criterion” ([@kelly-criterion-book], [@kelly-1956]) to determine $\nu_n$ so as to maximize the expected value of the logarithm of the capital growth under $\hat p_n$: $$\nu_n \ : \ E_{\hat p_n}[\log(1+\nu(x_n-p_n))]
\ \rightarrow \ \max.$$ Writing $$E_{\hat p_n}[\log(1+\nu(x_n-p_n))]
=\hat p_n \log (1+\nu(1-p_n)) + (1-\hat p_n) \log (1-\nu p_n)$$ and differentiating this with respect to $\nu$, the unique maximizer $\nu_n$ is obtained as $$\label{eq:nu_n}
\nu_n = \frac{\hat p_n - p_n}{p_n(1-p_n)} = \frac{\hat p_n}{p_n} - \frac{1-\hat p_n}{1-p_n}.$$ With this choice of $\nu_n$ we have $$\begin{aligned}
1+\nu_n (x_n-p_n) &= \begin{cases} \hat p_n/p_n  & \text{if} \ x_n=1\\
                                (1-\hat p_n)/(1-p_n) & \text{if}\ x_n=0
                              \end{cases}\\
&=\frac{\hat p_n^{x_n} (1-\hat p_n)^{1-x_n}}{p_n^{x_n} (1-p_n)^{1-x_n}} .\end{aligned}$$ Hence is written as $${{\cal K}}_n = \frac{\prod_{i=1}^n \hat p_i^{x_i} (1-\hat p_i)^{1-x_i}}
        {\prod_{i=1}^n p_i^{x_i} (1-p_i)^{1-x_i}}.$$

In the case that Skeptic models the joint probability $\hat p(x_1, \dots, x_n)$ of Reality’s moves, $\hat p_n$ is given as the conditional probability $$\hat p_n = \frac{\hat p(x_1, \dots, x_{n-1},1)}{\hat p(x_1,\dots, x_{n-1})}.$$ In this case $$\hat p_n^{x_n} (1-\hat p_n)^{1-x_n} = \frac{\hat p(x_1, \dots, x_{n-1},x_n)}{\hat p(x_1,\dots, x_{n-1})}, \qquad
x_n=0,1,$$ and ${{\cal K}}_n$ is written as $$\label{eq:joint-probability-formulation}
{{\cal K}}_n = \frac{\hat p (x_1, \dots, x_n)}{\prod_{i=1}^n p_i^{x_i}(1-p_i)^{1-x_i}}.$$

For the rest of this section we introduce some terminology of game-theoretic probability. An infinite sequence of 