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:

$B_k=\exp (20\ast m_k)$
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