How much difference does it make?
by Wim van Vugt
(posted 21 July 2004)
Wim van Vugt (photo by Frank van der Wolf)

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 3stage 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been consigned to the past and has been replaced by short onetoone
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 socalled 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 wellknown Gaussian bellcurve
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:
group size  requirements to reach the upper 23% zone 
critical score  required MGP score  TPR+ 
9  65.37%  68.8%  140 
13  60.29%  62.5%  90 
17  58.11%  59.4%  67 
21  56.83%  57.4%  53 
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 ICCFXXI.3/4final, Section 2 (by email) which group size was 13 players:


score 
percentage 
1.

Pankratov

8.0

66.7%

2.

Akesson

8.0

66.7%

3.

Nickel

7.5

62.5%

4.

Rakay

7.5

62.5%

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
mowingfield. 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.
