Improved Dynamic Grading
April 2013: See Rebirth of Dynamic Grading for updates on this method [editor].
A modified algorithm for calculation of modulators is proposed. It is simpler to understand and to code and yet gives a more efficient system.
In Dynamic Grading the grade of a player is updated after every game by adding an amount M_W * LPW in case of the winner and subtracting an amount M_L * LPW in case of the loser. Here LPW denotes the Loser’s Win Probability while M_W and M_L are respectively the modulator of the winner and the loser (see "Plain English introduction to Dynamic Grading" for more details). These modulators are calculated for every player just when they are needed for the postgame update.
In the new algorithm, M_W is calculated via the expression
M_W = 9.56 + 0.58 * RSD_W
where RSD_W is the Recent Standard Deviation for the winner, i.e. the Standard Deviation of the preceding 35 grade values. It is given by the square root of the average of squared differences between grade value and average grade value. Similarly for the loser: M_L = 9.56 + 0.58 * RSD_L. Thus the modulator depends on the preceding 35 grade values in a uniform way (the order of the values do not matter and each grade value carries the same weight). This contrasts with the way modulators are calculated in the original version. There (see "Introduction to Dynamic Grading") the modulator depended on the preceding 37 grade values in a non-uniform way, involving more stages of calculation via more complicated expressions.
Actually, the above new expressions for M_W and M_L apply only from game 35 onwards. For the first 34 games the constant modulator value M = 24 is used for all players. A further restriction applies throughout: a modulator is not allowed to exceed the value 37.7.
Despite the greater simplicity and transparency there is no loss of efficiency. Indeed, the important Grade Deviation statistic for the improved system (iDG) compare as follows to its value for the original DG-system (oDG) and some others (mentioned only to add perspective):
where the calculation is over the set of test games described in "Introduction to Dynamic Grading". The statistic GDev measures how closely grade-differences correspond to win probabilities.
The statistic GDev measures how closely grade-differences correspond to win probabilities. It is also an important design tool - it enables immediate inspection of what effect is produced by changing a parameter. For example, if any one of the parameters introduced is increased or decreased it immediately results in a worse GDev value. This applies in particular to the following:
So these parameter values are not lucky guesses. They are experimentally determined through trial and error to give the best possible (smallest) GDev value.
For a better idea of differences and similarities between the original DG and its proposed improvement, it is useful to look at the effect produced on specific players. Rapid improver Robert Fletcher is a good choice for this purpose. The table to follow compares his record under oDG and iDG over games 33 to 59. The non-obvious headings are as follows: GIS = Games In System, oM, iM = modulators for oDG and iDG respectively, RPT = Recent Performance Trend = sign of iDG – A, where A = average of the preceding 35 iDG-values. Thus RPT = "+" when iDG > A and RPT = "-" otherwise. This is more comprehensible than its counterpart pdt in the original system. A large iM together with "+" indicates rapid improvement and together with "-" indicates rapid decline. An iM near the minimal value 9.56 indicates steady performance.
The plot to follow compares oDG to iDG over games 33 to 59
Next follows a plot to compare the modulators over the same range:
There is no lack of smoothness in the variation of the new modulator iM, despite the fact that it is calculated from the raw statistic RSD without any smoothing process.
For a longer term comparison, let us look at corresponding plots for the next 100 games.
At the time of most rapid improvement (roughly from game 66 to 126) iM reached the maximum allowed value of 37.7. During this period iDG remained about 16 grade points higher than oDG. A look at the further history (not shown here) reveals that oDG and iDG were never far apart and repeatedly came together again. At game 414 they were identical.
Why was iDG not chosen in the first place? The development of Dynamic Grading was like navigating unchartered waters without a guide. The Observed Wins minus Expected Wins approach, highly successful for the chi-squared statistic, seemed natural enough to use. However, it led to a statistic which did not vary smoothly. Its smoothed version (PDT) did give good results. In a recent return to a contemplation of these things a search was undertaken for a statistic that would smoothly track recent performance steadiness. After many failures RSD was found. It worked like a charm. The various parameters were then easily chosen so as to give the smallest GDev statistic.
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