Elo Expected Score

Arpad Elo was a physics professor and a strong amateur player who was asked, in the late 1950s, to fix the American chess federation's rating system. What he produced is now used for chess, football, video games, university admissions research and the ranking of large language models, and almost all of it is contained in one curve: the expected score between two players depends on nothing but the difference between their ratings.

Here is a piece of provenance almost nobody knows. Elo's own book proposes a NORMAL distribution. His model was that a player's performance on any given day is normally distributed around their true strength, and that the expected score is the probability that one such draw exceeds another — which, for two normals, is again a normal cumulative function of the difference. The logistic curve that everyone now calls "the Elo formula" is not his. It was substituted later, by the USCF, because it fitted the observed results better: real players lose to much weaker opponents somewhat more often than a normal distribution predicts, and the logistic has fatter tails. FIDE kept a tabulated normal for years afterwards. So the equation on this page is Elo's system with someone else's curve inside it, and the two disagree by a few percent at large rating gaps.

The 400 is pure convention. There is nothing physical in it. Elo chose the scale so that a 200-point difference — roughly the gap he associated with one class interval — would give an expected score near 0.76, and so that the numbers would land in a familiar range for players used to the older Harkness system. Divide every rating by four and multiply the 400 by the same factor and not a single prediction changes. The scale is a unit, not a measurement.

Notice the assumption hiding in "the difference alone". The model says 2400 against 2200 is the same match as 1200 against 1000, and that is a modelling choice rather than an observed fact. Whether the rating scale is genuinely uniform in this sense has been argued for fifty years, and the evidence is that it is not quite — the curve is somewhat flatter at the top, where a 200-point gap is worth a little less than it is in the middle of the pool.

Two practical cautions. The result is an EXPECTED SCORE, not a win probability: a draw counts half, and in chess at master level most of an expected score of 0.76 is made of draws rather than wins. And the curve is symmetric — the expected scores of the two players always add to exactly one, which is what makes the rating system zero-sum.

One last piece of intellectual history worth having. In 1952, six years before Elo began, Ralph Bradley and Milton Terry published a general model of paired comparisons in Biometrika: give each item a positive strength parameter, and let the chance one beats another be its share of the pair's total strength. Take the logarithm of that strength, scale it by 400 divided by the natural log of ten, and you have Elo's logistic curve exactly. The rating system chess uses is the Bradley-Terry model with a particular choice of units, discovered independently and for entirely different reasons.