Problem $3($Joint Project$)$

Two people are engaged in a joint project. If each person i puts in the effort $x_{i}in[0,1],$ the outcome of the project is worth $f(x_1,x_2)$ and the cost to player $i$ is $c(x_i).$ The worth of the project is split equally between the two people, regardless of their effort levels. Each player aims to maximize the value they get from the project, i$.$e$.,f\frac{x_1,x_2}{2},$ minus the cost.

$($a$)$ Formulate this situation as a strategic game.

$($b$)$ Find the set of pure strategy Nash equilibria for each of the following cases:

$1$

i$.f(x_1,x_2)=3x_1{x}_2,c_i(x_i)=x_i^2,i=1,2$

ii$.f(x_1,x_2)=4x_1{x}_2,c_i(x_i)=x_i,i=1,2$