I have an increasing function on $[0,1]$, $p \mapsto \Pi(p)$, that has the following properties. $$\Pi(0) = 1 - \Pi(1) = 0$$ $$\Pi(p) + \Pi(1-p) < 1 \quad \forall{p} \in (0,1)$$ $$\Pi(p) > p \quad \forall{p} \in (0,\varepsilon)$$ where $0 < \epsilon < 0.5$ is a given small number. Furthermore, $\Pi$ is smooth.
Is it possible to characterize the class of functions that satisfy these properties beyond these functional relations?