OK, I’ll correct you.
First of all, as you correctly point out, this is not a problem with the “standard formula” used for players with 9 or more previous games. In that case, the minimum you can gain from a win is 0 (or actually, a small fraction greater than zero), and the maximum is your K-factor (minus a small fraction). Likewise, for a loss you will lose between 0 and K points, and for a draw you will gain or lose at most K/2 points.
As far as the “special formula” is concerned, if you look at “Approximating Formulas for the USCF Rating System” at http://math.bu.edu/people/mg/ratings/approx/approx.html you will find the following statement:
“It should be noted that, from the approximating formula, a player could gain rating points by losing to a high rated player, or lose rating points with a win over a low rated player. The actual rating procedure corrects for these possibilities.”
And, if you look at the more technical version of “The USCF Rating System” at http://math.bu.edu/people/mg/ratings/rs/rs2.html you will find the concept of a “provisional winning expectancy” along with the statement that “the goal, then, is to determine the value of R such that … the sum of provisional winning expectancies [ is the same as ] the actual attained score when a player is rated R … The procedure to find R is iterative …”. (Pardon me, Prof. Glickman, if I have butchered your mathematical prose here.)
The “provisional winning expectancy” is pretty much the same as the regular winning expectancy (the formula involving base-10 anti-logarithms), except that the provisional version is a crude approximation.
Your provisional winning expectancy is defined as follows:
(1) 0 if your opponent is rated 400 or more points higher than you.
(2) 1 if your opponent is rated 400 or more points lower than you.
(3) linearly between 0 and 1 if your opponent is rated between these limits.
For example, your expectancy is 1/2 if you and your opponent have the same rating, 1/4 if he is 200 points higher than you, or 3/4 if he is 200 points lower than you.
As far as I can tell, the only reason they use the “provisional winning expectancy” instead of the real McCoy is to keep it in line with the highly advertised, and extremely familiar, plus-or-minus-400 method of calculating new ratings.
So now let’s look at one of these cases where a previously unrated player “loses points” (i.e. ends up lower than he otherwise would) by winning a game. Let’s say he draws four players rated 1800, and defeats a 1300:
D1800
D1800
D1800
D1800
W1300
This gives him four 1800 performances and a 1700 performance, which averages out to 1780. He would be better off without the win, i.e.
D1800
D1800
D1800
D1800
which would average out to his “intuitive” rating of 1800.
The USCF paper goes through a humongously complex iteration process (which I have not yet attempted to understand) to arrive at this “intuitive” rating, but it seems to me that the following would work just as well in most cases:
-
Calculate the new player’s rating using the familiar add-400-for-each-win, subtract-400-for-each-loss method.
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If the player’s rating, thus calculated, comes out more than 400 points higher than a player he defeated, or more than 400 points lower than a player he lost to, throw out that result and do the calculation again.
-
Repeat step 2 as often as necessary. (It could happen that, once the rating is re-calculated, there will now be more results that should be thrown out.)
In particular, this would mean that a player who loses ALL his games would come out 400 points lower than his LOWEST opponent, rather than 400 points lower than his average opponent. In fact, USCF says this is what happens anyway.
Bill Smythe
