Binomial distribution $\pi$

For a binomial distribution $\pi$ with $\pi\in\{\pi_1,\pi_2\}$ , $p(\pi_1)=p, p(\pi_2)=q$ gain $G$ can be readily found:

$$\alpha_\mathrm{opt} = \frac{-p \pi _1-q \pi _2}{\pi _1 \pi _2}$$ (13)

and

$$G[\pi] = p \log \left(p\,\frac{\pi_2-\pi_1}{\pi_2}\right)+q\,\log\left(q\frac{\pi_2-\pi_1}{-\pi_1}\right)$$ (14)

The fair odds distribution $\tilde\pi$ on $\{\pi_1,\pi_2\}$ , for which ${\mathsf{E}}\tilde\pi=0$ is

$$\tilde p = p(\pi_1)=\frac{\pi_2}{\pi_2-\pi_1}, \tilde q = p(\pi_2)=\frac{-\pi_1}{\pi_2-\pi_1}$$ (15)

And therefore

$$G[\pi] = p \log\frac{p}{\tilde p} + q \log\frac{q}{\tilde q} = \mathrm{KL}(\tilde q\vert\vert p)$$ (16)