9. Percentage Expectancy

The basic fallacy of the Elo System is that a probability function can be applied as a rating formula without compromising its original meaning.  Syntactically, the Percentage Expectancy Curve may be identical to the logistic function or to normal distribution.  Semantically, it is worlds apart.  The fallacy appears to have escaped notice because it is thrown together with probability concepts which make sense in the context of rating systems.  Among these is the concept of percentage score against a given opposition as a sample of long-term percentage score.  The long-term percentage is essentially a probability and may appropriately be called percentage expectancy.  The percentage score based on a finite sample provides an estimate of this probability.

Performance, in the sense of an ordinary percentage score, is clearly a random variable.  This observation is the starting point for inferences about sampling error and was claimed by Elo as the basic axiom of his system.  The function that emerges as the Percentage Expectancy Curve is not, however, a distribution of percentage scores.  It makes little sense to speak of the probability associated with a rating difference unless this probability is explicitly assigned. Percentage expectancy in this sense is not a random variable at all, but a function of the rating difference (an inverse function, to be precise, since ratings are usually calculated from percentage scores).

A principle that is taken for granted in this fallacious interpretation of percentage expectancy is transitivity of probabilities. (For an amusing counterexample see "Nontransitive Dice and Other Paradoxes" [G].)  It is a commonplace observation in chess and other games that results are not transitive.  If x defeats y, and y defeats z, it does not follow that x defeats z, even though the latter result may be in some sense expected.  This general expectation of transitivity leads erroneously to a probability interpretation, wherein

          P(x defeats z) 

may be inferred from 

          P(x defeats y) and P(y defeats z).  

Elo begins the development of his ratio system by citing an obscure reference to the effect that the odds of x to score over z are

          (P_xy / P_yx) (P_yz / P_zy)  =  P_xz / P_zx

where P_xy is the probability of x scoring over y, etc. [E1, 8.33].  One implication is that if results between x and y are equal, that is, if 

            P_xy  =  P_{yx ,}

their respective results against z will be equal, that is,

             P_xz  =  P_{yz ,} 

 which would be a rare form of justice in chess.

The term percentage expectancy nevertheless has its uses in rating theory.  Despite its probability connotations, it generally denotes nothing more than the expectation that ratings will remain fairly constant, which is useful for extrapolations.  For a given pair of ratings percentage expectancy is the hypothetical result that produces no change in the ratings. It is calculated in any rating system as the inverse of its basic formula, solving for percentage score. 

The inverse of the basic ratio formula [5.2] is

[9.1]        P  =  R / (R + ER_c).

Consequently, the percentage expectancy for any pair of ratings, R and R_c, in a logarithmic system is

[9.2]       P_e  =   antilog_a(R) / [antilog_a(R) + antilog_a(R_c)] .

The base a in the Elo System has been previously cited as 1.005773..., but the antilog is more conveniently calculated for the respective ratings as

             antilog₁₀ (R / 400) .

Reducing the ratio in [9.2] by antilog_a(R) gives

[9.3]       P_e  =   1 / [1 + antilog_a(R_c - R)] ,  

which is the Percentage Expectancy Curve for a ratio system.  

An early criticism of the Elo System was that it applied arithmetic averaging, a linear process, to its nonlinear percentage expectancy function.  For nonlinear systems in general, the expected score W_e against an average opposition rating may not precisely equal the sum of percentage expectancies over the individual ratings, and this leads to small rating errors.  It is probably best to follow USCF practice in this matter and to avoid rating averages in the calculation of percentage expectancy, using instead percentage expectancies against individual opponents.  This, of course, makes the calculation of percentage expectancy, already rather cumbersome in the Elo System, even more cumbersome.