11. Methods of Calculation

 

  


The modern electronic computer does not care whether rating calculations are streamlined or cumbersome, simple or difficult.  Nevertheless, examination of the ways that humans grapple with these calculations can provide insight into rating theory.  We begin with the concept of percentage expectancy, which turns out to be quite useful in calculations if we avoid embroilments in probability theory.  Consider the ordinary percentage score

[11.1]          Po  =  Wo / No ,

which in chess has the special meaning of Wo points scored (including draws) in No games.  The subscripts indicate original values, as we are considering the process by which percentage score changes.  For the next event, in which W points are scored out of N games, the new percentage is

[11.2]          Pn  =  (Wo + W) / (No + N) . 

The original percentage may be written

[11.3]         Po  =  (Wo + NPo) / (No  + N) ,

which makes it easy to subtract the old rating from the new, giving

[11.4]         DP  =  (W - NPo) / (No  + N) .

This formula can be applied in recursive fashion from one event to the next, with No growing by accumulation of successive values of N.  A good approximation is obtained by holding No constant once it has become reasonably large, as  

[11.5]          DP  =  (W - NPo) / No .

If we define percentage score for an individual event as

 [11.6]           Pi  =  W / N ,

we can write

[11.7]          DP  =  (Pi - Po) (N / No) ,

which is analogous to Elo's change formula [7.6].  Let us suppose that this formula is applied with the expectation that percentage score will remain fairly constant from one event to the next.  Then the original percentage score, Po, will be the "percentage expectancy," and [11.5] may be written

[11.8]              DPe  =  (W - NPe) / No .

A method for calculating rating changes from changes in percentage expectancy can now be demonstrated.  We will use interval (linear) ratings by way of illustration.  Suppose that a player rated R faces an opposition rated ERc.  His percentage expectancy is calculated from [7.2].  As he scores W points, the change in his percentage expectancy is calculated from [11.8], and his new percentage expectancy is

[11.9]               Pn  =  Pe  +  DPe .

This new value may be substituted into the basic formula to give the new rating.

[11.10]             Rn   =  ERc + K(2Pn - 1) .

For linear ratings this is a somewhat roundabout method since a change formula is easily derived. From the basic formula we have the original rating as 

[11.11]            Ro  =  ERc + K(2Pe - 1) .

Subtracting from [11.10],

[11.12]            DR  =  2K(Pn - Pe) ,

and substituting by [11.8],

[11.13]           DR  =  2K(W - NPe) / No .

The method illustrated by [11.10] may have its uses in more complicated nonlinear ratings.  Another method of calculation uses [11.8] and the derivative of the basic rating formula with respect to P in a differential equation.  This was the method used by Elo to derive his established formula (Established Ratings).  In linear ratings the derivative is constant, and [11.13] illustrates the method well enough with 2K as the derivative.  In nonlinear ratings, the differential is an approximation.