E-Rating
Each Rating has Anti-Rating
Letter "E" may mean "Economic" or
"Enhanced" or something other proper word. While
working with E-Rating I saw that I have two absolutely
equivalent ways to calculate rating.
Usually we calculate rating as measure of strength. But in the
same way we could calculate "anti-rating" as measure of weakness.
For this we have to replace "win" with "loss"
and "loss" with "win". As a result of this
operation we will obtain another rank-list. Ranking by
anti-rating will not exactly be congruent with the Ranking by
rating.
I think that "rating mirror" could be calculated for
all ranking systems. As well as both rank-list are equivalent the
problem is how to join rank-lists by "rating" and
"anti-rating".
It depends on rating system. In general case we have to
calculate two probabilities to win using both rating and
anti-rating and then calculate mean value.
Pw + (1-Pl)
P = -------------;
2 Ra
Pw = ----------;
Ra + Rb
ARa
Pl = -----------
ARa + ARb
R - rating; AR - anti-rating;
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This way gives us probabilities but it does not give unit
rank-list. Another way is to create artificial value such as sqr(Ra/ARa).
Kenneth Massey proposed it when we discuss this problem.
Economical approach
Let's forget mathematical point of view on ranking problems
and try to solve it on other way.
Let's think that results of matches are goods. If teams wins
it means that this team buys a game. If teams lose it means that
it sells a game. Naturally that price for win and loss depends on
strength of team.
| "Seller
model" |
"Buyers
model" |
| Main assumption that
we made in this model is "Price for selling game
(loss) depends on "Seller"
only and does not depends on "Buyer". |
Main assumption that
we made in this model is "Price for selling game
(loss) depends on "Buyer"
and does not depends on "Seller". |
| If during
the some competition team "A" loss some games - it
sells them. "Receipt" from this operation will
be product of the number of losses (La) by
"price" for one game for team "A"
(Ra): Receipt "A" = Ra * La
Team "A" have to spend "Receipt"
to"buying" games (win) from others teams..
"Expenses" for purchase wins will be sum of
products of number of losses (Lia) by "price"
for one loss for team "I" (Ria)
Expenses "A" = Sum (Ri * Lia)
So we have equivalence:
Receipt "A" = Expenses "A"
or
Ra * La = Sum (Ri * Lia)
|
If during
the some competition team "A" wins some games - it
buys them. "Expenses" for this operation will
be product of the number of wins (Wa) by
"price" for one game for team "A"
(ARa): Expenses "A" = ARa * Wa
Team "A" have to covered
"Expenses" by "selling" games (loss)
to other teams. "Receipt" from selling loss
will be sum of products of number of wins (Wia) by
"price" for one win for team "I"
(Ria)
Receipt "A" = Sum (Ri * Wia)
So we have equivalence:
Expenses "A" = Receipt"A"
or
ARa * Wa = Sum (ARi * Wia)
|
| We can
create N such equivalencies. N-1 of them are linearly
independent. Additional equation we can obtain if we suppose
that Sum of all prices is constant. For example it may be
equal to 1 or N or N*1000. (I have to recognize that I do
not know why it must be constant during given competition
"auction"... After three days I came to
conclusion that this normalization means that only one
kind of goods is on this market. It is games) So we
obtain system of N linear independent equations with N
unknown values Ra. You can see that it is the same that we
have in E-Rating.
|
| "Seller-Buyer model" |
| And now we have to join
this to model. I think that proper way to do it is
subtract price for "buying" - ARa from price
for "selling" - Ra. So we have E-Rating
E-Rating
= Ra - ARa
You can see that E-Rating can have positive and
negative values. It is very understandable how to take
into account HOME-GUEST factor. If team "buy"
game (win) at home it pay less "money" because
it easy than "buy" in guest.
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