Following a discussion on TCCMB Wim van Vugt of the Netherlands wrote the following article. Wim has previously written the "On the Square" articles Numeric or alphanumeric - The final verdict, where he carefully analyzed the pros and cons of notation systems, and Freedom, inequality or brotherhood?, where he tackled the difficult subjects of female chess performance and the ethics of consultation. Now he tackles the questions raised in the discussion of World Champion Tunc Hamarat's public statements about how the world championship process has been changed by ICCF to make it easier to achieve the title. This article is an English version of his Dutch language column "De Vliegende Hollander (13)" which appeared on the Nederlandse Bond van Correspondentieschakers (NBC) web site (follow the "Columns en Artikelen" link in the top navigation bar to his column dated 2004-07-21).
-- J. Franklin Campbell

ABSTRACT

It has been shown that group size may severely influence the playing level of the qualifiers. Even if a constant percentage of 25% is taken as a standard, the smaller the group size the more dominant the role will be for the luck factor. For a change from 17 to 13 players the advantage for the lucky birds turns out to be 23 ELO points decrease in requirement for their TPR. For a 3-stage tournament this results in a decrease of 70 ELO points of the general playing level in the final round.

Last week the discussion was focused once again on the changes in the world championship cycle and, in particular, the changes within correspondence chess. It certainly has much to do with the awful demonstration of FIDE in Libya, where a solid match scheme of 24 games has been consigned to the past and has been replaced by short one-to-one matches, or even rapid games, and if needed a few blitz games. In correspondence chess such a development is hardly conceivable, where a mean reflection time of 6 days per move still is the standard. Also, the new ICCF chess server is going to apply this time schedule. The disputed point publicized by the current world champion Tunc Hamarat is the reduction of the group size from 17 to 13 players.

It was asserted that a smaller group size would move the luck factor to the foreground. Becoming a lucky number one was not the issue but being able to swindle a qualification causing a decrease of level in the finals was the problem, according to Tunc Hamarat. He had concluded that he would prefer not to participate any further in such a kind of Mickey Mouse championship. He had struggled for many years in one qualification after another to end up in the finals and at last getting the world title. Now it would appear that in a few years every man and his wife will walk into the finals. That was just a bit too much for him.

Of course, personally it doesn't matter me because I am far too weak a player ever to sojourn in those regions. So what am I worrying about 13 or 17? My interest is purely academic. What caught me was the question whether it was really true that the luck factor was going to play a much greater role and that Tunc's scenario could be a realistic concern. Or was it only based on feelings and some indescribable intuitive idea? And if it were true then certainly it must be possible to determine this by statistical calculations or some simulation models. And if the effect then proves to be not significant and only based on intuition, then nothing would be able to convince him to get it out of his mind. So I started to try to find the truth of this matter by setting up a thorough statistical analysis of the problem.

The first hurdle to take is how to define the "luck factor". What is good fortune and what is bad luck? And what is a "deserved" qualification? I have tried to solve that as follows: assume one has a group of all equally strong players. Still then there will be a random distribution of scores in which there is an upper half and a lower half. To base any promotion and relegation on this distribution would be ridiculous, although it could happen. Now in order to prove that it's Me the Great Player (MGP) I must clearly enough distinguish myself from the mobs in the middle. And so I must perform considerably better than the random score of the middle man and satisfy at least the so-called MGP score. That in a nutshell is the approach I have taken to find an answer to the question of what the luck factor is.

The random distribution that arises can be approximated by the well-known Gaussian bell-curve for which tables can be found in literature. Herewith can be determined what score is needed to belong to the upper 20% or 25% group, needed for a qualification.

For instance, for a certain group size it then could happen that a score of 63% (or more) is required to get into the upper 20% zone. That critical score of 63% turns out to be dependent of the group size and/or the number of games and also depends on the requirement imposed for qualification: 20% or 25% or any other percentage. For this example shown in the figure above it's 63% that is the MGP score.

The discussion was about the group size in the world championship cycle: groups of 17 players from which four could qualify against groups of 13 players with three qualifiers. In both cases it's 23% of the players. We could stick to this figure and make comparisons to other group sizes as well:

• groups of 9 players with 2 qualifiers
• groups of 13 players with 3 qualifiers
• groups of 17 players with 4 qualifiers
• groups of 21 players with 5 qualifiers.

What has to be done is the following:
Determine for every group size the critical score needed to arrive in the qualifiers zone (the MGP score). In order to do that I have used the following model applying to players of equal strength:

• the probability for a win for white is 30%
• the probability for a draw for white is 50%
• the probability for a loss for white is 20%

The general score for white therewith is 0.30x1 + 0.50x½ + 0.20x0 = 0.55, which fits very well with the value of 55% which we can easily find in all chess magazines. The high percentage of draws (50%) is for the higher playing levels around that figure or even above. For the more common levels of the average player the drawing percentage lies around 30% or lower. Remarkable fact is that the general score of 55% for white turns out to be independent of playing strength (which is however another survey I have done recently). Using this model and accounting for equal numbers of white and black games it follows that the mean score per game is ½ with variance 1/8N (for N games).

Herewith I was able to make the desired calculations, of which the results are summed up in the following table:

The required MGP score always lies higher than the theoretical critical score.

Example: In a group of 13 Me the Great Player must score at least 0.6029x12=7.23 points from 12 games. And so I must make 7½ out of 12, which is 62.5%. This can be translated to a TPR+, or a tournament performance rating which (for a group of 13 players) is 90 ELO points higher than the group average. In the same way the calculations have been made for the other group sizes.

As a sample test from practice I take ICCF-XXI.3/4-final, Section 2 (by email) which group size was 13 players:

All these four qualifiers have satisfied the requirement of the critical value of 62.5%.

Let's return for a moment to the group of 13 players and the critical score of 62.5% which can be translated to a TPR that's 90 ELO points above the group mean rating. One could consider this figure as the statistical noise that arises from the random process. In other words, the noise being 90 ELO points forces me to perform better than that, otherwise I can't stick my head above the mowing-field. This is a very heavy requirement of which one probably hasn't been so aware before! For instance, if I am only 30 ELO stronger than the average playing field then the probability for me to qualify will be rather low and hardly any greater than it is for all the other players in the group: 23%.

For a group size of 17 the noise would be 67 ELO points. This still is considerable and is only 23 points less demanding. So one could circumscribe the luck factor for the change from 17 to 13 as an increase of 23 ELO points in one's TPR in order to qualify and distinguish oneself of the lucky birds. In other words, for a player not especially expected to qualify it has been made 23 rating points easier to qualify with luck.

In itself this doesn't seem to be such a large effect and in the beginning I was inclined to conclude that it wasn't that important at all, the change from 17 to 13. And so it looked as if Tunc had been kind of whining and exaggerating somewhat with his assertions about a grave decrease of playing level of the finals. But looking into it somewhat deeper and more thoroughly I have to come to another conclusion now. It's clear that the qualifiers will be about 23 ELO points weaker in strength on average. And because it goes about a cycle of several (three) rounds, the same 23 ELO points will be missed for every next round again. In short, this multiplier effect intensifies every round until the last round, so that the final could well be about 70 ELO points weaker. And so this effect isn't so innocuous as it seemed to me at first. Has Tunc then been right after all?

© 2004 Wim van Vugt. All rights reserved.