Principal Agent Question Private Benefit
If we try to directly optimize with respect to $t$, given this system (two unknwowns, one equation), we get a nonsensical result.

$$\frac{\partial P}{\partial t}: -s = 0$$
Which may not be true, and I'm assuming you are taking $s$ as given, not something to choose. You can only optimize with respect to $x$.

$$\frac{\partial P}{\partial x} = 1 - sx = 0 \implies \boxed{x^* = \frac{1}{s}}$$

What does this tell you about the optimal value of $t$ if anything?