﻿ 3D Billiards by FlyOrDie - Rating system
Based on the rating system developed by Prof. Árpád Élő

The rating can be used to determine the winning odds of a specific player against another specific player. The system has been worked out according to statistical and probability theory.
The rating is an integer, the minimum is 0. The starting rate of an unknown player is 0.

Rating differenceProbability of win
00.50
1000.64
2000.76
4000.91
8000.99

Ratings are amended each time you finish a game, by adding or subtracting a small amount from your previous rating value. The amount added or subtracted depends on the result of the game and the rating of the players.

### Calculation

The following formula is used:
Rn = Ro + K(W-We)

Rn is the new rating.
Ro the old (pre-event) rating.
K a constant (32 for 0-2099, 24 for 2100-2399, 16 for 2400 and above).
W the score in the event - i.e. for a single match win=1; draw=0.5; loose=0.
We the expected score (Win Expectancy), from the following formula:
We = 1/ (exp10(-dr/400) + 1)
dr equals the difference in ratings.

This formula has the property that the sum of the rating changes is zero. It turns out that if the rating difference is more then 719 points, then if the strong player wins, there is no change in either rating.

### Rating Reduction

Overall ratings are decreased each day by an amount depending on the current rating.

The following formula is used:
Rn = ( Ro * Ro ) / 125000
Rn is the new rating.
Ro the old (previous day) rating.

Rating Amount subtracted
below 353 1
354-500 2
501-612 3
613-707 4
708-790 5
791-866 6
867-935 7
936-1000 8

So players with higher rating will see their rating decrease quicker.
A 500 rating will reach zero in approximately 1.5 years without playing.

### Series of games

Playing a series of games against the same opponent within a given time interval will adjust the calculation as follows:
The expected score is calculated for the series of games from the pre-match rating-difference of the two players, and the ratings are amended based on the expected score and the actual score of the series of games.
These are basically the extensions of the above calculation for more games played.
However, as more matches will reflect the skill difference more precisely, the amendment is raised based on the number of games played.

Number of games in
a series of games
Maximum
rating change
1 16
2 24
3 28
4 30
5 31
infinite 32

In other words the constant K in the above equations is multiplied by a factor of 2-(1/2^(n-1)), where n is the number of games played.

Note that if you play a 3rd player in between playing the same opponent, it will still be treated as a series of games.

### Rating categories:

Highest ranked player

Above 700
600 - 699
550 - 599
500 - 549
450 - 499
400 - 449
350 - 399
300 - 349
250 - 299
200 - 249
150 - 199
100 - 149
67 - 99
34 - 66
0 - 33

Above 700
600 - 699
500 - 599
400 - 499
300 - 399
200 - 299
100 - 199
50 - 99
0 - 49

Above 600
500 - 599
400 - 499
300 - 399
200 - 299
100 - 199
0 - 99