Solving expected value problem with logistic function

[tlonist]

I have an expected value problem where z is a desired expected value and I want to reach and x is an amount I can vary.

There is a probability of success based on a logistic function ρ(x) with a reward of λx and failure with a probability of (1-ρ(x)) and loss of x. I am trying to solve for the correct value of x to reach an expected value z.

So:

$$z = p(x) \lambda x - (1-p(x)) x$$

$$z = \frac{\lambda x}{1+ e^{-a-bx} } + \frac{x}{1+ e^{-a-bx} } -x$$


I tried solving in Matlab but it says there is no explicit solution and I haven't been able to solve by hand.

What would be the next course of action to solve this? Is there a way to simplify?

 

[fresh_42]

It looks as if $p(x)=\dfrac{1}{1+e^{-a-bx}}$ is a solution.