Fixed points of logistic function

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?

Related post: Sensitivity of logistic regression prediction