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).

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).

Wlodzimierz Holsztynski (wlod) wrote:

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.

I forgot a scaling coefficient. The following
formula will work psychologically better:

ar(X) := 1000 + 200*(log(mR(X)) - log(1000))

except that more players would end up with a negative
rating :-) It'd happen to players whose multiplicative
Ockham rating would go below

1000^(1/5) .=. 3.981...

Such players would be losing to the average
players a bit worse than 250:1.

***

regards,

Wlod

Wlodzimierz Holsztynski

1. The Ockham function. Predictions
============================

Given a game between two players X Y with
the pre-game ratings A B, their after-game
ratings respectively a

A' := A + r*(a*B - b*A)
B' := B + r*(b*A - a*B)

hence A'+B' = A+B -- their total rating is constant;
as in the earlier posting: a:b is the result of the
game, i.e. 0 \ a \ 1 and a+b = 1 (hence 0 \ a \ 1);
also, r is the relevance factor of the game (0 r 1).

Their next game will have relevance factor s; consider
the case when the result is the same a:b -- then

A" = A' + s*(a*B - b*A)
B" = B' + s*(b*A - a*B)

A direct computation gives:

A" = A + t*(a*B - b*A)
B" = B + t*(b*A - a*B)

where

t := 1 - (1-r)*(1-s)

(an elegant linear algebra proof is given
at the end of this post). Observe the pattern:

r = 1 - (1-r)
t = 1 - (1-r)*(1-s)

Now, a simple induction extends this pattern:

***

THEOREM 1 Let r_1 ... r_n be the relevance
========= factors of the consecutive games
of a match of players X Y, with the pre-match
ratings A B. Let's assume that the result of each
game is the same a:b. Then the post-match ratings
A_n B_n are as follows:

A_n = A + (1 - Prod(1 - r_k : k=1...n)) * (a*B - b*A)
B_n = B + (1 - Prod(1 - r_k : k=1...n)) * (b*A - a*B)

***

If the relevant game factor were constant, r_k = r,
then, for R := 1-r, we would have:

A_n = A + (1 - R^n) * (a*B - b*A)
B_n = B + (1 - R^n) * (b*A - a*B)

Then, in the case of an infinite match, we would have:

lim A_n = a*(A+B)
lim B_n = b*(A+B)

for n -- oo. The same would be true if for all n,
with possible finitely many exceptions only, we
would have r_k 1/k. The full statement is:

***

THEOREM 2 If Sum(r_k : k=1 2 ...) = oo, then
=========

lim A_n = a*(A+B)
lim B_n = b*(A+B)

***

We see that even when the relevance of the match
games approaches zero, the limit quotient of rating
is still going top be a:b, granted that the relevance
does not converge to zero too fast.


2. The meaning of the relevance constant
================================

To get a feel for the Ockham rating let's first
make a simplifying assumption that the relevance
factor is constant, say p (where 0 p 1).

It's convenient to introduce also q := 1-p
(like in probability theory).

Let's answer the question: after how many
straight loses of the higher rated player, the
ratings of the players will get equal?

Thus let's assume that A B are the two pre-match
ratings of players X Y. Now let player X keep winning.

After n games the ratings are going to be:

A_n = A + (1 - q^n) * B
B_n = B - (1 - q^n) * B

The two are going to be equal if and only if

A + (1 - q^n) * B = B - (1 - q^n) * B

(1 - q^n)*B = (B-A) / 2

q^n = 1 - (B-A) / (2*B)

q^n = (A+B)/(2*B)

n = log ((A+B)/(2*B)) / log (q)

(the numerator and the denominator are both
negative, hence the solution is positive). The
solution, as a rule, is not an integer. After floor(n)
games the player X is still going to be rated lower
than player Y, but after the ceiling(n) games the
player X will overcome player Y.

EXAMPLE An extremely strong newcomer X
======= plays a match against player Y
rated 2000. The initial rating of X is 1000.
Thus it will take

n = log(3/4) / log(q)

for the two ratings to get equal. For instance, for
q = 3/4, i.e. for p = 1/4, it would take exactly
one win by X. In general, the equalization will
happen after n wins by X when:

q = (3/4)^(1/n)
i.e. for
p = 1 - (3/4)^(1/n)

If you are in charge of selecting the parameters
of the Ockham rating function, and if you feel that
equalization of God and 2000 rated player should
happen after 6 straight wins then you'd set

p := 0.0468...

or just below 1/20.

