announces his initial capital ${{\cal K}}_0=1$.\
> FOR $n=1,2,\ldots$:\
> Forecaster announces $p_n\in (0,1)$.\
> Skeptic announces $M_n\in\mathbb{R}$.\
> Reality announces $x_n\in \{0,1\}$.\
> ${{\cal K}}_n:={{\cal K}}_{n-1}+M_n(x_n-p_n)$.\
> **Collateral Duty**: Skeptic must keep ${{\cal K}}_n$ non-negative.

Forecaster is supposed to decide its forecast $p_n$ based on all relevant side information available at the time of announcement. We modify the above protocol so that Forecaster also discloses the relevant side information $c_n$, which is a $d$-dimensional column vector, together with the probability $p_n$. Furthermore we define auxiliary capital processes ${{\cal S}}_n$ and ${{\cal V}}_n$.

> [Binary Probability Forecasting With Side Information]{} (BPFSI)\
> **Protocol**:\
> ${{\cal K}}_0:=1, {{\cal S}}_0:=0, {{\cal V}}_0:=0$.\
> FOR $n=1,2,\ldots$:\
> Forecaster announces $p_n\in (0,1)$ and $c_n\in \mathbb{R}^d$.\
> Skeptic announces $M_n$.\
> Reality announces $x_n\in \{0,1\}$.\
> ${{\cal K}}_n:={{\cal K}}_{n-1}+M_n(x_n-p_n)$.\
> ${{\cal S}}_n:={{\cal S}}_{n-1}+c_n(x_n-p_n)$.\
> ${{\cal V}}_n:={{\cal V}}_{n-1}+ c_n c_n{'}\; p_n(1-p_n)$.\
> **Collateral Duty**: Skeptic must keep ${{\cal K}}_n$ non-negative.

In the protocol, $c_n{'}$ denotes the transpose of $c_n$, ${{\cal K}}_n$ is a scalar, ${{\cal S}}_n$ is a $d$-dimensional column vector and ${{\cal V}}_n$ is a $d\times d$ symmetric matrix.

If $d=1$ and $c_n\equiv 1$, then ${{\cal S}}_n = \sum_{i=1}^n (x_i - p_i)$. When we study 