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Lexington, MA Scrabble® Club Rating System

This long page explains the workings of the Lexington Club Rating System. It has been in effect since early 1981 and has been a successful indicator of the relative strengths of the players. The system was designed primarily by Alan Frank, with assistance from a few others, including: Steve Root, Mike Wolfberg, Eric Albert, and Frank Voss. Mike authored the computer program which does the job of supporting this system. That program has undergone changes a few times, especially as it has undergone changes in the programming language used, but the basic mathematics of this rating system have remained unchanged.

The basic idea of the system is that each player acquires a numerical rating which is an indication of the player's relative strength in the club. The rating numbers have no intrinsic meaning; they are just meaningful in how they relate to the other ratings. The system was set up to have the top player rated around 1800; we expected the lowest rated players would still be above 1000, and so a low-looking 3-digit number would not be used.

Unlike the national rating system, in which only win/loss is important, the Lexington Club Rating System does take into account game scores. Rather than focussing on point spreads or pure scores, the quantity of importance is the percentage of total points a player scores. Thus a point spread of 10 points is considered more important in a low-scoring game than in a high-scoring game. In the computations, the winner's percentage of total points is boosted by 4% (and the loser's percentage is reduced by 4%) to give a premium to the win. The boost is not used in a tied game.

When a player first shows up at the club, during the first ten games, the rating a player gets is based on performance against the other players, who, for the most part, have established ratings. The new player's conditional rating is based on the game scores in relation to that player's opponents. For the purpose of computing the new player's opponents' ratings, the new player is assigned a guessed initial rating by the rating statistician, based on initial performance. This is typically 25-50 rating points higher than the first night's numbers say. This has the effect of pumping more points into the system when new players show up. It also gives the opponents of new players a small advantage in getting their own ratings increased, since the assigned rating may be a bit higher than the real level of the new player. On the other hand, a serious new player will likely increase in strength even over the first ten games played in the club, so the extra boost is also a prediction.

The mathematics for a new player are sufficiently complex that this presentation will skip this. If there is some demand for details on this subject, its presentation will be included in the future. Let us go on to explain the workings of the rating system for those players who have played at least 10 games at the club, as remembered in the club data. Each player is remembered in the club data for up to 99 weeks of inactivity. Once that time has passed, a returning player starts all over in the developing of statistics.

The club operates on a fiscal year which restarts each September. For the year (from September through August) each player's number of wins, number of losses, and average scores are computed. As the new season begins, these are restarted afresh, but the ratings are carried across the season boundary. An exception to this is when all ratings of the club members are boosted. This has been done only once since the inception of the system. It can easily be done again at a season boundary if and when we notice a general degradation of the ratings. The clue for this will be the observation that the highest rated player is far below 1800.

The idea of the rating system is that any two players in the club can play each other - even the strongest and the weakest, and the system provides a kind of handicapping mechanism. Based on the difference between the two players' ratings, there are expected outcomes of the games. For example, if the highest and lowest rated players were to play on 23-Jul-98 and have a game in which the higher-rated winner gets 433 and the opponent gets 286, then that is below par for the winner. For that game total, the score is predicted to be 470-249. The winning percentage should be 69.3% (including the 4% boost), but the game scores indicate the winner got 64.2% (boost included). This causes the winner to lose 3 rating points and the opponent to gain 3 rating points. Some real-life examples are given later on this page.

The following table indicates par game scores for various rating differences between the two players. For example, if strong player (with rating 1805) plays a weaker player (with rating 1605), the rating difference is 200. An expected score may be 405-295 (when their total points are 700). If this or some other game scores on the 200 row was achieved, their ratings do not change.

