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Elo Rating

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Elo Rating System

A method for calculating the relative skill level of players in competitive games, where ratings rise or fall based on match outcomes weighted by the opponent's own rating.

Definition

The Elo rating system assigns every player a numerical rating that estimates their relative skill, then updates that number after every match based on the actual result compared to what the rating predicted. Beat someone rated higher than you, and your rating jumps more than beating someone rated lower, the system is built around surprise, not just wins and losses.

Originally developed for chess by Arpad Elo, the same underlying formula now shows up in esports rankings, sports forecasting models, and various matchmaking systems. Convert between rating scales or estimate outcome probabilities with the Elo Rating Converter.

Formula

Expected Score: E_A = 1 ÷ (1 + 10^((R_B − R_A) ÷ 400))

New Rating: R_A′ = R_A + K × (S_A − E_A)

Where R_A and R_B are the current ratings of the two players, S_A is the actual result (1 for a win, 0.5 for a draw, 0 for a loss), and K is a constant that controls how much a single result moves the rating.

Worked Example

Player A is rated 1600, Player B is rated 1400. Player A's expected score is 1 ÷ (1 + 10^((1400−1600)/400)) ≈ 0.76, a roughly 76% predicted win chance given the rating gap. If Player A wins with a K-factor of 32, their new rating becomes 1600 + 32 × (1 − 0.76) ≈ 1608, a small gain since a win was already the likely outcome.

Key Things to Know

  • The formula is zero-sum between the two players. Whatever rating one player gains, the opponent loses the equivalent amount, keeping the overall rating pool balanced across a large player base.
  • A 400-point gap corresponds to roughly a 10-to-1 predicted win probability, which is why Elo differences of a few hundred points already represent a large practical skill gap.
  • The K-factor is a deliberate trade-off between stability and responsiveness. A system tuned to react quickly to recent form will also swing more on a single unusual result than one built for long-term stability.
  • Elo assumes transitive skill, meaning if A tends to beat B and B tends to beat C, the model assumes A tends to beat C too, real-world matchups don't always follow this cleanly, which is one of the system's known limitations.

Frequently Asked Questions

Because the expected score against a stronger opponent is lower to begin with, so an actual win represents a bigger surprise relative to what the system predicted. Beating someone rated 200 points above you gains far more rating than beating someone 200 points below you, even though both are single wins.
It controls how much a single result moves your rating, a higher K-factor (like 32) makes ratings react faster to recent results, while a lower K-factor (like 16) makes ratings more stable and slower to change. Many chess federations use a higher K-factor for newer players and a lower one once a rating stabilises.
Yes, Elo and its variants are widely used outside chess, including in competitive video games, sports prediction models, and even some dating and matchmaking algorithms. The core idea, that rating changes should reflect how surprising a result was, generalises well beyond its original chess application.
Extremely, a 1000-point gap between two players is enormous. In chess terms, that's roughly the difference between a strong club player and a grandmaster, and the expected win probability for the lower-rated player drops to under 1%.
Yes, a draw counts as a score of 0.5 rather than 1 or 0, so it still moves ratings, just less dramatically than a win or loss. A draw against a much weaker opponent actually lowers your rating, since your expected score against them was well above 0.5.