splitting a game into $2$ separate games

I have a mathematical problem which I would be grateful if anyone can guide me through.

I have a game with a utility function like ($$f(x)+f(y)$$- $$x$$ and $$y$$ are independent) which has got unique nash equilibrium. However, 'cause the utility function of each user is not symmetric, I can not prove whether Best response dynamics converges to it's nash or not. But, I can split the game into $$2$$ completely distinct and independent games, one of them is spermodular ($$f(x)$$) and has got unique nash, the other one is symmetric and has got unique nash ($$g(y)$$), too. Both games' utility functions ($$f(x)$$ and $$g(y)$$) are strictly concave and $$f(x)+g(y)$$ is also strictly concave. Is it possible to conclude that in order to find the unique nash of the game $$f(x)+g(y)$$ it is sufficient to find the unique nash of the $$2$$ games $$f(x)$$ and $$g(y)$$ and then add them together? Is there a theorem or proof for that? Thanks in advance!!!!

• Hi @ZARA and welcome to MSE! I think the question would be more attractive if you used $\LaTeX$, there is a fantastic tutorial here: math.meta.stackexchange.com/questions/5020/… – Dr. Mathva Mar 10 at 13:23
• For your question, it suffices to set $symbols before and after each mathematical expression such as f(x)for instance. You could so change f(x)by$f(x)\$ – Dr. Mathva Mar 10 at 13:24