TOTAL POINTS SCORED

RATING
DIFF.
450 500 550 600 650 700 750 800 850 900
0-37   225-225 250-250 275-275 300-300 325-325 350-350 375-375 400-400 425-425 450-450
40   226-224 252-248 277-273 302-298 327-323 352-348 377-373 402-398 428-422 453-447
50   230-220 255-245 281-269 306-294 332-318 357-343 383-367 408-392 434-416 459-441
75   236-214 263-237 289-261 315-285 341-309 368-332 394-356 420-380 446-404 473-427
100   242-208 269-231 296-254 323-277 350-300 377-323 404-346 430-370 457-393 484-416
125   247-203 275-225 302-248 330-270 357-293 385-315 412-338 440-360 467-383 495-405
150   252-198 280-220 308-242 336-264 364-286 392-308 420-330 448-352 476-374 504-396
175   256-194 285-215 313-237 342-258 370-280 399-301 427-323 456-344 484-366 513-387
200   260-190 289-211 318-232 347-253 376-274 405-295 434-316 463-337 492-358 521-379
225   264-186 294-206 323-227 352-248 382-268 411-289 440-310 470-330 499-351 528-372
250   268-182 298-202 327-223 357-243 387-263 417-283 446-304 476-324 506-344 536-364
275   271-179 301-199 331-219 362-238 392-258 422-278 452-298 482-318 512-338 542-358
300   275-175 305-195 336-214 366-234 397-253 427-273 458-292 488-312 519-331 549-351
325   278-172 309-191 339-211 370-230 401-249 432-268 463-287 494-306 524-326 555-345
350   281-169 312-188 343-207 374-226 405-245 437-263 468-282 499-301 530-320 561-339
375   284-166 315-185 347-203 378-222 410-240 441-259 473-277 504-296 536-314 567-333
400   286-164 318-182 350-200 382-218 414-236 446-254 477-273 509-291 541-309 573-327
425   289-161 321-179 353-197 386-214 418-232 450-250 482-268 514-286 546-304 578-322
450   292-158 324-176 357-193 389-211 422-228 454-246 486-264 519-281 551-299 584-316
475   294-156 327-173 360-190 393-207 425-225 458-242 491-259 523-277 556-294 589-311
500   297-153 330-170 363-187 396-204 429-221 462-238 495-255 528-272 561-289 594-306
525   299-151 333-167 366-184 399-201 433-217 466-234 499-251 532-268 566-284 599-301
550   302-148 335-165 369-181 403-197 436-214 470-230 503-247 537-263 570-280 604-296
575   304-146 338-162 372-178 406-194 439-211 473-227 507-243 541-259 575-275 608-292
600   307-143 341-159 375-175 409-191 443-207 477-223 511-239 545-255 579-271 613-287
625   309-141 343-157 377-173 412-188 446-204 480-220 515-235 549-251 583-267 618-282
650   311-139 346-154 380-170 415-185 449-201 484-216 518-232 553-247 587-263 622-278
675   313-137 348-152 383-167 418-182 452-198 487-213 522-228 557-243 592-258 626-274
700   315-135 350-150 385-165 420-180 455-195 491-209 526-224 561-239 596-254 631-269

When two rated players have played, this is how the rating change is computed. First, compute the quantity WINNER-PERCENT, as:

    WINNER-PERCENT = 100 * WINNER-SCORE / (WINNER-SCORE + LOSER-SCORE)

but then it is boosted by 4 when the game is not tied, so

    if (WINNER-SCORE is not equal to LOSER-SCORE) then
        WINNER-PERCENT = WINNER-PERCENT + 4.0


Then compute the expected percentage difference by this formula (originally determined by looking at game data):

    RATING-DIFF = abs(WINNER-OLD-RATING - LOSER-OLD-RATING)
    EXPECTED-PERCENT = sqrt(RATING-DIFF + 6.25) + 47.5
    if (LOSER-OLD-RATING is greater than WINNER-OLD-RATING) then
        EXPECTED-PERCENT = 100 - EXPECTED-PERCENT


The following table presents the expected percentage of the total score the higher rated player is expected to achieve given the rating difference of the two players.

