Elo Matchmaking Calculator
The Elo rating system, developed by Arpad Elo, is the most widely used mathematical framework for ranking players in competitive games. It assigns each player a numeric rating and uses a logistic function to predict expected outcomes based on the rating difference between two opponents. This calculator shows the expected win probability for both players and computes the rating change after an actual game result using the K-factor update rule.
Elo expected score formula
E(A) = 1 / (1 + 10^((R(B) - R(A)) / 400))
E(B) = 1 - E(A)
New R(A) = R(A) + K * (S(A) - E(A))
E(A) is the expected score (win probability) for player A. S(A) is the actual score (1 for win, 0.5 for draw, 0 for loss). K is the K-factor controlling rating sensitivity.
Elo ratings in competitive games
- The Elo system was originally designed for chess but is now used in esports, online games, sports analytics, and even academic ranking systems.
- A 200-point Elo difference corresponds to roughly 76% expected win rate for the higher-rated player.
- Many games use Elo variants such as TrueSkill (Microsoft) or Glicko-2, which add uncertainty estimates to the base Elo framework.
- Matchmaking systems target an expected score near 0.50 for both players to ensure fair and competitive matches.
Elo matchmaking: frequently asked questions
What is the Elo expected score formula?
Expected score for player A is E(A) = 1 / (1 + 10^((R(B) - R(A)) / 400)), where R(A) and R(B) are the Elo ratings. E(A) is the expected fraction of the game won (1 for win, 0.5 for draw, 0 for loss), averaged over many games at these ratings.
What does a 200-point Elo difference mean?
A 200-point Elo gap corresponds to the higher-rated player winning about 76% of games (expected score 0.76). A 400-point gap gives approximately 91% expected score. The relationship is sigmoidal, not linear.
Is the 400-point divisor standard?
The standard Elo system uses 400 as the divisor in the logistic function. Some variants (Glicko, USCF) may use different parameters, but the base formula with 400 is the most widely used in chess and most competitive games.
How do rating updates work after a game?
After a game, each player's rating changes by K * (S - E), where K is the K-factor (sensitivity), S is the actual score (1 win, 0.5 draw, 0 loss), and E is the expected score. New players use a higher K-factor for faster convergence.
What K-factor should I use for my game?
FIDE chess uses K=40 for new players, K=20 for established players, and K=10 for grandmasters. Online games often use K=32. Higher K-factors make ratings more volatile; lower K-factors make them more stable.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: itl.nist.gov/div898/handbook/.
- FIDE Rating Regulations: fide.com.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.