Actually, already after 4 wins it is clear that
player X is (almost certainly) at least as strong
as player Y. Thus perhaps

p := 1 - (3/4)^(1/4) .=. 0.0694

is even better. However, it is important, that
a new player plays different players, so that
s/he will not "punish" just one--it's more fair
this way. Also, when players mix well then
the relevance constant may be lower because,
for instance, the new player will soon play
strong players instead of the same one, whose
rating would go down. Let's study this issue in
the section below. Now let me mention that
with p = 0.0694 our super-strong novice X
will go from 1000 to 1500 ratinig by playing
four games against a 2000 player (who will
go down to 1500). Then X may win 4 games
against a 3000 player and by doing so X
would gain extra half of its new rating, etc.
By playing this way 4 game matches against
stronger and stronger opponents, whose pre-match
rating each time would be the double of the
rating of X, her/his rating after n matches
would be 1000 * (1.5)^n; e.g. after 10 matches
(40 won games in a row) it'd be 57665.
Now you see that the additive representation
may be psychologically easier on your eyes.

***

Regards,

Wlod (Wlodzimierz Holsztynski)

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.

I couldn't have stated it better myself.


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

1. The Ockham function. Predictions
============================

Given a game between two players X Y with
the pre-game ratings A B, their after-game
ratings respectively a

A' := A + r*(a*B - b*A)
B' := B + r*(b*A - a*B)

hence A'+B' = A+B -- their total rating is constant;
as in the earlier posting: a:b is the result of the
game, i.e. 0 \ a \ 1 and a+b = 1 (hence 0 \ a \ 1);
also, r is the relevance factor of the game (0 r 1).

Their next game will have relevance factor s; consider
the case when the result is the same a:b -- then

A" = A' + s*(a*B - b*A)
B" = B' + s*(b*A - a*B)

A direct computation gives:

A" = A + t*(a*B - b*A)
B" = B + t*(b*A - a*B)

where

t := 1 - (1-r)*(1-s)

(an elegant linear algebra proof is given
at the end of this post). Observe the pattern:

r = 1 - (1-r)
t = 1 - (1-r)*(1-s)

Now, a simple induction extends this pattern:

***

THEOREM 1 Let r_1 ... r_n be the relevance
========= factors of the consecutive games
of a match of players X Y, with the pre-match
ratings A B. Let's assume that the result of each
game is the same a:b. Then the post-match ratings
A_n B_n are as follows:

A_n = A + (1 - Prod(1 - r_k : k=1...n)) * (a*B - b*A)
B_n = B + (1 - Prod(1 - r_k : k=1...n)) * (b*A - a*B)

***

If the relevant game factor were constant, r_k = r,
then, for R := 1-r, we would have:

A_n = A + (1 - R^n) * (a*B - b*A)
B_n = B + (1 - R^n) * (b*A - a*B)

Then, in the case of an infinite match, we would have:

lim A_n = a*(A+B)
lim B_n = b*(A+B)

for n -- oo. The same would be true if for all n,
with possible finitely many exceptions only, we
would have r_k 1/k. The full statement is:

***

THEOREM 2 If Sum(r_k : k=1 2 ...) = oo, then
=========

lim A_n = a*(A+B)
lim B_n = b*(A+B)

***

We see that even when the relevance of the match
games approaches zero, the limit quotient of rating
is still going top be a:b, granted that the relevance
does not converge to zero too fast.


2. The meaning of the relevance constant
================================

To get a feel for the Ockham rating let's first
make a simplifying assumption that the relevance
factor is constant, say p (where 0 p 1).

It's convenient to introduce also q := 1-p
(like in probability theory).

Let's answer the question: after how many
straight loses of the higher rated player, the
ratings of the players will get equal?

Thus let's assume that A B are the two pre-match
ratings of players X Y. Now let player X keep winning.

After n games the ratings are going to be:

A_n = A + (1 - q^n) * B
B_n = B - (1 - q^n) * B

The two are going to be equal if and only if

A + (1 - q^n) * B = B - (1 - q^n) * B

(1 - q^n)*B = (B-A) / 2

q^n = 1 - (B-A) / (2*B)

q^n = (A+B)/(2*B)

n = log ((A+B)/(2*B)) / log (q)

(the numerator and the denominator are both
negative, hence the solution is positive). The
solution, as a rule, is not an integer. After floor(n)
games the player X is still going to be rated lower
than player Y, but after the ceiling(n) games the
player X will overcome player Y.

