Author: James B. Shearer
Date: 12:02:01 10/27/99
Go up one level in this thread
On October 27, 1999 at 13:50:46, Dann Corbit wrote:
>On October 27, 1999 at 11:47:14, Robert Hyatt wrote:
>[snip]
>>If you play someone that far under you, you had better win every game, or else
>>your rating _only_ can go down.
>
>But if they are really that far under you, you will win the expected number on
>average, and your number will stay the same.
>Consider the following table:
>Win expectency for a difference of 0 points is 0.5
>Win expectency for a difference of 100 points is 0.359935
>Win expectency for a difference of 200 points is 0.240253
>Win expectency for a difference of 300 points is 0.15098
>Win expectency for a difference of 400 points is 0.0909091
>Win expectency for a difference of 500 points is 0.0532402
>Win expectency for a difference of 600 points is 0.0306534
>Win expectency for a difference of 700 points is 0.0174721
>Win expectency for a difference of 800 points is 0.00990099
>[snip]
>Win expectency for a difference of 2000 points is 9.9999e-006
>
>If you play someone one hundred points lower than you, they will get 36% of the
>points and you will get 64% and your rating will stay the same.
>If you play someone 200 points lower, you will get 76% of the points and they
>will get 24% and your rating will stay the same. If you play someone 300 points
>lower, you will get 85% of the points. If you play someone 400 points lower you
>will get 91% of the points. At a 500 point difference, you will win 95% of the
>points. At a 600 point difference, you will get 97% of the points. At a 2000
>point difference, you will win 99.99% of the points. In all cases, your rating
>will stay the same. Playing lower rated players should not (in theory) change
>your rating at all. The very rare draw or extremely rare loss to a low-rated
>player will be balanced out by a bazillion wins. On the other hand, games
>against players hundreds of points beneath you are not really very exciting
>[imo]. Who would gather around to watch Kasparov play me? If he played Anand
>or Adams or some highly skilled player, that would be something people want to
>watch. The reason is that I have basically no chance of winning so the outcome
>is pretty well known even before we start. So from a point of interest, I don't
>think it makes a lot of sense to play opponents that are miles beneath.
>
>Those that claim your rating can be inflated by choosing opponents are not aware
>of how the math works. And (let's suppose) that you have played someone ten
>times and lost them all. You might think that -noplay would be good for you.
>But look at all the recent SSDF contests where one program had a big lead and
>suddenly lost it. Without a huge number of games, there is really no way to
>know what the win expectancy would be, and once we know it accurately, then it
>will only reflect upon our true rating.
>[snip]
This is just wrong.
First you ignored the point of Hyatts post. The Elo system assumes
ratings are real numbers but on ICC they are rounded to the nearest integer.
This introduces a bias particularly for opponents with very different ratings.
Suppose for example you are playing someone 800 points below you. Suppose you
should be getting .4 rating points for every win and losing 31.6 rating points
for every loss, 15.6 rating points for every draw (these are not the actual
exact values). However because ICC rounds ratings to the nearest integer you
will in fact gain 0 points for a win and lose 32 points for a loss, 16 points
for a draw. So in effect you are unjustly losing .4 rating points every time
you play this guy.
Second you are ignoring the fact that the ELO system is based on a
model of chess strength which is just an approximation of reality. For example
the ELO system predicts that if A beats B 2/3 of the time and if B beats C 2/3
of the time then A will beat C 4/5 of the time. There is no theoretical reason
for this to be the case, it is just an empirical approximation. You could use a
model which predicted A would beat C 3/4 of the time or 5/6 of the time and
derive a different rating system based on it. To the extent that a win
probability model differs from reality a rating system based on it will have
systematic anomalies where the actual win probability for certain rating
differences differs from the predicted win probability since it will be
impossible to get the win probabilities right for all rating differences. These
anomalies are likely to be particularly apparent at large rating differences
since the ratings will adjust to try to get the common (fairly small rating
difference) cases more nearly correct at the expense of the rare cases.
James B. Shearer
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