Let $x_n=0,1$ be the indicator variable for the event, i.e., $x_n=1$ if the event occurs and $x_n=0$ otherwise. We suppose that a player “Reality” decides the binary outcome $x_n$. When Forecaster announces $p_n$, it also sells a ticket with the price of $p_n$ per ticket. The ticket pays one monetary unit when the event occurs in day $n$, i.e., the value of the ticket at the end of day $n$ is $x_n$. A bettor or gambler, called “Skeptic”, buys $M_n$ tickets with the price of $p_n$ per ticket. Then the payoff to Skeptic in day $n$ is $M_n (x_n - p_n)$. We allow $M_n$ to be negative, so that Skeptic can bet also on the non-occurrence of the event. If the probability announced by the agency is appropriate, it is hard for Skeptic to make money in this game. On the other hand, if the probability is biased in some way, Skeptic may be able to increase his capital denoted by ${{\cal K}}_n$. Hence we can measure the quality of probability forecasting in terms of ${{\cal K}}_n$.

We now give a protocol of the game, following the notational convention of [@ShaVov01].

> [Binary Probability Forecasting]{} (BPF)\
> **Protocol**:\
> Skeptic 