Elo Rating Change After a Game

This is the entire update rule, and its economy is the reason the system spread. Compare what happened with what was expected, scale the surprise, and add. No history is kept, no games are re-examined, and the arithmetic can be done on the back of a scoresheet — which mattered enormously in 1960 and is part of why Elo won.

It is also exactly zero-sum. Your opponent's expected score is one minus yours and their actual score is one minus yours, so whatever you gain they lose, provided you share a K-factor. The total number of rating points in a closed pool is therefore fixed, which has a consequence federations spend real effort on: as improving juniors take points from established players and those players eventually leave with their points, the pool can deflate or inflate over decades. Every large rating system has some mechanism bolted on to manage this, and none of them are in the equation.

The K-factor is a choice, and it is the only free parameter Elo has. It is not a constant of the system and there is no correct value for it. A large K makes the rating responsive: it tracks a player who is genuinely improving, and it also swings wildly on a bad weekend. A small K makes it stable: it resists noise, and it is slow to notice that someone has got much better. Every rating body picks its own compromise, and picks differently for different players — FIDE uses a larger K for newcomers and for juniors, whose true strength is genuinely uncertain, and a smaller one for established players at the top, whose ratings rest on hundreds of games. The USCF, national federations and online servers all use different schemes again.

So a K chosen for one pool means nothing in another, and neither do the rating changes it produces. "I gained 40 points last night" is not a comparable statement across two systems, and neither is "I am rated 1900". This is the point on which almost all cross-site rating arguments founder.

A rating is a position in a pool, not a measurement of a person. It is calibrated against the other players in the same pool and against nothing else. Two 1900s from populations that never play each other are two numbers that happen to share three digits; there is no arithmetic anywhere on this site, or anywhere else, that makes them comparable. The number does not measure how well you understand the game, and it does not travel with you when you change communities.

The K-factor also carries the system's biggest known weakness. Because the step size never shrinks, a rating that has been earned over five hundred games moves exactly as far on one result as a rating based on five games. That is obviously wrong — the five-hundred-game rating is far better evidence — and fixing it is precisely what Glicko was invented to do.