Dear Wlod, I take the liberty of cross posting this to chess misc since i'm
not sure all ratings afficionados and mathematicals fellows or even
Englishmen read chess.politics. Thank you for making this beginning to the
subject. For myself I would like to see 2 things happen:-

1) you example a series of games played among a group of players rated every
100 points between 1500 and 2000, and
2) i was intrigued by the back-of-envelop British system of yore, and should
like to see a comparison of that system with this [since, at least, in that
system you could calculate your own resulting rating on-the-fly.]

Phil Innes

"Wlodzimierz Holsztynski (wlod)" wrote in message
oups.com...
Wlodzimierz Holsztynski

The Ockham rating function
=====================

Index:

1. Introduction
2. Success function sc(a A B)
3. The game's relevance factor; the rating function
4. The simplest relevance factor
5. The strength quotient factor
6. The activity factor
7. SUMMARY


The Ockham rating function is the crucial
component of my Ockham rating system.

This function applies for two-party games
(like chess, weiqi, tennis, soccer, ...).
Each party has a rating before the game.
Their rating after the game will depend on
the result of the game and on their ratings
just before the game, and to some extent
also on other factors like the difference of
playing strength and activity of the two partners
(opponents :-).

(I'll address the issue of the parallel games,
as in the correspondence chess, elsewhere).

I assume that the result of each game is a
real number a \in [0;1], i.e. non-negative and
not greater than 1. This is meant to be the
result of the first player; then the result of the
second player is b := 1 - a.

I will describe certain multiplicative rating
functions. For such functions each player's
rating is always a positive real number (which
may change after each game).

Multiplicative rating has the following property:

Let A B be the ratings of two players
just before their long match. If the running
average score a_n (and b_n := 1 - a_n)
oscillates around a (resp. b := 1-a) then
the ratings of the players will oscillate
near the values:

Aoo := a*(A+B) and Boo := b*(A+B)

respectively. We see that the ultimate ratings
A:=Aoo B:=Boo will be proportional to the
average scores a b:

A/B = a/b

(Certain oscillation is due to the discrete nature
of the problem; the rating should reflect the
temporary state of affairs and not an infinite
time interval).

Let me now introduce the auxiliary notion of
success and relevance, followed by the rating
formula.

2. Success function sc(a A B)
=======================

I define the success function of the first player
in a game as:

sc(a A B) := a*B - b*A

where A B are the ratings of the players
before the game, and a b are the scores
of the players, where b := 1-a, i.e. a+b=1
(remember that a \in [0;1], hence b \in [0;1]).

The success of the second player is:

sc(b B A) = b*A - a*B = -sc(a A B)

The success of the first player is positive,
zero or negative according to the inequalities:

a/b A/B == sc(a A B) 0
a/b = A/B == sc(a A B) = 0
a/b A/B == sc(a A B) 0



3. The game's relevance factor; the rating function
=======================================

For a game between two players of similar skill and
activity the relevance factor is about maximal p, allowed
for the given list. In general, the relevance factor is a
positive number, never exceeding p, where the list constant
parameter p is less 1. In general, the relevance of a game
may depend on the two players X Y (not just on their
ratings but on other factors as well), and on the time
of the game:

rev(game) = function(X Y date)

The new ratings A' B' of two players, rated A B
before the game, are

A' = A + rev(game) * sc(a A B)
B' = B + rev(game) * sc(b A B)

where a b are the game scores of the two players,
hence a + b = 1. It follows that the sum of their
ratings is preserved:

***************
A' + B' = A+B
***************

The higher the relevance factor the higher the
impact of the game on the two partners' rating.

4. The simplest relevance factor
=========================

For an amateur rating list the relevance
factor may be simply a constant like p=1/8.
When p is high (1/8 is perhaps high) then
ratings are sensitive to the temporary fluctuations
of the player's strength. If p is low then ratings
are more stable.

5. The strength quotient factor
=======================

Even for amateur lists one may introduce
the strength quotient factor as, for instance,
the cubical root of the quotient of ratings:

q(A B) := (min(A B) / max(A B)) ^ (1/3)

where A B are the two players' ratings (before
their game). Thus a more sophisticated rating
function will have rev := p*q(A B). Thus a game
between players who in an 18-game match are
expected to end with a score 16:2 the quotient
factor would be 1/2, and it would make the impact
of the game two times smaller than if it were played
between equal partners.

Remark: q(A B) = q(B A)

6. The activity factor
===============

It is also important to take into account
how active the two players are. Thus let me
introduce a player activity coefficient pA(X date)
of a player X at a given moment:

let g1 and g2 be the number of the games
rated for the given list, played by player X
during the year and 2 years respectively,
before the given moment "date"; then:

pA(X date) := min(12 g1) * min(20 g2) / 240

Now the activity relevance of a game between
players X Y at time "date" can be defined as:

act(X Y date) :=

max(1/100, 1 - |pa(X date) - pA(Y date)|)

Then the relevance factor may be defined as:

rev(X Y date) := p * q(A B) * act(X Y date)

Observe that act(X Y date) = act(Y X date), hence

rev(X Y date) = rev(Y X date)


7. SUMMARY
===========

The Ockham rating function is a part of the Ockham
rating system, which can and should be the vital
componet of the organized professional chess world.

