ArenaRating

# Arena Rating System

This document is valid for Arena competitions on or after August 6, 2023.

## Arena Rating

The rating $R$ is determined by the number $r$ related to the rating, which is:

$R = \begin{cases} r & \text{if } r\ge 400 \\ \max \left\{ 1, \frac{400}{\exp \left(\frac{400-r}{400}\right)} \right\} & \text{if } r<400\end{cases}$

Therefore, negative ratings do not exist, and if you have entered the arena at least once, your rating will be an integer greater than $1$.

### Expected Performance

A competitor's expected performance before the start of the arena $E$ is a weighted average of the competitor's performances in several previous competitions.

#### If you have not participated in any Arena contests before

The initial $E$ is based on the participant's AC Rating $A$.

$E = 1\,000 + \left\lfloor \frac{A}{2} \right\rfloor$

If the participant has no AC rating, this value is $1\,000$.

We only use this value if we have never participated in an arena contest, so after the first contest, your AC rating will not affect the performance calculations.

#### If there is an arena contest you have participated in before

$E$ is the weighted average of the participant's performance in all arenas in which they have previously participated.

Suppose a participant has participated in $N$ arenas. Among the participating arenas, if the value related to the performance described below was $p_i$ in the $i$-th arena in the latest starting time, and the starting time of that arena was $T_i$, For competitions starting at start time $C$, the weight $W_i$ for the $i$th competition is calculated as follows:

$W_i = \min\left\{0.8^i, 0.25^{\left[\left(C-T_i\right)/365\text{ days}\right]}\right\}$

Therefore, the more recently played competitions, the higher the percentage of them reflected in expected performance. With this weight, $E$ is the weighted average of the values $p_i$ related to the performance of previous arenas.

$E = \frac{\sum_{i=1}^{N} p_i\times W_i}{\sum_{i=1}^{N} W_i}$

### Actual Performance

A participant's actual performance $P$ after the arena is over will be calculated from the participant's ranking and the expected performances of other participants.

If the expected performance of the $i$th participant is $E_i$ and there are $t$ participants above $k$, their performance-related value $p_i$ is the value of $X$ satisfying the following expression:

$\sum_{i=1}^{N} \frac{1}{1+10^{\left(X-E_i\right)/400}} = k-1+0.5t$

Depending on the difficulty of the competition, there is an upper limit on actual performance. To be precise, actual performance is determined by the smaller value of the value related to performance and the constant $B$ related to the difficulty of the competition.

$P_i = \min\left\{p_i, B\right\}$

The value $B$ is set according to the maximum value of the participation range of the competition.

Rated range upper bound$B$
X$\infty$
SSS ~ SSS+$3\,400$
SS ~ SS+$3\,000$
S ~ S+$2\,600$
A ~ A+$2\,200$
C ~ B+$1\,800$

### Rating

The rating calculation is similar to the expected performance calculation, but the weight calculation time for each arena contest is based on the time of the last arena you participated in. In other words, the weight formula from the expected performance is used as it is, but it is calculated with $C=T_1$. Let us denote the weight of the newly calculated $i$ arena as $W_i^\prime$.

Also, to make the rating start at $0$ for the first few arenas, we subtract a function $d$ from the rating that starts at $1\,200$ and gradually converges to $0$ depending on the number of participating arenas. $d$ is calculated as:

$d\left(N\right) = 1\,800 \times \frac{\sqrt{\sum_{i=1}^{N} 0.64^i}}{\sum_{i=1}^{N} 0.8^i} - 600$

In addition, the following performance correction function $c$ is placed with the intention of making a large difference between very good and moderately good performance, but reducing the difference between moderately bad and very bad performance.

$c(P) = 2^{P/800}$

Taken together, the number $r$ related to the rating is calculated as:

$r = c^{-1} \left(\frac{\sum_{i=1}^k c\left(P_i\right) \times W_i^{\prime}}{\sum_{i=1}^k W_i^{\prime}}\right) - d\left(N\right)$

## Arena tier

Your Arena Tier is determined by your Arena Rating. These values are likely to change several times during the early contests of the Arena system.

TierRating
X3000 -
SSS+2600 - 2999
SSS2400 - 2599
SS+2200 - 2399
SS2000 - 2199
S+1800 - 1999
S1600 - 1799
A+1400 - 1599
A1200 - 1399
B+1000 - 1199
B800 - 999
C+400 - 799
C1 - 399