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)