*up to the Lexington Scrabble® Club*

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.

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This page, maintained by Mike Wolfberg, was last updated on December 14, 2012. |