I have a complete information static game with a concave twice differentiable objective function $F_i(x_1,...,x_n;\beta)$ with player $i=1...n$ choosing non negative arguments $x_i\geq 0$, where $\beta$ are parameters. The system of necessary and sufficient first order conditions are:
$f_1(x_1,...,x_n;\beta)=0,....,f_n(x_1,...,x_n;\beta)=0$
I know that this system of equations has a unique solution (unique pure strategy Nash equilibrium), implying that I can write each $x_i=h_i(\beta)$, where $h_i$ is a function.
Now I add a constant $\alpha>0$ to each equation:
$f_1(x_1,...,x_n;\beta)+\alpha=0,....,f_n(x_1,...,x_n;\beta)+\alpha=0$
My question is whether this system still has a unique solution, meaning whether I can write $x_i=g_i(\beta,\alpha)$, where $g_i$ is a function.
I have tried this on a few games like Cournot oligopoly and I'm still looking for some counterexample. I'm unsure of how to go about a proof if it's correct as working with fixed points is difficult.
(This is self-study, not a homework question).