Help with shared constraints (game theory)

[ruzbayhhi]

I've been struggling with shared constraints problems for a while now. I have a game between two players with a shared constraint. For example, player 1 is trying to maximize f(x,y) by choosing x, and player 2 is trying to maximize g(x,y) by choosing y. The players are competing in a non-cooperative game.
There is a shared constraint such that x+y<= a.

If I understand the source below correctly (p. 18 onward), what I should do is form a Lagrangian: L1 = f(x,y)+λ(a-x-y) and L2=g(x,y)+λ(a-x-y)? (i.e., λ for player 1 is the same as for player 2) and then solve the FOC? Is this the right way to approach this?

Reference: http://www.dis.uniroma1.it/~facchinei/didattica/giochi/Lettura_3.pdf

 

[fresh_42]

It is one way to approach it since both share this constraint. There might be other approaches if $f,g$ are known, i.e. have a certain form as linearity.