# Rosen's uniqueness theorem for concave payoff functions and linear inequality constraints

Can Rosen's existence and uniqueness of equilibrium theory be applied to a game with $n$ players with each player $i$ having the following payoff maximization model:

$$\max f(x_i,x_{-i})$$ $$\text{s.t.}$$ $$x_i \le \bar{x}$$

where $f$ is a strictly concave function. And the constraints are decoupled. Each $x_i$ is smaller than the same scalar, $\bar{x}$.

Can I use Rosen's conditions to prove that there is a unique Nash equilibrium to this game? Thank you.

By counter example, no. Let's compactify strategy spaces by supposing for all $i$, $x_i \in [0, 1]$. Suppose utilities are given by: $$f_i(x_i ,x_{-i}) = x_i\sqrt{x_{-i}} -\frac{x_i^2}{2}$$ Then best replies are given by: $$x_i^*(x_{-i}) = \sqrt{x_{-i}}$$ yielding pure strategy equilibria at $(0,0)$ and $(1,1)$.