How are numerical chess rankings calculated for different ranking systems?

Question

I know there are several different ranking systems for chess, but how do they work in general? How exactly do they approximate your skill and how you will fare against other players? How can I figure out my own rating? Does the jump between 2000 and 2100 require a lot more practice/improvement than the jump between 1000 and 1100?

I know I phrased the question to be generic to all systems, but feel free to go into the specifics of any one rating system if it helps answer the question. :D

Answer 1

In general, the idea is to give an approximation of relative strength among players, based on their history in rated competition in that type of chess (normal, blitz, correspondence). The greater the difference, the higher the probability that the higher-rated player will win.

Ratings are modified based on the difference in rating between the players and the expected result. Ratings generally go up for a win and down for a loss; the rating change is bigger if the difference in ratings as bigger. There is typically a damping factor that reduces or removes the penalty for losing to a much higher-rated player.

There are typically different rules for players with established ratings and players who haven't yet played enough games to establish a rating. Provisional ratings fluctuate more and have less predictive value for future results.

In general, yes, once you have established a rating, it is harder to increase your rating by 100 points from 2000 than it is to increase it by 100 points from 100. That does depend on your skill level: if you have a knack for chess, you may never have a rating in the low 1000s.

Below are summaries for most of the major ratings I have been able to identify; apologies to those who use a different rating system.

Elo system

The formula for calculating your Elo rating is this:

new rating = old rating + K×(W-We), where K=10, W=actual score, and We=expected score

In tournament play, the difference in ratings between you and each of your opponents is calculated first. From this, an expected score against each opponent is derived from a table. The sum of those expected scores is compared to your actual score, and thus your new score is calculated.

ECF system

As of 2009, a player's grade is calculated by taking the opponent's grade and adding 50 points for a win, subtracting 50 points for a loss, and taking the opponent's grade as it stands for a draw. For grading purposes it is assumed that the opponent's grade is never more than 40 points above or below one's own. An ECF grade can be approximated to an Elo rating by multiplying by 8 and adding 600*.

For a further explanation, per the ECF website:

In the interval between the end of a grading period and publication of the new grades, the "current" grade for calculation purposes is the new, as yet unpublished, grade.

The Grade is calculated by dividing the total number of points scored by the number of games played. If there are at least 30 games in the current period, then the Grade is based on these games alone. If there are not, results are brought forward from the previous periods as required (see 'Category' above). In no case does calculation go back more than 36 months.

Results are brought forward in two different ways, depending whether the Grade is Rapid or Standard. With Rapidplay, any games brought forward from a previous period will be the most recent games in that period. This is possible because the dates of Rapid games are (almost) always known. With Standardplay, unfortunately, this is not the case. So, instead, the required number of (notional) games is brought forward at the average score for the period.

For ungraded players, a grade is estimated this way:

A Rapid Grade, where available, will be used in default of a Standard Grade; and vice versa. If the player has no Grade at all, a starting grade is calculated as follows.

Stage 1 is to calculate a 'grade' for each ungraded player on his games against graded opponents. The 40-point rule is not used. If all his opponents are graded, it stops there and the result will be used as his starting grade.

Stage 2 brings in games between the ungraded players. Once again the 40-point rule is not used. The players are 'graded' on all their games, using as starting grades the figures obtained from Stage 1.

The resulting 'grades' will not be very accurate. So they are fed in again as new starting grades, and Stage 2 is repeated. This continues, with increasing accuracy each time, until the figures (more or less) stop changing. The starting grades can then be considered accurate.

These starting grades are then used in the grading proper.

Glicko system

The Glicko system is an improvement on the Elo system, introducing a concept called Ratings Deviation.

The rating system itself is beyond the ability of this site to display; without the add-in that Math.SE has, the formulas would be nearly impossible to follow.

The RD measures the accuracy of a player's rating. For example, a player with a rating of 1500 and an RD of 50 has a real strength between 1400 and 1600 with 95% confidence. Twice the RD is added and subtracted from their rating to calculate this range. After a game, the amount the rating changes depends on the RD: the change is smaller when the player's RD is low (since their rating is already considered accurate), and also when their opponent's RD is high (since the opponent's true rating is not well known, so little information is being gained). The RD itself decreases after playing a game, but it will increase slowly over time of inactivity.

USCF Correspondence

The USCF used a linear approximation of the Elo method for calculating correspondence chess methods ... this is how it works now.

Established players

To calculate a new rating after a game, use the following formula:

Rn = Ro + .04(ED) +/- 16

This means that a new rating (Rn) is determined by taking the old rating (Ro), adding or subtracting 4 percent of the difference in ratings between opponents (.04(ED)), and adding or subtracting 16 points.

Rating differences that exceed 350 points are figured as 350 points.
For players rated 2100-2399, the formula Rn = Ro + .03(ED) +/- 12 is used.
For players rated 2400 and above, the formula Rn = Ro + .02(ED) +/- 8 is used.

Provisional players

During your first 25 games as a correspondence chess player, your rating is calculated as the average of your game results.

Each game result is as follows:

• 400 points plus the opponent's rating for a win, unless the opponent's rating is more than 400 points less than yours.
• The opponent's rating minus 400 points for a loss, unless the opponent's rating is more than 400 points greater than yours.
• The opponent's rating for a draw, regardless of the difference in rating.
• Your current rating in all other cases.

For your first result, because you do not yet have a rating, the 400-point differences do not apply: you get the opponent's rating for a draw, +400 for a win, -400 for a loss. If your opponent also doesn't have a rating, then it's 1700 for the winner and 1300 for the loser, or 1500 for both if you draw.

There is also a provision for forfeits under certain circumstances, like the death of an opponent. (Yes, this is a drawback to correspondence chess ... sometimes you suddenly don't have an opponent any more.)

EDIT : The ELO system is not that simple : here is the FIDE handbook that regulates the rating system : http://www.fide.com/component/handbook/?id=172&view=article.

For those who don't want to read it, here goes (for already rated players) :

For each game played against a rated player, calculate the rating difference D

A difference in rating of more than 400 points shall be counted for rating purposes as though it were a difference of 400 points.

• Use a table to find the player's score probability PD
• dR = score - PD (score is 0, 0.5, 1 for each game)
• The rating change is : the sum of each dR multiplied by K

K is the development coefficient, defined as follows :

• K = 40 for a player new to the rating list until he has completed events with at least 30 games
• K = 20 as long as a player's rating remains under 2400.
• K = 10 once a player's published rating has reached 2400 and remains at that level subsequently, even if the rating drops below 2400.
• K = 40 for all players until their 18th birthday, as long as their rating remains under 2300.
  The Rating Change is rounded to the nearest whole number. 0.5 is rounded up ( whether the change is positive or negative).

Hope it is useful for everyone that wondered.