Solving expected value problem with logistic function

  1. 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 probabilty 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:

    [tex] z = p(x) \lambda x - (1-p(x)) x [/tex]

    [tex] z = \frac{\lambda x}{1+ e^{-a-bx} } + \frac{x}{1+ e^{-a-bx} } -x[/tex]


    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?
     
  2. jcsd
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