Glicko g Attenuation Factor

This little factor is where Glicko earns its keep on the prediction side. Suppose you are rated 1700 and your opponent is rated 1500 — but their 1500 was assigned after two games and could plausibly be anything from 1100 to 1900. What is your expected score? Elo says 0.76 and does not blink. Glicko says: you do not know, so you should be less confident, and here is exactly how much less.

The mechanism is straightforward once you see it. If the opponent's true strength is uncertain, you should average the Elo prediction over all the strengths they might have. That average is not the prediction at their mean rating, because the curve bends — and for a curve shaped like this one, bending downward above the middle and upward below it, averaging over a spread always pulls the answer toward 0.5. The wider the spread, the harder the pull.

What comes out, when the logistic curve is integrated against a normal distribution of the opponent's strength, is very nearly the same curve with the rating difference multiplied by a shrinking factor. That factor is $g$. It is 1 when the opponent's rating is exact and falls toward zero as the deviation grows; at the system's ceiling deviation of 350 it is about 0.67, so a third of any rating advantage is thrown away. The $3/\pi^{2}$ is what the integral leaves behind, and Glickman is careful to note that it is an approximation to that integral rather than an exact evaluation of it — the exact one has no elementary form.

Only the OPPONENT'S deviation appears here. Your own is absent, and that is not an oversight. Your uncertainty about yourself does not change what you should expect to score — your expectation is already an average over your own possible strengths — but it does change how much the result should move your rating afterwards. That belongs in the update, not in the prediction, and it is one of the tidiest divisions of labour in the whole system.

Read as a piece of ordinary reasoning rather than as algebra, $g$ says something people find easy to agree with and hard to act on: if you do not know how strong someone is, you do not know how likely you are to beat them. A confident prediction requires a well-established rating on both sides of the board, and no amount of certainty about your own strength substitutes for ignorance about theirs.