USCF 30 pt versus Fischer's 9:9

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wlod

USCF 30 pt versus Fischer's 9:9

Post by wlod »

A USCF tournament game is always about 30 points. When you play a stronger opponent then you may win 20pt or you may lose 10pt--the difference is 30pt, or against a somewhat weaker opponent you are risking 18pt, while you may win 12pt--the difference is again 30pt, etc. That's roughly how it is. On the other hand, Fischer claimed that 10:9 match advantage is inessential and worthy of being awarded the world champion title at the expense of the old champion. This in effect suggests that in agreement with the Fischer's view, when the quotient of m-ad ratings (never mind that Fischer never was aware of them) of two players are in proportion 9:10 then one game won by the lower rated player should allow the rating of that player to catch up with that of the other or that the proportion should even become 10:9 (it was 9:10). I am curious to what extent this is true (or false).

Withing the r-ad system, to make Fisher's claim valid is a question of setting properly the dynamic coesfficient dyn. Let A B be the pre-game ratings, where A/B = 9/10. The post-game ratings will be:

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        A'  =  A + dyn*(a*B - b*A)

        B'  =  A + dyn*(b*A - a*B)

where a (0 \< a \< 1) is the result of the game, and b:=1-a. When the game is won by the first (lower rated) player, then the result of the game is a=1 (and b=0). Then

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        A'  =  A + dyn*B

        B'  =  B - dyn*B

The ratings become equal when dyn satisfies the equation:
    • A + dyn*B = B - dyn*B
i.e. when
    • dyn = (B-A)/(2*B)
which, for the Fischer's proportion A/B = 9/10, means that
    • dyn = 1/20
Thus the m-ad system side of the issue is clear. Now I am curious about the interpretation of the 30pt of the USCF rating.

For the sake of completeness, let's compute dyn under which the proportion of the ratings of the two players as above would get inversed, so that we will have A'/B' = 10/9 (where A/B = 9/10). Then
    • 9*(A + dyn*B) = 10*(B - dyn*B)
i.e.
    • dyn = (10*B-9*A)/(19*B)
which, for the Fischer's proportion A/B = 9/10, means that
    • dyn = 1/10
Thus, if we insist on a harmony with Fischer's view, the dynamic constant should be set somewhere between 1/20 and 1/10; say dyn := 1/16 sounds good :).

Regards,
  • Wlod
wlod

Re: USCF 30 pt versus Fischer's 9:9

Post by wlod »

wlod wrote:[...] Fischer claimed that 10:9 match advantage is inessential and worthy of being awarded the world champion title at the expense of the old champion.
* ... and NOT worth of being awarded ...