How a rating system works
A rating is a measurement, not a verdict. It is a number that tries to predict how a game will go, updated a little after every result. Here is the whole idea in three steps.
1 · The expected score (the logistic curve)
Before two players meet, the rating gap predicts how often each should score:
EA = 1 ⁄ (1 + 10(RB − RA) ⁄ 400)
- Equal ratings → E = 0.5 (a coin-flip).
- A 400-point edge → E ≈ 0.909 (the stronger player should score about 91%).
- It is symmetric: your expected score plus your opponent’s always adds to exactly 1.
2 · The update (surprise moves the number)
After the game, compare what happened to what was expected, and nudge the rating by that surprise:
R′ = R + K · (result − E)
result is 1 for a win, ½ for a draw, 0 for a loss. If you did exactly as expected, the surprise is 0 and your rating barely moves. Beat someone you were expected to lose to, and the surprise is large — so you gain a lot. That is why an underdog who beats a much stronger player can jump by a chunk in one game, while the favourite who wins gains almost nothing. It is the same formula both ways — nobody is being judged.
3 · The K-factor (fast vs steady)
K is how loud each surprise is allowed to shout.
- Big K → ratings react fast to new results, but they wobble — a couple of unlucky games can swing you.
- Small K → ratings are steady and hard to fool, but slow to catch a genuinely improving player.
Real systems tune K by how much they trust a game (new players get a bigger K; established ones a smaller K). In the lab, drag K and watch the trade-off live.
Why a season converges
One game is noisy. But over a whole season of games, the surprises average out, and the ratings sort the players into their true order. We measure how well the season did with a rank correlation, ρ (rho): ρ = 1 means the displayed order matches the true order perfectly. A wide field (clearly different strengths) converges fast; a tight pack stays noisy longer and needs more games. That is not a flaw — a measurement of nearly-equal players is genuinely harder.
Beyond Elo
Elo tracks a single number. Real modern systems also track how sure they are — a player who has played only a few games has a wide uncertainty band; it narrows as they play. That is the idea behind Glicko’s rating deviation and TrueSkill’s multiplayer ranks. The lab shows a simplified “± how sure” column that shrinks with games — the honest version of that idea.
Honest note: a rating measures how a model of games is doing — not a person’s worth, intelligence, or potential. It is a running prediction, nothing more.