M-ad rating can be applied to any game or even any collection of competitions, between two players, where the result of each game or encounter can be represented by a real number a between 0 and 1. The result of the game is understood to be the result of the first player. The second player result of the same game is 1-a.
Rating may depend on different factors, but the basic m-ad rating function
- Df = Df(A B a)
- Df(A B a) + Df(B A 1-a) = 0
- A' := A + Df(A B a)
B' := B + Df(B A 1-a)
- A' := A + Df(A B a)
- A' + B' = A + B
To keep things simple and clear, each new to the rating list player will get the same initial rating, and this initial rating will always and forever be the average rating of the whole given list, i.e. it will be the average mean of all ratings at any time, it will be always the same. Thus a m-ad chess list open for all players may have the initial, i.e. average rating set to, say, 1400 points. Another list, which admits only strong players, may have the initial/average rating set higher, say to 2000 points, just to tell it easily apart from the other list. Or, perhaps, all lists can set their average rating to 1000, and that's it.
I'll write more about the related issues, which are partly social, perhaps on the other forum.
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When we think about the games between chess players then we feel that some games are more important than others. And for the purpose of rating, the most important games are between the players who are similar with respect to their degree of being in practice, or rusty, or new; and also with respect to their relative playing strength. The games between players of about equal strength should have more weight than between players whose strength is widely apart.
Thus a m-ad rating system for chess players has an auxiliary function
- 0 < wgt(P Q) \< 1[/list
- diff(P Q a) := wgt(P Q) * Df(A B a)
- A' := A + diff(P Q game)
B' := B + diff(Q P 1-a)
- A' := A + diff(P Q game)
I need to run now, but I will continue (I hope),
- Wlod