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# A Proposed Rating System

There have been some interesting proposals to revise the system by which chess players are given ratings points based on their performance in serious events.

For example, two of them currently under consideration are the Chessmetrics and Glicko-2 systems.

The system proposed here is not as novel as either of these. Instead, it is closely based on the existing Elo system, but with modifications to address some of its perceived weaknesses. Elements of the older English Chess Federation rating system are also included, with more extensive modification, as a way of further addressing these issues by exploiting the weaknesses found in that system.

### Preliminaries

Associated with each player, there will be two rating numbers, E and H. E is the player's actual rating. H is a leading indicator of the player's rating, based on recent performances.

The ratings formulas are parametrized, and the parameters have the following values:

```                         K      Q      P      J      M
Established Players     24      2     0.1    0.02    0
Ordinary Players        32      3     0.2    0.05    0
Beginning Players       48      8     0.25   0.1     0.2
```

The three ranks of players can be defined as done by FIDE or by the USCF.

When a rated game of chess is played, three steps take place, based on the result.

The value W equals 1 if the player whose rating is being updated won that game, 0.5 if the game was a draw, and 0 if that player lost.

### Updating H

The formula for updating H after a game of chess is the following:

```H' = (1-Y) * H + Y * (S + 800 * W - 400)
```

The expression 800*W-400 equals 400 if the player won, 0 if the game was a draw, and -400 if the player lost.

Y is equal to P if the game was a win or a loss, and equal to J if the game was a draw.

If the game was a draw, S is the opponent's rating; that is the value of E for the opponent, not H.

If the player won, S is either the opponent's rating, or the player's own rating minus 320 points, whichever is greater.

If the player lost, S is either the opponent's rating, or the player's own rating plus 320 points, whichever is less.

Thus, S, plus or minus 400 points if the game was not a draw, is the player's single-game performance rating; that is combined with the former value of H for the player in proportions governed by P.

S is limited so that it does not differ by more than 320 points from the player's own rating, when otherwise it would be so high as to cause a player to gain in rating from losing a game, or so low as to cause a player to lose in rating from winning a game.

### Updating E, main step.

The player's actual rating is maintained by same basic formula as in the Elo rating system:

```                                  100
E' = E + Z * ( (100 * W) - ----------------- )
R - E
(-------)
400
10          + 1
```

but the values placed in it are somewhat different.

Z is equal to K in the table above if the game was a win or a loss. If the game was a draw, Z is equal to Q instead. The purpose of this change is to take into account the fact that chess games often end in draws, and defensive play can be used to obtain a draw even against somewhat stronger opposition.

R represents the opponent's rating. It is either equal to E or to H (not to H', but to the value of H before it was updated by the current game) based on the following rules:

If the game was a win for the player being rated, and the opponent's E and H ratings are both higher than the E rating of the player, then whichever of E or H for the opponent is lower is to be used. If the opponent's E is higher, and H is lower, than the player's E, then the player's own E rating is used. Otherwise, the opponent's E is used.

If the game was a loss for the player being rated, and the opponent's E and H ratings are both lower than the E rating of the player, then whichever of E or H for the opponent is higher is to be used. If the opponent's E is lower, and H is higher, than the player's E, then the player's own E rating is used. Otherwise, the opponent's E is used.

If the game was a draw, and the opponent's E and H were either both higher or both lower than the player's E rating, then whichever one is closest to the player's E rating is used. Otherwise, if one is lower and the other higher, the player's rating is not changed as the result of the draw.

The purpose of this change is that a player's rating, as indicated by E, may not always be an accurate indication of a player's true strength. A player who is rapidly rising in the ranks may be stronger than his E rating indicates.

Thus, if a player loses to such a player, in order that the change in rating will be more accurate, the H rating, which is a rapidly-changing indicator of a player's strength based on recent performances is used in such a case, so the loss is counted as one against a player as strong as that player might be, as shown by the H rating which reflects recent performance quickly, rather than as strong as that player is said to be based on the slower-changing but usually more accurate E rating.

Similarly, less credit is given for defeating a player who seems to be on the decline - the H rating is used instead of the E rating in that case as well.

### The Mixing Step

Finally, for players who are beginning rated play, one additional step takes place in the calculation of E:

```E'' = (1 - M) * E' + M * H'
```

That is, the faster-changing H rating is mixed in with the E rating so that the ratings of beginning players will more quickly be brought to their correct value through play.

### Assignment of Initial Ratings

A player playing one of his first ten games of rated play will be a special case.

The results of those first ten games will be used to assign the player's rating based on the Harkness system.

Thus, the initial rating will be the average of the ratings of the players he played against, plus 10 points, or minus 10 points, for each percentage point by which his winning percentage differs from 50%.

During this time, if he plays against another player who is playing one of his first ten games, that player's rating will be taken as 1000 for purposes of establishing an initial rating.

When a rated player plays against a player playing one of his first ten games:

• The applicable value of K for calculating adjustments to the rated player's E rating is cut in half;
• Neither the rated player's E rating or H rating is adjusted if a draw takes place;
• If the rated player loses, the beginning player's ratings are taken to be E=1400 and H=1500; and
• If the rated player wins, the beginning player's ratings are taken to be E=1000 and H=1000.

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