Elo rating calculator

The Elo rating system is a method for calculating the relative skill levels of players in zero-sum competitive games. Developed by Hungarian-American physics professor Arpad Elo in 1960 and adopted by the International Chess Federation (FIDE) in 1970, it has since spread far beyond chess into nearly every domain of competitive gaming: League of Legends, Rocket League, Dota 2, chess.com, Lichess, Go servers, and even some professional sports leagues use Elo or Elo-derived systems to rank players and match opponents of similar ability.

The core insight of the Elo system is that a player's rating is not an absolute measure of skill but a relative one — it reflects your performance compared to the population of players you face. Beating a much stronger opponent earns far more points than beating a weaker one, while losing to a stronger player costs barely anything. This elegant asymmetry ensures that ratings converge toward each player's true skill level over time, making it one of the most robust and widely-used ranking systems in the history of competitive games.

How does the Elo formula work?

The calculation proceeds in two steps: first compute the expected score based on both players' current ratings, then update the rating based on the difference between actual and expected outcome. Four variables define the system.

• Player A rating (Ra): the current rating of the player whose score change you are computing. For example, 1400 for an intermediate player on Lichess or chess.com.
• Player B rating (Rb): the opponent's current rating. The difference Rb – Ra directly determines the expected outcome and thus the magnitude of rating change.
• Result (Sa): 1 for a win, 0.5 for a draw, 0 for a loss. This is the actual outcome compared against the predicted expected score Ea.
• K-factor: the sensitivity multiplier. FIDE uses K = 40 for new players (<30 rated games), K = 20 for established players, K = 10 for players rated ≥ 2400. Online platforms often use different values; the higher the K, the faster ratings change.

Formula: Expected score Ea = 1 / (1 + 10^((Rb – Ra) / 400)). Updated rating: Ra' = Ra + K × (Sa – Ea). The constant 400 in the denominator calibrates the scale: a player rated 400 points higher is expected to win approximately 91% of the time.

Real-world example: the underdog wins

Consider Player A rated 1400 against Player B rated 1600 — a 200-point gap in B's favor. Player A's expected score: Ea = 1 / (1 + 10^((1600 – 1400) / 400)) = 1 / (1 + 10^0.5) = 1 / (1 + 3.162) = 1 / 4.162 ≈ 0.240. Theory predicts Player A wins only 24% of the time. With FIDE's K = 20 for established players, if A wins (Sa = 1): gain = K × (Sa – Ea) = 20 × (1 – 0.240) = 20 × 0.760 = +15.2 points → A's new rating: 1415. Player B loses: 20 × (0 – 0.760) = –15.2 points → B drops to 1585. If instead B had won as expected (Sa for A = 0): A loses K × (0 – 0.240) = –4.8 points → drops to 1395. B gains 4.8 → rises to 1605. For a draw: A gains 20 × (0.5 – 0.240) = +5.2 → rises to 1405; B loses 5.2 → drops to 1595. Notice how an upset win is worth three times as many points as holding a draw against a higher-rated opponent — this asymmetry is what makes the Elo system self-correcting and resistant to inflation over time.

Frequently asked questions about the Elo rating system