## The League Rating

### In a Nutshell

The League Rating is based on the Elo-System, which gives a scoring probability for a given rating difference between 2 players.

The sum of the scoring probabilities for a player against all his opponents would be his Expectation. It then calculates a Performance, which is the rating where the Expectation equals the actual achieved score in the tournament. Finally, it constructs the new rating from the old rating, a Constancy-Factor (C) and the number of games played.

Special treatment is given to New Players and cases where players score 100% or 0% in a tournament.

#### Lookup Table

Rtg Dif | H | L | Rtg Dif | H | L | Rtg Dif | H | L | Rtg Dif | H | L | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0-3 | 0.50 | 0.50 | 92-98 | 0.63 | 0.37 | 198-206 | 0.76 | 0.24 | 345-357 | 0.89 | 0.11 | |||

4-10 | 0.51 | 0.49 | 99-106 | 0.64 | 0.36 | 207-215 | 0.77 | 0.23 | 358-374 | 0.90 | 0.10 | |||

11-17 | 0.52 | 0.48 | 107-113 | 0.65 | 0.35 | 216-225 | 0.78 | 0.22 | 375-391 | 0.91 | 0.09 | |||

18-25 | 0.53 | 0.47 | 114-121 | 0.66 | 0.34 | 226-235 | 0.79 | 0.21 | 392-411 | 0.92 | 0.08 | |||

26-32 | 0.54 | 0.46 | 122-129 | 0.67 | 0.33 | 236-245 | 0.80 | 0.20 | 412-432 | 0.93 | 0.07 | |||

33-39 | 0.55 | 0.45 | 130-137 | 0.68 | 0.32 | 246-256 | 0.81 | 0.19 | 433-456 | 0.94 | 0.06 | |||

40-46 | 0.56 | 0.44 | 138-145 | 0.69 | 0.31 | 257-267 | 0.82 | 0.18 | 457-484 | 0.95 | 0.05 | |||

47-53 | 0.57 | 0.43 | 146-153 | 0.70 | 0.30 | 268-278 | 0.83 | 0.17 | 485-517 | 0.96 | 0.04 | |||

54-61 | 0.58 | 0.42 | 154-162 | 0.71 | 0.29 | 279-290 | 0.84 | 0.16 | 518-559 | 0.97 | 0.03 | |||

62-68 | 0.59 | 0.41 | 163-170 | 0.72 | 0.28 | 291-302 | 0.85 | 0.15 | 560-619 | 0.98 | 0.02 | |||

69-76 | 0.60 | 0.40 | 171-179 | 0.73 | 0.27 | 303-315 | 0.86 | 0.14 | 620-735 | 0.99 | 0.01 | |||

77-83 | 0.61 | 0.39 | 180-188 | 0.74 | 0.26 | 316-328 | 0.87 | 0.13 | over 735 | 1.00 | 0.00 | |||

84-91 | 0.62 | 0.38 | 189-197 | 0.75 | 0.25 | 329-344 | 0.88 | 0.12 |

### Calculating the Expectation

For each opponent of a player, take the rating difference and look up the scoring probability in the table above.

Example: If he is rated 150 points higher than his opponent, 0.70 points are expected. The sum of these is the expected total score.

If a player scores exactly as he is "expected" to do, his Performance is identical to his old rating.

### Calculating the Performance

The Performance is done by an iterative process which finds the rating where the expectation is the score a player actually made.

For example, if a player rated 1600 had an expectation of 3 points, but scored 4.5, you would calculate the expectation for a higher rating, let's say 1800. If the expectation for that is still lower than 4.5, try another higher rating; if it is higher, try a rating between 1600 and 1800. Continue iterating until expectation equals 4.5, and you have the performance. (Admittedly this would be tedious to do manually, but it is easy enough for a computer).

### The New Rating

is determined by the following formula:

r_new = | C * r_old + n * performance |

C + n |

Where r_old is the previous rating, n is the number of games, and C is the constancy-factor. A higher C means a more stable rating. It is currently 20 for players at their 4th+ tournament, and 16, 11 and 6 for the 3rd, 2nd and 1st tournaments, respectively.

### Special cases

#### Perfect and zero scores

If a player scores 100% of the available points or zero points, the performance is extrapolated from the two neighboring possible scores. Example: For 6 out of 6, the Performances for 5 and 5.5 points are taken and the difference between them added to the performance for 5.5 points.

The extrapolation is necessary because the performance would in theory be infinite (and due to the limits of the lookup table still 735 above the rating of the highest opponent), where in reality the sample is just too small. The correct solution would be to postpone making the rating for this player until he finally stops winning, but that doesn't suit our needs.

#### ...with only one game

If in the above case there is only one game to rate, the new rating will be made with ICC's rating formula (with K's of 48, 32, 24 and 16 for 1, 2, 3, 4+ tournaments).

#### New Players

For every tournament, new players (those who have no previous League Rating) are rated first. Their new LR is made from their fixed rating and the performance. Their LR is then used instead of their FR for the calculations of the established players. This is done to reduce the effects of first fixed ratings, which sometimes turn out to be way off.#### Maximum rating loss limit

There is a maximum allowed rating loss per tournament of 75 points for well established players, 100 points for those who are in their 3rd tournament, 150 points for those in their 2nd, and 200 points for players in their 1st tournament).