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.