EXAMPLE An extremely strong newcomer X
======= plays a match against player Y
rated 2000. The initial rating of X is 1000.
Thus it will take

n = log(3/4) / log(q)

for the two ratings to get equal. For instance, for
q = 3/4, i.e. for p = 1/4, it would take exactly
one win by X. In general, the equalization will
happen after n wins by X when:

q = (3/4)^(1/n)
i.e. for
p = 1 - (3/4)^(1/n)

If you are in charge of selecting the parameters
of the Ockham rating function, and if you feel that
equalization of God and 2000 rated player should
happen after 6 straight wins then you'd set

p := 0.0468...

or just below 1/20.

Actually, already after 4 wins it is clear that
player X is (almost certainly) at least as strong
as player Y. Thus perhaps

p := 1 - (3/4)^(1/4) .=. 0.0694

is even better. However, it is important, that
a new player plays different players, so that
s/he will not "punish" just one--it's more fair
this way. Also, when players mix well then
the relevance constant may be lower because,
for instance, the new player will soon play
strong players instead of the same one, whose
rating would go down. Let's study this issue in
the section below. Now let me mention that
with p = 0.0694 our super-strong novice X
will go from 1000 to 1500 ratinig by playing
four games against a 2000 player (who will
go down to 1500). Then X may win 4 games
against a 3000 player and by doing so X
would gain extra half of its new rating, etc.
By playing this way 4 game matches against
stronger and stronger opponents, whose pre-match
rating each time would be the double of the
rating of X, her/his rating after n matches
would be 1000 * (1.5)^n; e.g. after 10 matches
(40 won games in a row) it'd be 57665.
Now you see that the additive representation
may be psychologically easier on your eyes.

***

Regards,

Wlod (Wlodzimierz Holsztynski)

"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:


Their next game will have relevance factor s; consider
the case when the result is the same a:b -- then [...]

A direct computation gives:

A" = A + t*(a*B - b*A)
B" = B + t*(b*A - a*B)

where

t := 1 - (1-r)*(1-s)

(an elegant linear algebra proof is given
at the end of this post). Observe the pattern:

r = 1 - (1-r)
t = 1 - (1-r)*(1-s)

Oops, here you go:

Given a game between two players X Y with
the pre-game ratings A B, with result a:b,
their after-game ratings respectively are (by definition):

A' := A + r*(a*B - b*A)
B' := B + r*(b*A - a*B)

It is assumed that 0 \ a \ 1, and b := 1-a,
i.e. a+b = 1. And r is the relevance factor.

Let E be the identity matrix:

1 0

0 1

Let matrix J be the transposition of vector [1 -1].
Let G := [-b a] be called the game result vector.
Thus matrix M := J*G looks like this:

-b a
b -a

Let the pre-game rating value vector V be the
transposition of [A B], and the post-game value
vector V' be the transposition of [A' B'] (the
post-game ratings). Then

V' = (E + r*M) * V

Observe also that:

G*J = -1

(this matrix product does not depend on the
result of the game! It's always -1), and:

M*M = -M

Now consider another, next game between the
same players (played right after the previous one).
Let it have the same result a:b, i.e. the same
game vector G, hence matrix M := J*G stays the
same, while let the relevance factor this time be s.
Then

V" = (E + s*M) * V'

Thus:

V" = (E + s*M) * (E + r*M) * V

= (E + (r + s - r*s)*M) * V

= (E + (1 - (1-r)*(1-s))*M) * V

i.e.

V" = (E + (1 - (1-r)*(1-s))*M) * V

Now, a simple induction gives:

THEOREM 1 Let r_1 ... r_n be the relevance
========= factors of the consecutive games of
a match between players X Y, with the pre-match
ratings A B. Let's assume that the result of each
game is the same a:b. Then the post-match
rating value vector V_n is as follows:

V_n = (E + (1- Prod(1 - r_k : k=1...n))*M) * V_0

(where V_0 is the pre-match rating value vector).

***

Note that the rating "value vector at infinity"

V_oo := (E + M)*V_0

is the transposition of vector (A+B) * [a b],
proportional to vector [a b]. Vector V_oo
is the limit of V_n whenever the infinite sum
of the relevance factors r_1 + r_2 + ... is
infinite. Then it does not depend on the initial
ratings of the two players (otherwise, when
the sum of the relevance factors is finmite,
it does).

**************

Regards,

Wlod (Wlodzimierz Holsztynski)

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