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### The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 3:22 pm**

by **John Upham**

I suspect this has been covered previously but can anyone enlighten me as to the reasoning that took place in 18?? or whenever it was to put the so-called 40 point rule in place?

I'm not agreeing or disagreeing but merely mathematically and statistically curious.

Don't worry Chris : I won't be proposing a modernisation of this facet of the system!

John

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 4:59 pm**

by **Mike Gunn**

Hi, John. It is so your grade doesn't go up when you lose to a much stronger player (and vice versa). Consider player A (grade 100) playing player B (grade 160). B beats A as expected) but only scores 100+50 grading points, i.e. 150 for the game). A scores 160-50=110 points. So A's grade goes up (although he's lost) and B's grade goes down (although he has won).

The 40 point rule says you treat any player who is graded more 40 points different to you as if the grading difference is exactly 40. Thus in the above example when grading the game from B's point of view A has a grade of 120 and A scores 170 points whereas from A's point of view A's grade is taken as 140 and he scores 90 points.

I have an article somewhere (from the 60s or 70s) where Richard Clarke says this was not needed in the early days of BCF grading, but this change is about to be introduced because more players of widely differing abilities are playing each other than previously. There is some kind of mathematical support for this in that you are matching Elo's curve by two straight lines rather than a single straight line (not that Elo's curve is actually right, as others have said).

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 5:15 pm**

by **John Upham**

Mike,

I understand precisely the effect of the 40 point rule and its consequences. I have implemented it for the performance calculations on the Surrey Border League web site.

My point is how was the figure of 40 arrived (rather than 30 or 50) or pie**4 or some other random constant.

I assume it is some kind of fudge factor arrived at to dampen down rating changes.

Recently, Simon McCullough (166) drew with Simon Williams (232: SW was lucky not loose and the clock saved him)

Why should Simon M. only get 166+40=206 for that game rather than SWs rating? Can someone attempt to justify this please?

The key words artificial and non-rigorous spring to mind.

I look forward to becoming enlightened!

John

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 5:51 pm**

by **Roger de Coverly**

*My point is how was the figure of 40 arrived (rather than 30 or 50) or pie**4 or some other random constant. *

It's the point at which the stronger player is supposed to score 90% or the weaker 10%. The word decile springs to mind for some reason.

Actually it's probably Stewart Reuben's fault - I can remember the rule being introduced (1970 maybe?) at about the time when mega swiss open tournaments started to be popular. It's always been a bit unjust - the rich (in a grading sense) stay rich and the poor stay poor. Of course, the ELO international system has a very similar feature with a cutoff at 320 points.

If you didn't have such a rule, then players looking to defend their rating might refuse to play against lower rated opponents.

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 6:33 pm**

by **Mike Gunn**

This is what Sir Richard Clarke wrote about the BCF grading scheme in 1969 (it is part of a longer article where he also discusses other grading schemes).

"The B.C.F. Grading System

The B.C.F. scheme followed Harkness. As soon as news of this plan reached here, the B.C.F. Development Committee began to consider the possibility of a grading system: we did not then know of the existence of the Ingo-system. I had published an article in the April, 1953, "B.C.M." on the grading of grandmasters; so then a sub-committee of Gilchrist, Buckle, Wade, and myself was set up, reported in September, 1953, and led to the first B.C.F. Grading List in March, 1954.

We adopted the Harkness notation and the basic Ingo/Harkness principle. But our situation was different, for play in this country is mainly in team events, and not in tournaÂ¬ments. So instead of treating the tournament as a unit, like Ingo and Harkness, our unit had to be the individual game. We had to aggregate every individual's play, and calculate the average grading number of his opponents, and apply to this the premium or discount according to the difference between his percentage score and 50 per cent. Our system must reckon performance over a specified period, for players' club and county results are in units of the season's play. So we calculate and publish the results once a year, instead of continuously from tournament to tournament. We reckon (and the Americans agree) that one needs thirty games to establish a reliable grading number. We began with a three-year grading period. We quickly reduced it to two years, as we organized the flow of material. Now we grade on a one-year period, where thirty games have been played; and if the number in the year is less, fill in the difference from the previous year's performance.

When there is enough material to grade large numbers on one year's play, there is advantage in having a specified period. In the Ingo and U.S.C.F. systems, the period covered is different for every player, and is indeterminate. The grading numbers as published are very up-to-date for very active players; but for those who play less often the figure will represent performance over many years. Thus, the regular official list will be a mixture of very short periods for very active players and quite substantial periods for those who play less.

The B.C.F. scheme has graded huge numbers of players since 1954. The number appearing in the B.C.F. and Union lists, down to grade 11b, is of the order of 3,000. This is the tip of the iceberg: there are four times as many who play in a season but not enough to qualify;and the total number on the graders' books is about 20,000. Our operations are decenÂ¬tralized, and we rely entirely on the devoted work of scores of graders and sub-graders.

After ten years' experience, we conducted a major review. The numbers in the upper grades were falling, and we needed to check whether this was a true trend or whether there was an inherent deflationary tendency: again, there was evidence of a growing regional imbalance, the North appearing to be undervalued and the West overvalued in relation to the South and Midlands. The Grading Committee's review took some years, and the reforms have been introduced as they came.

