Ratings

# How the NSA Rating System Works

This is an informal description of how the rating system works, and may differ from the actual calculations in minor details. The reference implementation of the rating system is the Perl module Ratings::Elo, part of the tsh software distribution. This module is used by the NSA Rating Calculator, an online tool which you can use to compute your own ratings.

At the end of each tournament, we calculate the number of games that you were statistically expected to win, then subtract the number of games you actually did win to get what’s called your excess. We multiply the excess by a number (your multiplier, which depends on the number of games you have played and your current rating), to find out by how much your rating should change. If you outperform your expectation, your excess is positive and your rating rises; if you don't reach your expectation, your excess is negative and your rating falls.

Here is how your expected number of wins is calculated. You have a fractional number of expected wins against each of your opponents, which is calculated as 1 - 1/(1+exp(0.0031879*delta)), where delta is the difference between your rating and your opponent’s. For example, if you are rated 1800 and your opponent is rated 1700, delta=1800-1700=100, and you are expected to win 0.57903 games against this opponent for each game you play. The fractional number of expected wins against each of your opponents are added together to make your total number of expected wins at your tournament.

Your expectation should be 0.6 wins at a difference of 127 ratings points, 0.7 at 266, 0.8 at 435 and 0.9 at 689. Prior to 2009, we used a different formula whose corresponding differences were 72, 148, 239 and 362, which were thought to be too low.

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 1584, 1584, 1723, 1977, 1977, 2116 and 2116, 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.

## 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.

## What a Local Club Tournament is

A Local Club Tournament differs from an Open Rated Tournament in that it is not listed in the NSA Tournament Calendar, has a different rating fee structure, all of its players must be current NSA members, and any changes to player ratings are reduced by a factor of three.