How to find Nash equilibria through KKT conditions (convex optimization)?

I have a game defined by the following utility functions (example with two agents):

$v_1 = u(w_1 - x_1 + \alpha(x_1 + x_2)) + \beta_{12} u(w_2 - x_2 + \alpha(x_1 + x_2))$ $v_2 = u(w_2 - x_2 + \alpha(x_1 + x_2)) + \beta_{21} u(w_1 - x_1 + \alpha(x_1 + x_2))$

where, $x_1, x_2 \geq 0$, $x_1 \leq w_1, x_2 \leq w_2$, $\alpha \in (0.5, 1)$, $\beta_{12}\in (0,1)$ and $\beta_{21}\in (0,1)$. $u$ is a concave, twice differentiable function. Individuals have to choose contribution $x_i$ to maximize utility.

It can be actually generalised for $n$ players as $\boldsymbol{v} = \boldsymbol{B}\boldsymbol{u}(\boldsymbol{w} + \boldsymbol{A}\boldsymbol{x})$. Because $u$ is concave, finding pure strategy Nash equilibria should be a convex optimization problem, and I should be able to use KKT conditions to solve it (see Theorem 2.7 here).

Here is where I start to be insecure and would need some guidance.

(1) How exactly would the KKT conditions look like?

• what would be the objective function? a sum of the individual utilities?
• What are the constraints?

(2) How come that Nash equilibria are the local maxima of the objective function?

I am sure there are some cooler convex optimization techniques to do this, if you can give a precise example of how to apply them in this case, great.

Mathematica or R examples are welcome.

• In general, finding a Nash equilibrium is a fixed point problem. Each agent is supposed to maximise his/her utility function, given opponents' choices. – mlc Nov 19 '17 at 22:21
• @mlc What is the practical implication of it being a "fixed point problem"? I understand the individual motivations here, maybe you could elaborate on how it can get me on the right path regarding my question? Thanks. – František Kaláb Nov 20 '17 at 10:54