RATING
DIFF.
   EXPECTED
PERCENTAGE
0 - 2   50
3 - 9   51
10 - 18   52
19 - 29   53
30 - 42   54
43 - 57   55
58 - 74   56
75 - 93   57
94 - 114   58
115 - 137   59
138 - 162   60
163 - 189   61
190 - 218   62
219 - 249   63
250 - 282   64
283 - 317   65
318 - 354   66
355 - 393   67
394 - 434   68
435 - 477   69
478 - 522   70
523 - 569   71
570 - 618   72
619 - 669   73
670 - 722   74
723 - 777   75
778 - 834   76
835 - 893   77
894 - 954   78
etc.

Then compute the rating change from the percentage difference:

    PERCENT-DIFF = WINNER-PERCENT - EXPECTED-PERCENT
    if (abs(PERCENT-DIFF) is less than or equal to 10.0) then
        RATING-CHANGE = PERCENT-DIFF
    else
        RATING-CHANGE = (log(abs(PERCENT-DIFF)) * 10.0) - 13.0


The "log" in the above computation is the natural logarithm, sometimes denoted as "ln", such as in the MS Windows Calculator program. It is named "log" in the C programming language run-time library.

The following table presents the full rating change as a result of a game given the percentage difference between the expected and actual percentages.

PERCENT
DIFF.
   RATING
CHANGE
0 - 12   same
13 - 14   13
  15     14
16 - 17   15
18 - 19   16
20 - 21   17
22 - 23   18
24 - 25   19
26 - 28   20
29 - 31   21
32 - 34   22
35 - 38   23
39 - 42   24
43 - 46   25
47 - 51   26
52 - 57   27
58 - 63   28
64 - 70   29
71 - 77   30
78 - 85   31
86 - 94   32
> 94     33

The winner is boosted by RATING-CHANGE (rounded to the nearest integer) when that player has played less than 50 games at the club; otherwise, the winner is boosted half of that (rounded to the nearest integer). The idea is that until a player is established in the club's records, that player's rating changes more quickly. It is assumed veteran players' ratings deserve to change more slowly.

Similarly, the loser is reduced by either RATING-CHANGE or half of RATING-CHANGE, depending on the number of games the loser has played.

Once a player has played at least ten games in the club, the above rules apply. The old ratings used in the computations are the ones published on the news sheet, so rating changes for each game are accumulated for the entire session before they are used to update the current data. This implies the order of the recorded games does not matter (for players who have played at least 10 games).

Here are a few examples of how the system works. They are based on real data representing games played 23-Jul-98. Let us say these folks have played more than 50 games in the club and have these ratings:

PLAYER    RATING
A   1824
B   1805
C   1713
D   1708
E   1610
F   1588

Now, here are games they played. The "PAR GAME" column indicates the expected game score when the total points scored were the same as the actual total. The presented percentages include the 4% adjustments to the players. You can predict the rating change by computing the difference in the percentages. For example, in the first game, the difference is approximately 8. For numbers in this range, the change in rating is the same as the percentage difference. Half the rating change is used for players who have played more than 50 games since they began at the club, you see a rating change of 4 (half of 8). Rating changes are made in whole numbers.

WINNER    LOSER    PAR
GAME
EXPECTED
WINNER
PERCENT
ACTUAL
WINNER
PERCENT
WINNER
RATING
CHANGE
LOSER
RATING
CHANGE
A 459   D 272   399-332 58.6% 66.8% + 4 - 4
C 440   A 399   383-456 41.7% 56.4% + 7 - 7
A 429   E 325   440-314 62.3% 60.9% - 1 + 1
D 424   E 314   396-342 57.7% 61.5% + 2 - 2
C 512   E 267   420-359 58.0% 69.7% + 6 - 6
A 421   E 236   383-274 62.3% 68.1% + 3 - 3
C 354   B 326   317-363 42.6% 56.1% + 7 - 7
B 419   F 297   418-298 62.4% 62.5% 0 0

You can find out the rating changes by providing the game score and players' ratings by using the Lexington Scrabble® Club Rating Changes Calculator here on this web site. There is a similar program (named DeltaRat) on the laptop PC usually operating at club sessions.


back to the top of this page This page, maintained by Mike Wolfberg, was last updated on December 14, 2012.