After each game rated for a given list between two
players X Y rated A B, the new ratings A' B' of the
players X Y are given by:


A' = A + rev(game) * sc(a A B)
B' = B + rev(game) * sc(b A B)

where sc(A B) is the result of the game,
and rev(game) := rev(X Y date) is the
relevance factor of the game. It is a multiplicative
rating function. The rating list will have a constant
average of rating equal always to 1000.

For psychological reasons the additive translation
of the multiplicative rating will be provided. It is
obtained by the formula

ar(X) := log(mR(X)) + 1000 - log(1000)

where aR(X) mR(X) are respectively the additive
and the multiplicative rating of player X.

The relevance factor will emphasize the games
between the players who are in a similar situation
(whose rating is similar, and their activity).

Each new member of any Ockham rating list
will start with rating equal 1000.

The combination of

the class rating lists + relevance factor + equal entry

solves the problem of accepting new rated players
on a list.

*******

Wlod (Wlodzimierz Holsztynski)

PS. I'll write about the predictive properties
of the Ockham rating function separately
(including a precise meaning of constant p;
see the above sections about the relevance).

Chess One wrote:

Dear Wlod, I take the liberty of cross posting this
to chess misc since i'm not sure all ratings afficionados
and mathematicals fellows or even Englishmen read
chess.politics.

Thank you, Phil (I meant misc anyway but ended on
*.politics accidentally). First of all thank you for encouragement,
and to Jerzy for remembering.

For myself I would like to see 2 things happen:-

1) you example a series of games played
among a group of players rated every
100 points between 1500 and 2000,

I'll try. If there was more iinterest in my Ockham
approach then I would write a simulation in perl
(less work) ot in C++ (more work), so that one could
answer many questions. The pseudo-random constructions
would provide a possibility of simulating massive data.


I hope that some of the participants of rgc*
will see the harmony of the proposed system:
a predictable, simple, elegant rating function plus
the rating lists.

You may also turn your attention to my treatment
of the relevance factor. For instance, the players
new to the list would tend to establish their rating
to much extent by playing first among themselves
(their games with the established members would
hgave less relevance). This is fair. Playing new
players is like lottery. It's better to reduce such
fluctuations of the established ratings. (There is
more to it).

2) i was intrigued by the back-of-envelope British
system of yore, and should like to see a comparison
of that system with this [since, at least, in that
system you could calculate your own resulting rating on-the-fly.]

Phil Innes

You can do use the back-of-envolope to compute
the Ockham rating too. In the simpplest case you need
know only the ratings of the two players. In the case
of the more subtle function, you need to get their
current activity coefficients (from the Internet site of
the rating agent), but the computation is still simple.
(The activity coeffcient of the players would be continuously
computed by the rating agent or rather by their computer).

Regards,

Wlod

PS. Where can I read about the "Brittish
system of yore"?

PPS. The topic of rating or of comparing and
voting and ordering is quite extensive. Perhaps
I'll write how it's done in economy and other
applications (years ago I have rediscovered
their best approach); in chess it would apply
the best to any group of players who played
each other a lot (enough to establish direct
pairwise comparisons). Indeed, for the method
of the pairwise comparisons you need the
pairwise comparisons :-) Such comparisons
are absent in a large group of players.

"Wlodzimierz Holsztynski (Wlod)" wrote in message
ups.com...

Thanks for offering to sample a range - my suggestion are maybe not the best
ones [too close together?] and maybe better to take 1000, 1200, 1500, 1500,
1700 and 2,000 as sample players. Your comments on new player
intedeterminate strength / fluctations, noted.

PS. Where can I read about the "Brittish
system of yore"?

Try fellow mathematicos here - eg David Richerby, or maybe D. N. Walker
knows too?

PPS. The topic of rating or of comparing and
voting and ordering is quite extensive. Perhaps
I'll write how it's done in economy and other
applications (years ago I have rediscovered
their best approach); in chess it would apply
the best to any group of players who played
each other a lot (enough to establish direct
pairwise comparisons). Indeed, for the method
of the pairwise comparisons you need the
pairwise comparisons :-) Such comparisons
are absent in a large group of players.

Wlodzimierz Holsztynski (Wlod) wrote:
In addition to the new players there is also the issue of inactive
players (vide Kamsky). I have addressed these problems, as well as
inflation/deflation--no such thing in the case of the Ockham rating
lists! In particular, to qualify for the next higher rating list, a
player has to be in the top 1/3 of the players from that higher
list, for say 12 consecutive games, when compared oh her/his current
highest rating list (see the details in one of my earlier postings).

That doesn't stop inflation. The primary cause of inflation is, as I
understand it, new players coming in, giving a bunch of rating points
to the established players and then giving up. Your system does
nothing to stop this, as far as I can see, though the idea of having
stratified lists should slow the propagation of these inflationary
points.


