One equation, one widget, and the reason a win can pay nothing.
Every rating system starts from the same idea: a player's number is anestimate of hidden strength, and we test it by predicting the next result. You are not awarded points for winning; you are corrected only when reality differs from the prediction.
Elo turns the gap between two ratings into E, the share of the points you are forecast to take. Draws count as half a win, so E is an average score, not a win probability.
E = 1 / (1 + 10^((R_opp − R_you) / 400))The 400 is a scale constant: a 400-point lead means the stronger player is expected to score 10× the weaker player's points — about 91%. A 200-point lead is roughly 76%.
After the game, move each rating by how far the actual score S (1 win / ½ draw / 0 loss) surprised the forecast:
R′ = R + K · (S − E)Meeting expectation (S = E) changes nothing. Over-performing gains; under-performing loses. K sets how far one game can move you.
Expected score E = 0.984
Rating change = +0.5 (+0 on the board)
Why "won twice, gained nothing" happens. E ≈ 0.984; a win is worth only K·(1 − E), which rounds to 0 on the board. The maths is correct — the payout design fails the player.
E ≈ 0.984. A win is worth only K·(1 − 0.984) — at K=32 that is +0.5, which rounds to 0 on the board. The maths is working exactly as designed; the design is what fails the player. Fixing the payout (a floor, a clamp, or a bigger unit) isPricing a win.K is a dial between two failure modes. Large K finds true skill fast but makes one lucky night swing the number. Small K is stable but leaves new and improving players stuck near their seed. Chess uses tiered K (higher for juniors and new players, lower for masters). Convocados' Season Rank uses K=64 for provisional players and 32 after.
Ninety percent. A 400-point gap is a 10:1 points ratio, i.e. about 91%.
Change the rating more. Larger K scales the whole surprise term, so one game moves further.
Exactly as expected. S − E ≈ 0 when the favourite wins, so the update is near zero.
Read"How Elo Ratings Actually Work"(Chess.com) for the friendliest derivation, then skim theWikipedia Elo article for the formulas and worked example. Keep theglossary open as you go.