I am just recently studying game theory, and I am pretty confused about the derivation of the Best-response function in the case of the piecewise utility function with linear functional form where we just end up with a constant partial derivative. Can someone please help explain to me how we obtain the best response function $b_1(x_2)$ below from $u_1(x_1, x_2)$?
Example
Given $x_1, x_2\in [0,10]$, $u_1(x_1, x_2) = 0$ if $x_1 = 0$, $= 4x_1-4$ if $0<x_1\leq x_2$, $= 6x_2 - 2x_1 - 4$ if $x_1 > x_2$. Then we obtain $b_1(x_2) = \left\{0\right\}$ if $x_2 < 1$, $\left\{0, x_2\right\}$ if $x_2 = 1$ and $\left\{x_2\right\}$ if $x_2 > 1$.
What I am confused is that shouldn't $b_1(x_2) = \partial{u_1}/\partial{x_1} = 0$ if $x_1 = 0$, $=4$ if $0< x_1 \leq x_2$, $= -2$ if $x_1 > x_2$??
Further thought
We have: $\partial u_1/\partial x_1 = 0$ if $x_1=0$, $=4$ if $0<x_1\leq x_2$, $= -2$ if $x_1 > x_2$. Since $u_1(x_1=0, x_2) = 0$. Now, since $u_1$ is strictly increasing over the interval $0< x_1\leq x_2$, $u_1(x_1=x_2, x_2) = 4x_2 - 4$ is the maximum value. Finally, since $u_1(x_1, x_2)$ is strictly decreasing over the interval $x_1 > x_2$, $u_1(x_1, x_2)$ is max at $x_1\rightarrow x_2^{+} = 4x_2 - 4$.
Finally, due to the continuity and non-negativity requirements of utility function, we need $x_2\geq 1$ for the case of $x_1>0$. Furthermore, when $x_2 = 1$, then max $u_1(x_1, x_2=1) = 0$ over $0< x_1 < x_2$. But $u_1(x_1 = 0, x_2) = 0$. So $b_1(x_2) = \left\{0, x_2\right\}$ if $x_2 = 1$. Now, for $x_2> 1$, then $u_1(x_1=x_2,x_2) = 4x_2 - 4$ is the maximum, so $b_1(x_2) = \left\{x_2\right\}$ is the best-response function. Finally, if $x_2 < 1$, then $u_1(x_1, x_2<1) = 0$ is the maximum value of $u_1(x_1, x_2)$ for any $x_1$. Thus, $b_1(x_2) = 0$ if $x_2 < 1$.
Is this a correct argument??