Glicko Rating Update (One Game)

Elo's update multiplies the surprise by a K-factor that somebody chose. Glicko's multiplies it by a step size that falls out of the arithmetic, and the whole of the difference is in the denominator: the sum of what was already known and what this game adds.

Read the two terms. The first, $1/RD^{2}$, is the precision of the existing rating — large when the rating is well established, small when it is a guess. The second, $q^{2}g^{2}E(1-E)$, is the information the game itself supplies. Their sum is the precision after the game, and the step size is the inverse of it. When prior knowledge dominates, the rating barely moves; when the game dominates, the result is nearly all the evidence there is and the rating moves a long way. This is Bayesian updating in one line, and it is the correct answer to the objection that Elo moves a five-hundred-game rating as far as a five-game one.

Look at the $E(1-E)$ as well, because it is easy to miss and it says something real. A game between evenly matched players carries the most information — its outcome was genuinely in doubt — while a game against someone far weaker or far stronger carries very little whichever way it goes. So a lopsided pairing shifts a rating less than an even one does, even after allowing for the expected score. Elo has nothing corresponding to this.

The step size here is an OUTPUT. Divide it by the surprise and you get the effective K-factor for that one game, and it is worth doing: a player with a wide deviation of 200 can see an effective K in the hundreds, while a well-established player sees one in the teens. Nobody chose either number.

What this page cannot show is the other half of the update. Playing a game always narrows the deviation, by exactly the same denominator — the new RD is the inverse square root of the summed precision — and that shrinkage is as much a part of Glicko as the rating move. Nor does this page attempt Glicko-2, which adds a third parameter, the volatility, describing how erratic a player's results are. That step is genuinely outside what a solver like this can do: the volatility update has no closed form and is found with an iterative root-finder, which is a program rather than a formula. Multi-opponent updating is out of reach for the same structural reason as on the expected-score page — it is a sum over however many games were played, and the number of inputs here is fixed.