Glicko Expected Score Against One Opponent

This is the Elo curve with the rating difference shrunk first. Set the attenuation to one — an opponent whose rating carries no uncertainty at all — and the two are identical, which is the right relationship for a system meant to extend Elo rather than replace it. Everything Glicko adds to the prediction lives in that one factor.

Work an example. You are 1700, your opponent is 1500. If their rating is solid, you expect 0.760. If their rating is a two-game guess with a deviation of 350, you expect 0.684 — a substantial reduction, and it happens without either rating changing by a point. The movement is always toward 0.5 and never away from it, because uncertainty can only make a forecast less confident.

Two things this does not do, and both are worth being clear about.

Your own rating deviation does not appear. That is correct and is explained on the attenuation page: your uncertainty about yourself affects how much the result should move your rating, not what you expect the result to be.

And this is the single-opponent form. Real Glicko processes a whole rating period at once, working out an expected score against each opponent faced and combining them, with each one's attenuation depending on that particular opponent's deviation. There is no way to express a sum over a variable number of opponents in a solver with a fixed set of inputs, so this page gives you the one-opponent case and leaves the summing to you. For a single game — which is what most readers arrive here with — it is the whole answer.

The same caution that governs every page in this shard governs this one. A rating is a position in a pool. Glicko attaches an honest error bar to that position, which is a real improvement, but an error bar on a pool-relative quantity is still pool-relative. It tells you how confidently the system places you among the people you have played, and it tells you nothing at all about how you would fare against a population it has never seen.