System Documentation - POLY IMPULSE

DOCUMENTATION

Tiers & Algorithm

Season 1 Season 0

Global Tiers Requirement

DEITY

Rank 1

(Min: 77,297 pts)

DEMIGOD

Rank 2 - 3

(Min: 68,328 pts)

CELESTIAL

Rank 4 - 10

(Min: 17,153 pts)

ETHERIAL

Rank 11 - 25

(Min: 6,395 pts)

ASCENDANT

Rank 26 - 50

(Min: 3,177 pts)

ACOLYTE

Rank 51 - 100

(Min: 1,318 pts)

UNRANKED

Rank 101+

(Min: 0 pts)

Points System Algorithm

The system awards points for both improvements in rank and improvements in time. Scores are awarded between 10000 points (for the rank 1 player) and 0 points.

It calculates how many "adjusted deviations" away from the world record a player's time is. A "leaderboard volatility" metric is calculated for each map. Stages with a high spread place greater importance on rank; maps with a small spread emphasize time.

Inputs

$R$ is the player's rank.
$t$ is the player's time.
$\mathbf{T} = (T_1, T_2, \dots, T_{100})$ is the vector of the top 100 times on a given map.

Parameters

$\alpha = 1.48$ is the points curvature parameter.
$\beta = 5.5$ is the weight parameter.
$\gamma = 1.22$ is the deviation adjustment parameter.
$\kappa = 0.38$ is the rank scaling parameter.
$\rho = 0.82$ is the rank decay parameter.
$w_0 = 0.4$ is the baseline weight parameter (lowest possible weight on ranking is 40%).

Algorithm Steps

1. Compute the standard deviation $ \sigma $ of $\mathbf{T}$.

$$ \sigma = \sqrt{\frac{1}{100} \sum_{j=1}^{100} (T_j - \mu)^2} $$

where $ \mu = \frac{1}{100} \sum_{j=1}^{100} t_i $ is the average time on the leaderboard.

(1)

2. Compute the standardised volatility $ V $:

$$ V = \frac{\sigma}{T_1} $$

where $ T_1 $ is the minimum time on the leaderboard (i.e. the world record).

(2)

3. Compute the transformed time deviation metric $ M $:

$$ M = \gamma \times \frac{t - T_1}{\sigma} $$

(3)

4. Compute the rank adjustment $ Q $:

$$ Q = \kappa (R - 1)^\rho $$

(4)

5. Compute the dynamic rank weight $ \omega $:

$$ \omega = 1 - \left( V + [1 - w_0]^{-\frac{1}{\beta}} \right)^{-\beta} $$

(5)

6. Compute the weighted deviation metric $ D $:

$$ D = \omega Q + (1 - \omega)M $$

(6)

7. Finally, calculate the player's points $ P $:

$$ P = \text{floor}\left( 10000 \times \left[ 1 - \left( 0.5 + 0.5 \, \text{erf}\left( \frac{\ln(D)}{\alpha} \right) \right) \right] \right) $$

Note that $ \text{erf} $ is the Gaussian Error function.

(7)