Standard rating formula

This algorithm is to be used for players with $N > 8$ who have not had either all wins or all losses in every previous rated game.

Define the ``Standard winning expectancy,'' $\We$, between a player rated $R$ and his/her $i$-th opponent rated $R_i$ to be

$$\We(R,R_i) = \frac{1}{1 + 10^{-(R-R_i)/400}}$$

The value of $K$, which used to take on the values 32, 24 or 16, depending only on a player's pre-event rating, is now defined as

$$K = \left\{ \begin{array}{ll} 800/(N' + m) & \mbox{if the ev... ... m/2) & \mbox{if the event is \lq\lq half-$K$''} \end{array}\right.$$

where $N'$ is the effective number of games, and $m$ is the number of games the player completed in the event. The following are example values of $K$ for particular values of $N'$ and $m$.

		Value of $K$
$N'$	$m$	Full-K	Half-K
6	4	80	50
6	6	66.67	44.44
6	10	50	36.36
20	4	33.33	18.18
20	6	30.77	17.39
20	10	26.67	16
50	4	14.81	7.69
50	6	14.29	7.54
50	10	13.33	7.27

If $m<3$, or if the player competes against any opponent more than twice, the ``standard'' rating formula that results in $R_s$ is given by

$$R_s = R_0 + K (S - E)$$

where the player scores a total of $S$ points (1 for each win, 0 for each loss, and 0.5 for each draw), and where the total winning expectancy $E = \sum_{i=1}^m \We(R_0,R_i)$.

If both $m\geq 3$ and the player competes against no player more than twice, then the ``standard'' rating formula that results in $R_s$ is given by

$$R_s = R_0 + K (S - E) + \max(0, \; K(S - E) - 10\sqrt{m'})$$

where $m'=\max(m,4)$ (3-round events are treated as 4-round events when computing this extra term). The quantity

$$\max(0, \; K(S - E) - 10\sqrt{m'})$$

is, in effect, a bonus amount for a player who performs unusually better than expected. Note that the value ``10'' in the above formula will change to ``16'' in January 2003.

The resulting value of $R_s$ is the rating produced by the ``standard'' rating algorithm.