# intro

## key points

- we incentivize **early prediction accuracy** by giving more weight to such submissions.
- we introduce a **differentiation mechanism** to reward more top miners along the obvious fact that (in terms of Brier score) an accuracy of 0.80 is much better than 0.76 - this mechanism is currently **linear**. It will be made exponential soon after to also reflect the also obvious fact that improving from 0.89 to 0.90 is exponentially harder than improving from 0.75 to 0.76.

## definitions

Let K be the amount of miners, E an event, $I$ the first time $E$ is listed on our network, D the cutoff, and $T$ a threshold.
Let $S(p,o_E)$ be the Brier score of a prediction $p$ sent for the event $E$ which resolution was $o_E$ (1 if it occured and 0 otherwise).
Let $(p_{t,k})_{t \leq D}$ be the time series of all the predictions submitted by miner $k$ for $E$ before cutoff.

# The scoring rule

We want each prediction to have a corresponding weight depending on the time of the submission $t$. Let then $w_{t,D}$ be the weight of the prediction $p_{t,k}$. It is a function of the submission date $t$ and the cutoff date $D$.

When the event $E$ resolves we compute the Brier score of each submission and we multiply this score by the corresponding weight. We then obtain the following weighted time series:
$(w_{t,D} \times S(p_{t,k}, o_E))_{t \leq D}$

The final scoring rule is obtained by averaging this quantity. We detail all the steps below as well as add an additional discretization step.

## weights

We choose exponentially decreasing weights along the intuition that predicting gets exponentially harder as one goes back in time.

We divide the time segment $[I,D]$ into $n$ intervals $[t_j, t_{j-1}]$ of equal length (currently 4 hours). Then for the interval $[t_j, t_{j-1}]$ we set the weight $w_{j,D} = e^{-\frac{n}{j}+1}$ where $t_n = I$ and $t_0 = D$ and where $j$ decreases from $n$ to $1$.

## averaging per window

Each $p_j$ is the arithmetic average of the miner's predictions in a given time window $[t_j, t_{j+1}]$ i.e

$p_j = \frac{\sum_{t' \in [t_j, t_{j+1}]} p_{t'}}{\sum_{t' \in [t_j, t_{j+1}]} 1}$

## main scoring component

Given our time series $(w_{j,D} \times S(p_{j,k}, o_E))_{j \leq n}$
we compute the time weighted average $m_k$:

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

<!--
**Penalising low information.**
Sending $\frac{1}{2}$ does not bring any information so we penalize that with:
$m_{k,E} = \max(m_k - 0.75,0)$ -->

<!--
## scoring component 2

Among the new scores of the miners $(m_{i,E})_{k \leq K}$, we identify the minimum $\min m_{i,E}$ and the maximum $\max(m_{i,E})$. We then compute a bonus term:

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

-->

## the full scoring rule

We add an exponential to obtain the final scoring rule with parameters $\alpha$:

$$\exp(\alpha \times m_{k,E})$$

We have $\alpha$ set currently close to $25$ so that top miners close to $0.9$ average Brier score would be able to get **10%** of the reward. We expect to increase $\alpha$ moving forward.

If one then denotes:
$L_k = \exp(\alpha*m_{k,E})$

We normalise the scores across all miners using the $l1$ norm:

$s_k = \frac{L_k}{\sum L_j}$

We increment a daily average $s_{ki}$ denotes the above score in the case of event $i$, we have $S_{daily} = \frac{1}{n} \sum_k s_{ki}$

This resets every 24 hours.

We continuously compute an EMA

$\alpha S_{daily} + (1- \alpha) S_{prev}$