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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)+Î»(a-x-y) and L2=g(x,y)+Î»(a-x-y)? (i.e., Î» 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