Forum Archive : Ratings

Strange behavior with large rating difference

From:   Ron Karr
Address:   karr@best.com
Date:   26 May 1996
Subject:   Re: rating calculation
Forum:   rec.games.backgammon
Google:   4oa652$jk9@nntp1.best.com

Sheldon Richter wrote:
> I really do not understand the ratings calculations.
> I thought that the ratings would change more for
> longer matches, and less for shorter matches.
> Can anyone please explain the logic of the following:
> rating calculation:
> rating difference D=1190.934235
> match length      N=64
> Experience: magic_one 6961 - wes 7871
> Probability that underdog wins: Pu=1/(10^(D*sqrt(N)/2000)+1)=0.000017
> P=0.000017 is 1-Pu if underdog wins and Pu if favorite wins
> K=max(1 , -Experience/100+5) for magic_one: 1.000000
> change for magic_one: 4*K*sqrt(N)*P=0.000551
> K=max(1 , -Experience/100+5) for wes: 1.000000
> change for wes: -4*K*sqrt(N)*P=-0.000551
> rating calculation:
> rating difference D=1187.363751
> match length      N=3
> Experience: magic_one 6967 - wes 7874
> Probability that underdog wins: Pu=1/(10^(D*sqrt(N)/2000)+1)=0.085668
> P=0.085668 is 1-Pu if underdog wins and Pu if favorite wins
> K=max(1 , -Experience/100+5) for magic_one: 1.000000
> change for magic_one: 4*K*sqrt(N)*P=0.593523
> K=max(1 , -Experience/100+5) for wes: 1.000000
> change for wes: -4*K*sqrt(N)*P=-0.593523

The theory is:  in a longer match, the lower-ranked player is less likely
to win than in a shorter match.  The points awarded to the favorite, if
he wins, are proportional to this probability.  It's also proportional to
the square root of the match length.  In most cases, the favorite will
get more points for winning longer matches; it's just that the amount
increases less than linearly as the match length increases.

This is a very odd case, where the ratings difference is unusually huge
(and wes is known as a player who deliberately throws matches).  So in a
64-point match, the odds of the "underdog" winning are calculated as
infinitesimal, so the favorite will not get very many points for winning
the match.  (If wes had won, he would have gotten a ton of points.) In
the 3 point match, the favorite "only" has an 85% chance of winning, so
even though the match is shorter, he gets more points for winning.

Technically, this could be considered a "flaw" in the ratings formula,
but in reality I wouldn't even try to have the formula be accurate in the
ridiculous case of an 1190-point ratings difference between players.
It's more important to be accurate in the normal cases of a few hundred
points difference.

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