Dave.


--
David Richerby Broken Poisonous Gnome (TM): it's
www.chiark.greenend.org.uk/~davidr/ like a smiling garden ornament but
it'll kill you in seconds and it
doesn't work!

David Richerby wrote:

Wlodzimierz Holsztynski (Wlod) wrote:

In addition to the new players there is also the issue of inactive
players (vide Kamsky). I have addressed these problems, as well as
inflation/deflation--no such thing in the case of the Ockham rating
lists! In particular, to qualify for the next higher rating list, a
player has to be in the top 1/3 of the players from that higher
list, for say 12 consecutive games, when compared on her/his current
highest rating list (see the details in one of my earlier postings).

That doesn't stop inflation. The primary cause of inflation is, as I
understand it, new players coming in, giving a bunch of rating points
to the established players and then giving up. Your system does
nothing to stop this, as far as I can see, though the idea of having
stratified lists should slow the propagation of these inflationary
points.

Dave, what's wrong with you or rather
with your comprehension??!!!

There are, in the increasing order of the strength,
rating lists L0 L1 L2 ...

Only the lowest list L0 is not fully under control.
But this does NOT cause any inflation of the higher
lists. The newcomers to the higher lists will be
STRONGER than the average active players of
those stronger lists due to the very criterium
of admission. If anything, one would worry
about deflation!

Let me recall for you the details of the
conditions under which a player from
list L_k can advance to the list L_(k+1)
(s/he will still belong to all lower lists).

Such a player is compared to the players
from the list L_(k+1) on the list L_k.

But only to the players whose rating on L_(k+1)
is at least 1000, which means that they are
at least average for the list L_(k+1), i.e. they
are in the top 50% of L_(k+1).

Now our candidate for 12 consecutive games
has to be, within the list L_k, in the top 2/3
of the active players from L_(k+1) who already
are at least average or better on list L_(k+1).
This means that our candidate has to be,
for 12 consecutive games, in the top 1/3
of the active players of the L_(k+1) list
(in the top 2/3 of the top 1/2, i.e. about in
the top 1/3).

I hope that this makes it clear. I am ready
o answer questions but please try to refrain
from fast and unfounded judgements.

******

The parameters like 12 games or top 2/3
can be adjusted, even repeatedly--no harm,
to keep inflation/deflation in check. One
would follow the rating of the newcomers
of each of the list L_k. This way only
minimal fluctuations would be allowed, and
they would be easily corrected.

Formally, inflation/deflation in my Ockham
system is not possible at all, because the
average rating on each list is always 1000.
Thus in this formal sense the situation is
ideal.

It is however important that the newcomers
to the lists L1 L2 ... (never mind now L0)
will be coming at a sensible rate, and that
they will be doing better but not much better
than the older members. The taking over by
the new generation should take place at
a reasonable pace. It should of course, and
it will, depend on the coming new talent.

The proposed Ockham system should do
it quite well. It is a robust concept, not overly
sensitive to minute changes of parameters.

Regards,

Wlod

PS. One could control L0 strictly too
but it's not too important, it's not worth it.
The weak players from L0 list will have
almost no influence on the rest of the
more advanced players. Some vague
social pressures and the cost should
already keep L0 reasonable, so that
the weakest players will be at least
at the USCF 1400-1500 level (with very
minor exceptions, especially among the
active players).

If you want a control already at the L0
level then a practical (and historically
tested) way would be to have certified
(trusted) chessplayers-examiners. A
candidate would have to score, say,
7.5 out of 12 against an examiner
rated 1000 or 6.5 out of 15 against
an examiner rated 2000 (it's a multiplpicative
scale)..., which would mean that the
candidate during the match is on a level
clearly ABOVE the average 1000 L0-points.

Perhaps it's too much.

Chess One wrote:

"Wlodzimierz Holsztynski (Wlod)" wrote in message
ups.com...

Thanks for offering to sample a range - my suggestion are maybe not the best
ones [too close together?] and maybe better to take 1000, 1200, 1500, 1500,
1700 and 2,000 as sample players. Your comments on new player
intedeterminate strength / fluctations, noted.

Phil, Elo at least got the translation invasriance,
meaning that the expected result of a match
depends only on the difference of the Elo ratings
of the two players and not on the ratings as such.

Thus if you are interested in the 100pt difference
or 200pt or 300pt difference... and if someone
will tell me the predicted match result then I can
tell you how fast the Ockham ratings will converge
from the initial ratings, which perhaps do not reflect
their relative strength properly, to the relatively
proper ratings when the two play a match.

Now, let someone try to do it for Elo :-) :-)
They would have to run a simulation :-)

I think that Elo would be extremely happy if he got
not just the translation invariance but also additivity
(which is equivalent to the multiplicativity). He would
do it if he knew how.

***

In the scientific applications (to economy etc)
the multiplicativity is actually called **consistency**
(not transitivity, as I said earlier).

***

Regards,

Wlod