ARENA: how to rate players

From: moudgill@cs.cornell.edu ( Mayan Moudgill)

Date: Tue, 14 Sep 1993 16:12:20 +0000

One of the best systems I've seen for rating players is that used in chess.
I've attatched the description from the rec.games.chess FAQ, below.
It isn't as complicated as it sounds...
:)
Mayan

*** For the first 20 games (provisional rating): ***

Take the rating of the opponent +400 if the player wins.
Take the rating of the opponent -400 if the player loses.
Take the rating of the opponent if the game is a draw.

Average these numbers.  (If unrated players play other unrated players, this
requires several iterations of the above.)

*** After 20 games (established rating): ***

The maximum amount a player can win or lose per game (called the "K" factor)
varies according to rating.  Players rated under 2100 have a 32-point
maximum; players rated 2100-2399 have a 24-point maximum, and players rated
2400 and up have a 16-point maximum.  (In a "1/4 K" tournament, divide these
maximums by four.)

If players of equal rating play, the loser loses half of the maximum, the
winner gains the same amount.  No change for a draw.

If players of unequal rating play, the higher-rated player gains fewer points
for a win, but loses more points for a loss.  (The lower-rated player does
the opposite, of course.)  A higher-rated player loses points for a draw; a
lower-rated player gains points.  For players rated 400 or so points apart,
the maximum rating change is used for an upset, and the minimum gain/loss is
1 point if the much higher-rated player wins.

The true formula for the number of points won/lost versus the ratings
difference is a curve, but a straight-line approximation for players with a K
factor of 32 points can be used, where every 25 points of ratings difference
is one additional rating point gained/lost starting from a beginning of 16
points for a win/loss, and from zero for a draw.  (I.e., for a 100-point
difference, the higher-rated player gains 16 - 4 = 12 points for a win, but
loses 16 + 4 = 20 points for a loss.  If a draw, the higher-rated player
loses 4 points, the lower-rated player gains 4.)

The actual formula is as follows:

K = K factor
delta_R = (Opponent's rating) - (Player's rating)
Expected_Wins = 1/(10^(delta_R / 400) + 1)
New_Rating = (Current rating) + K * ((Actual wins) - (Expected_Wins))

An established player's rating cannot drop below (his rating - 100) truncated
to the next lowest hundred (i.e., a 1571 player cannot drop below 1400).
This is called the rating's "floor."