How Elo ratings actually work, with the formula

The short answer: an Elo rating predicts a result rather than scoring one. The expected score is E = 1 / (1 + 10^((Rb - Ra) / 400)), and after a game your rating moves by K times the gap between what you actually scored and what was expected.

Elo is widely quoted and rarely explained. The underlying idea is simple, elegant, and worth knowing if you follow the game at all — and the same system now underpins ratings in everything from tennis to online video games.

Ratings predict results

The gap between two ratings implies an expected score. The formula is:

E = 1 / (1 + 10^((Rb − Ra) / 400))

Where Ra is your rating and Rb is your opponent’s. It produces a number between 0 and 1 — your expected share of the point.

Rating gap	Expected score for the stronger player
0	50%
50	~57%
100	~64%
200	~76%
400	~91%
800	~99%

The 400-point figure is not arbitrary. It is built into the formula as the gap at which the stronger player is expected to score roughly ten times as often as the weaker one — the constant that defines the whole scale.

Updates are surprise-driven

After a game, the rating change is:

New rating = Old rating + K × (actual score − expected score)

Beating someone you were expected to beat moves your rating barely at all, because the surprise was small. Losing to them moves it a lot.

Work an example. A 2600 player faces a 2400 player. Expected score: about 0.76.

• Wins (K=10): 2600 + 10 × (1 − 0.76) = +2.4 points
• Draws: 2600 + 10 × (0.5 − 0.76) = −2.6 points
• Loses: 2600 + 10 × (0 − 0.76) = −7.6 points

Note that a draw against a much weaker player costs rating. That is the system working correctly — a draw was a worse result than expected.

The K-factor

Player	K
New player, fewer than 30 rated games	40
Under 18 and rated below 2300	40
Most established players	20
Any player who has reached 2400	10

K controls how fast a rating moves. High K for newcomers lets the system find their true strength quickly, which matters because a new player’s initial rating is close to a guess. Low K at the top prevents a single bad tournament from erasing years of results.

The junior exception is deliberate: young players improve fast, and a low K would leave a rapidly improving fifteen-year-old badly underrated for years.

Titles and thresholds

Title	Rating requirement	Norms
Grandmaster (GM)	2500	3
International Master (IM)	2400	3
FIDE Master (FM)	2300	None
Candidate Master (CM)	2200	None

The rating must be achieved at some point — it is not required permanently, which is why titles are never revoked. Norms are strong performances in qualifying international events with specified numbers of foreign and titled opponents.

Both conditions must be met, which is why players sometimes hold the rating for years while hunting a final norm, and occasionally the reverse.

What Elo does badly

• Small samples. A rating based on twenty games is close to noise.
• Isolated pools. If a group of players only ever play each other, their ratings drift relative to everyone else with nothing to anchor them.
• Inactivity. A player who stops competing keeps their rating frozen while their actual strength changes.
• Cross-era comparison. Ratings are relative to the pool, and the pool has changed.

Why inflation arguments happen

Ratings are relative. Comparing a number from one era with a number from another is not straightforward, because the population has changed around them.

A player rated 2700 today is not automatically stronger than one rated 2650 thirty years ago. The rating tells you where someone sits among their contemporaries, and nothing more than that. Anyone arguing otherwise is treating a relative measurement as an absolute one.