The Elo System
Understanding how chess federations like FIDE evaluate relative player strength computationally.
The Mathematical Foundation
The Elo rating system, invented by Arpad Elo, is fundamentally a method for calculating the relative skill levels of players in zero-sum games. It doesn't measure absolute skill—only your expected performance against a specific opponent.
If Player A has rating $R_a$ and Player B has rating $R_b$, the exact expected score (win probability plus half the draw probability) for Player A is given by:
The K-Factor (Volatility)
The $K$-factor acts as a multiplier indicating how much a player's rating can fluctuate based on a single result.
- K = 40: Highly volatile. Used for new players whose "true rating" is not yet statistically established.
- K = 20: The standard multiplier for the vast majority of active tournament players.
- K = 10: The standard multiplier for Grandmasters and established masters (typically 2400+ rating), because their strength is heavily proven and shouldn't fluctuate wildly off a single upset.
Your actual rating update is simply: New Rating = Old Rating + K × (Actual Score - Expected Score).