Elo Rating Match Update

How the Elo rating system works

The Elo rating system came from Arpad Elo, a Hungarian-American physics professor who also happened to be a chess master, and his goal was to rank chess players. The US Chess Federation (USCF) picked it up in 1960, FIDE (the world chess body) followed in 1970, and these days it has spread well past chess into Go, esports, tennis, football and matchmaking systems. What makes it tick is one idea: the gap between two players' ratings tells you the expected result of a game.

How it works

Think of each player's expected score as a probability somewhere between 0 and 1. When player A faces player B, that score works out to E_A = 1 / (1 + 10^((R_B − R_A) / 400)), and B's mirrors it at E_B = 1 − E_A. The 400 in there is what sets the scale, so a player rated 400 points higher is expected to win 10 times for every loss. Once the game is over, the rating shifts by R'_A = R_A + K · (S_A − E_A), where S_A is the actual result — 1 for a win, 0.5 for a draw, 0 for a loss. The K-factor decides how quickly ratings move. FIDE starts new players at K = 40 for their first 30 games, drops that to K = 20 afterward, and uses K = 10 once someone hits 2400 and counts as a master. The USCF, for its part, historically ran K around 32 for typical players. Crank K up and ratings react fast but get jumpy; keep it low and they're steady but slow to catch up.

Worked example

Say player A is rated 1600, player B is rated 1400, and we set K = 20. A's expected score comes out to E_A = 1 / (1 + 10^((1400 − 1600)/400)) = 1 / (1 + 10^(−0.5)) ≈ 0.76. Should A win (S_A = 1), the new rating is R'_A = 1600 + 20·(1 − 0.76) = 1600 + 4.8 ≈ 1605, a modest bump because the win was the likely outcome anyway. Flip the result and have underdog B win, and B picks up roughly 15 points, the reward for pulling off something improbable. Whatever A gains, B loses by the same amount, so the total rating across the pool stays put.

FAQ

Why do I gain so few points beating a weaker player? Because the win was already what everyone expected of you. Elo only pays out for results that beat expectation, and beating a much weaker opponent is "priced in" before the game even starts.

Is a draw always neutral? Not at all. Draw with someone much stronger and your rating still climbs, because your expected score going in was under 0.5.