We decided that there was some downward bias in the system. The main difficulty was that the huge increase in junior play had introduced an element of deflation. In our system, the total strength of each player's opponents is the sum of their last year's grading numbers. If a large proportion of the players is improving fast, their opponents' strengths will be underÂ¬estimated. The juniors' grading numbers are not depressed: the damage is to the grading numbers of the people who play against there. We also found that some of the regions were handling new entrants differently. Adjustments have been made, and review will be done oftener in future.

The hard core of British players who play year after year and are the backbone of club and county chess is remarkably small. Much of the total of serious play consists of people who come in and play for a year or two and then drift away, and of school and university players who drop out when they start business. The rapid turnover of players makes grading very difficult, particularly of juniors. We have improved our practice as a result of the review, but we are not yet certain that we have compensated enough against the inherent deflationary characteristics of these systems.

Another possible deflation results from the invalidity of the Ingo/Harkness/B.C.F. system where there is a big difference between the strength of the players. When a man with grading number X beats one with Y, he scores 50 + Y, which can be expressed as 50 + X â€” (X â€” Y). If the difference between X and Y is 50, the stronger player gains nothing for winning, but is debited by getting X â€” 50 and X â€” 100 for a draw or a loss. If the difference exceeds 50, the weaker player actually gains (and the stronger loses) even if the stronger wins the game. Ingo and B.C.F. therefore omit games where the difference exceeds 50 whenever the stronger player wins. This is inevitable, but it is unfair to the stronger, who is on a "heads you win, tails I don't" fork.

The practical seriousness of this defect depends upon how often people with differences of 50 or moreâ€”six gradesâ€”play against one another. When B.C.F. grading began, it did not happen often, which was one reason why we accepted the anomaly. With the growth of the Swiss System, it is now more frequent. Nevertheless, our estimates show that this cannot be a serious reason for deflation, much less important than the under-valuation of juniors. We could in fact virtually eliminate the anomaly by treating all differences between players' gradings of over 40 points as if they were 40 points; but we decided that the added comÂ¬plexity of calculation for our graders outweighed the benefits from removing this anomaly. But we may revert to this later.

So the B.C.F. system is starting on its second fifteen years. It can now be regarded as satisfactory for its real purpose, which is the grading of thousands of players in a great variety of kinds of competition."

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 9:27 pm**

by **Chris Majer**

FIDE rating calculations have an equivalent rule that the maximum difference between players is treated as 350 points, i.e. 43.75 ECF points on the basis of the *8 conversion. Bearing in mind the BCF system was originally set up so that it can be done by local graders doing arithmetic, 40 points looks like a reasonable choice.

### Re: The Arbitrary 40 point Rule

Posted: **Thu Apr 24, 2008 10:04 pm**

by **Roger de Coverly**

**FIDE rating calculations have an equivalent rule that the maximum difference between players is treated as 350 points,**

I knew I should have looked it up!

The FIDE 350 cut off is where the probability hits 90% - so under the international system a player rated 2350 (219) is expected to score 9-1 against one rated 2000 (175). Under the ECF system you only need to be 215 to expect 90%.

It's worth noting that the ECF and international systems have slightly different models of how players are expected to perform to maintain their ratings.

### Re: The Arbitrary 40 point Rule

Posted: **Mon Apr 28, 2008 11:54 pm**

by **Anthony Higgs**

Mike Gunn wrote:
The 40 point rule says you treat any player who is graded more 40 points different to you as if the grading difference is exactly 40.

Hmm. I had thought this only applied when the higher rated player won the game, so that this player did not lose points as per the previous examples.

Does it also apply if the higher rated player loses? I (ECF 152) lost to a player graded 92 earlier this season and therefore assumed my grading performance for that game was 42. Should it actually be 62 (152-40-50)? Its similar to the Simon Williams game above - does the system help me here when I don't really deserve it?

Thanks,

Anthony

### Re: The Arbitrary 40 point Rule

Posted: **Tue Apr 29, 2008 12:40 am**

by **Chris Majer**

Anthony wrote

Hmm. I had thought this only applied when the higher rated player won the game, so that this player did not lose points as per the previous examples.

Does it also apply if the higher rated player loses? I (ECF 152) lost to a player graded 92 earlier this season and therefore assumed my grading performance for that game was 42. Should it actually be 62 (152-40-50)? Its similar to the Simon Williams game above - does the system help me here when I don't really deserve it?

The 40 point rule applies equally to both players and to all possible results. Thus the maximum one can gain or lose in any single game is 90 points. As you say 62 points is what will be assigned to you by the grading system for your unfortunate loss. The system is entirely fair so you will get your just deserts.

### Re: The Arbitrary 40 point Rule

Posted: **Wed Oct 22, 2008 5:38 pm**

by **Robert Jurjevic**

My understanding is that the 40-points rule is to better approximate (with the green line in the graphs in the post below) the logistic curve (which is red line in the graphs in the post below) than what would be the case with 50-point rule (blue line in the graphs in the post below)...

http://www.ecforum.org.uk/viewtopic.php ... &start=210
P.S.

Apparently, logistic curve approximates the relationship between expected performance 'p' and grade difference 'd' quite well.