# Fixed Points of Logistic Function

ModelingStatisticsposted by John Cook June 15, 2018 John Cook

Here’s an interesting problem that came out of a logistic regression application. The input variable was between 0 and 1, and someone asked when and where the logistic transformation

*f*(*x*) = 1/(1 + exp(*a* + *bx*))

has a **fixed point**, i.e. *f*(*x*) = *x*.

So given logistic regression parameters *a* and *b*, when does the logistic curve given by *y* = *f*(*x*) cross the line *y* = *x*? Do they always cross? Can they cross twice?

There’s always at least one solution. Because *f*(*x*) is strictly between 0 and 1, the function

*g*(*x*) = *f*(*x*) – *x*

is positive at 0 and negative at 1, and by the intermediate value theorem g(*x*) must be zero for some *x* between 0 and 1.

Sometimes *f* has only one fixed point. It may have two or three fixed points, as demonstrated in the graph below. The case of two fixed points is unstable: the logistic curve is tangent to the line *y* = *x* at one point, and a tiny change would turn this tangent point into either no crossing or two crossings.

If |*b*| < 1, then you can show that the function *f* is a **contraction map** on [0, 1]. In that case there is a unique solution to *f*(*x*) = *x*, and you can find it by starting with an arbitrary value for *x* and repeatedly applying *f* to it. For example, if *a*= 1 and *b* = 0.8 and we start with *x*= 0, after applying *f* ten times we get *x* = *f*(*x*) = 0.233790157.

There are a couple questions left to finish this up. How can you tell from *a* and *b* how many fixed points there will be? The condition |*b*| < 1 is sufficient for *f* to be a contraction map on [0, 1]. Can you find necessary and sufficient conditions?

*Original Source*