## exponential component (current update)

Recall that given the time series of the miner's predictions $(p_t)$ we are scoring the miner at settlement through exponentially decreasing weights $(w_t)$:

$$m_k = \frac{\sum_{j} w_{j,D} \times S(p_{j,k}, o_E)}{\sum_j w_{j,D}}$$

where $S$ is currently the Brier scoring rule.

Right now we are applying the following transformation to $m_k$:

$$B_{k} = \frac{m_{k} - \min(m_{i})}{\max(m_{i}) - \min(m_{i})}$$

and the score is a linear combination of $B_k$ and of $m_k$.

Instead now we will directly pass $m_k$ into an exponential:

$B_k = \exp(20*m_k)$
where $20$ is a parameter that might be adjusted later.

## logarithmic scoring (work in progress)

The problem with the Brier score is that it is not punishing mistaken confidence e.g sending 0 while the outcome is 1. It is always positive so the worst outcome for a miner is always zero.

Instead one can score a miner who predicted that the probability that a given event will happen is $p$:
- $1+\log(p)$ if the event occured
- $1+ \log(1-p)$ if not

The logarithm is equal minus infinity near zero so we would add a $\max(-, a)$ where $a$ is a negative number:
- $\max(1+\log(p), -0.5)$ if the event occured
- $\max(1+ \log(1-p), -0.5)$ if not