up to Scrabble Tournament Ratings

This document is obsolete as of 2009-01-01; when the NSA adopted a new rating system..

# How the NSA Ratings System Works

In response to a number of requests, I've written up this description from memory; it is not an official publication of the Ratings Committee, and corrections are welcome.

After each tournament, the number of games that you were expected to win in that tournament is calculated, and subtracted from the number of games that you actually did win. Your rating is then adjusted by the product of this difference with a multiplier which depends on the number of games you have played and your pre-tournament rating.

Your expected wins are calculated so that the probability density of winning is normally distributed with respect to the difference between your rating and your opponent's, with a mean of zero and a standard distribution of 400/sqrt(2). I use something like the following Perl code to calculate this fractional value:

```# \$y = &erf2(\$x) - unnormalized error function
sub erf2 { local(\$x)=\$_; \$x >= 4 ? return 1 : \$x <= -4 ? return 0 : 0;
local(\$n,\$sum,\$term,\$x2) = (0, 0.5, \$x/1.772453850905516, \$x*\$x);
while (\$term > 1E-10 || \$term < -1E-10)
{ \$n++; \$sum += \$term; \$term *= - (\$x2 * (\$n+\$n-1))/((\$n+\$n+1) * \$n); }
\$sum;
}

# \$expectation = &expected(\$ratings_difference)
sub expected { local(\$ratings_diff) = @_; &erf2(\$ratings_diff/400); }
```

Your expectation should be 0.6 wins at a difference of 72 ratings points, 0.7 at 148, 0.8 at 239 and 0.9 at 362. Many expert players believe that the latter expectations are too high.

Your multiplier is looked up in the following table:

```                  Games Played
1-49  50+
pre-      1-1799   30   20
tourney 1800-1999   24   16
rating  2000+       15   10
```

So if you're rated 1850 going into a tournament, and play 7 games against opponents rated 1702, 1702, 1778, 1922, 1922, 1998 and 1998, you're expected to win 0.7+0.7+0.6+0.4+0.4+0.3+0.3 = 3.4 of your games. If you actually win four and tie one, then you'll have won an excess of 4.5-3.4 = 1.1 games. If you had played 48 games prior to this tournament, your multiplier from the table would be 24, so your rating would increase by 24*1.1 = 26.4 to 1876 (rounding to the nearest integer).

Note that your rating is affected neither by the scores of your games, nor by how you did against individual opponents; only by your total number of wins, and the differences between your rating and your opponents'.

I've glossed over a few details in the above, which I'll delve into now. Skip to the next section if details don't interest you.

• If your rating would cross a multiplier table row after a tournament, i.e. it would go past 1800 or 2000, then you must calculate your ratings change in two steps using the two appropriate multipliers. (Use as many of your excess wins as it takes to get to the boundary point first.)
• If your rating increases by more than five points per game played (i.e. in the example above, by more than a total of 35 points), then the excess above those five points per game is doubled. These bonus points are sometimes called "acceleration points" and are intended to deal with players who are greatly underrated. If the basic calculation shows that you were supposed to gain 50 points at a seven-game tournament, then you'll gain those 50 points plus 50-35=15 acceleration points, for a total of 65.
• If you play an opponent who has earned acceleration points, then you are entitled to so-called "feedback points": one twentieth of a point per acceleration point per game played against that opponent. This is intended to mitigate the effect of playing a severely underrated player.

Here is a more detailed worked example, at the request of the player mentioned.

Austin Nichols had a pretournament rating of 1182. At the Big Apple Scrabble Tournament in October 2007 he played opponents rated: 1097 1025 1276 1277 1196 1220 1054 1018 1139 1182 1270 1256 1276 1294 1256. His fractional expected wins against them are respectively: 0.61811 0.71058 0.36982 0.36848 0.48026 0.44656 0.67456 0.71898 0.56042 0.50000 0.37785 0.39680 0.36982 0.34606 0.39680. These add up to his total expectation: 7.3351. He won 10 games, exceeding his expectation by 10 - 7.3351 = 2.6649 games. The base increase to his rating is 30 * 2.6649 = 79.947. One of his opponents earned 49.5 acceleration points, so Nichols earns 49.5/20 = 2.475 feedback points for each of the two occasions on which he played this opponent, bringing the increase to 79.947 + 2 * 2.475 = 84.897. This is greater than five times the number (15) of games Nichols played, so Nichols earns 84.897 - 5 * 15 = 9.897 acceleration points. His total increase is therefore 84.897 + 9.897 = 94.794, rounded to 95 points, giving him a post-tournament rating of 1182 + 95 = 1277.

## What a Performance Rating is

A tournament report usually tell you what your "performance rating" was for that tournament. This is the rating that you would have had to have had before the tournament, in order for your rating to have remained unchanged after the tournament. Performance ratings are most easily calculated by an iterative process. If you have won all your games or lost all your games, you're given a fractional fictitious loss or win to avoid giving you an infinite P.R.

## How You Get Your Initial Rating

You receive an initial rating after your first tournament, equal to your performance rating in that tournament. The iterative process involved in calculating performance ratings can get slightly hairy if too many unrated players play each other.