How the Elo system works
The Elo system, created by physicist Arpad Elo and adopted by the U.S. Chess Federation in 1960 and FIDE in 1970, rates players so that the difference between two ratings predicts the outcome of a game. After each game, the winner takes points from the loser, and the number of points depends on how surprising the result was.
The formulas
Expected score of player A against player B:
E = 1 ÷ (1 + 10^((Rb − Ra) ÷ 400))
New rating after the game:
R’ = Ra + K × (S − E)
where S is the result (1 for a win, 0.5 for a draw, 0 for a loss) and K the K-factor, which sets how fast ratings move.
Worked example
You are rated 1500 and play an opponent rated 1600, with K = 20:
| Result | Value |
|---|---|
| Your expected score | 36.0% |
| Change with a win | +12.8 |
| Change with a draw | +2.8 |
| Change with a loss | −7.2 |
After a win, your rating becomes 1512.8 and your opponent’s drops to 1587.2. Because you were the underdog, a win earns you more than a loss costs you.
What rating gaps mean
| Rating difference | Expected score of the stronger player |
|---|---|
| 0 | 50% |
| 100 | 64% |
| 200 | 76% |
| 400 | 91% |
| 800 | 99% |
K-factors
| Organization | K-factor |
|---|---|
| FIDE, fewer than 30 rated games, or under 18 and below 2300 | 40 |
| FIDE, rated below 2400 | 20 |
| FIDE, after reaching 2400 | 10 |
| US Chess | varies with rating and number of games |
A higher K makes ratings react faster, which suits new players whose strength is still uncertain.
Beyond chess
The same system, often with adjustments, is used to rank players and teams in Go, Scrabble, football (soccer) and video games. Many sites use variants such as Glicko, which also tracks how reliable a rating is.
Tips
- The expected score counts draws as half a point, so it is not a pure win probability.
- Ratings from different organizations or websites are not directly comparable, even when they use the same formula.
- Over a tournament, add up your expected scores and compare the total with your actual points: K × (points − total expected) is your rating change.