Convexity of the expectation of boolean functions

Question

Let $$f:\{-1,1\}^n \to \{-1,1\}$$ be a monotone, odd ($f(-x)=-f(x)$) Boolean function.

Let $$F:[0,1]\to[0,1]$$ denote the probability that $f(x_1,...,x_n)$ where $x_1,...,x_n$ are i.i.d. $\pm1$ R.V. with probability $p$ to be 1.

Our question is the following:

Is it true that $F$ is convex in $[0,1/2]$ for any such monotone odd $f$?

(It is easy to see that $F$ is symmetric around $1/2$.)

Another way to view this is that $dF/dp$ is increasing in $[0,1/2]$, meaning the bits become more influential as $p$ is closer to $1/2$.

So far, in the example we worked out (such as majority) this turned out to be the case, and we could not find any references, nor managed to prove it or find a counterexample. A related fact - when I encountered "graphs" of the probability that certain monotone graph properties are satisfied in $G(n,p)$ as a function of $p$, they were always "drawn" as convex till the threshold and concave afterwards, but I could not find any explicit references to this phenomena.

Answer 1

By Russo's formula, $F'(p)$ coincides with the expected number of pivotal elements at $p$ (where a an element $x_i$ is pivotal for a configuration in $\{-1,1\}^n$ if changing the sign of $x_i$ also will change the sign of $f$). So if $F'$ is increasing on $[0,1/2]$, then we expect to see most pivotal elements at $p = 1/2$.

I don't think that this is true in general. Instead of 'majority', think of 'weighted majority' where $x_1$ has disproportionally large weight. Then intuitively $x_1$ will be pivotal for most values of $p$, but any other $x_i$ is more likely to be pivotal for $p$ close to $0$ or close to $1$, so we'd expect fewer pivotal elements at $1/2$ than closer to $0$ or $1$.

If I'm not mistaken, the following example exhibits this behaviour. Let $$f = \mathrm{sign}(5 x_1 + \sum _{i=2}^7 x_i)$$ Then $f$ is $1$ exactly when either $x_1 = 1$ and at least one of the other $x_i$ is $1$, or $x_1 = -1$ and all other $x_i$ are equal to $1$. Hence $$F(p) = p(1-(1-p)^6)+ (1-p)p^6$$ whose second derivative assumes negative values.