title: Splitting a game into 2 separate games
body: <p>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!!!!</p>