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!!!!