The Elo rating system is a method for calculating the relative skill levels of players in competitor-versus-competitor games such as chess. It was invented by Arpad Elo, a Hungarian-American physics professor. The system is designed so that a player's rating changes only during a match, based on the opponent's rating and the outcome of the game. The core idea is that if a player beats a much higher-rated opponent, they gain more points than if they beat a lower-rated opponent. Conversely, losing to a much lower-rated opponent results in a larger loss of points.
The Mathematics Behind Elo
The Elo system uses a logistic curve to model the probability of a player winning a game. The expected score for player A against player B is calculated as:
E_A = 1 / (1 + 10^((R_B - R_A) / 400))
Where:
E_A is the expected score for player A.
R_A is the rating of player A.
R_B is the rating of player B.
The expected score is a value between 0 and 1, representing the probability of player A winning (plus half the probability of a draw). A draw counts as 0.5 for each player.
After a game, a player's new rating is calculated using the following formula:
R'_A = R_A + K * (S_A - E_A)
Where:
R'_A is the new rating for player A.
R_A is the old rating for player A.
K is the K-factor, which determines how much a rating changes. Higher K-factors mean more volatile ratings. For most players, a K-factor of 32 is common. For top players, it might be 10 or 24, and for new players, it might be higher (e.g., 40).
S_A is the actual score achieved by player A (1 for a win, 0.5 for a draw, 0 for a loss).
E_A is the expected score calculated earlier.
K-Factor Values
The K-factor is a crucial part of the Elo system, representing the maximum number of rating points a player can gain or lose in a single game. Different organizations use different K-factor values:
FIDE (International Chess Federation):
40 for players new to rated tournaments (until they complete 30 games and/or reach a rating of 2400).
20 for players with a rating under 2400.
10 for players who have achieved a rating of 2400 or more and have played at least 30 games.
US Chess: Uses a similar system but with slightly different thresholds.
Online Platforms (e.g., Chess.com, Lichess): Often use dynamic K-factors or variations like the Glicko system, which is an enhancement of Elo. For simplicity, this calculator uses a standard K-factor of 32.
How to Use This Calculator
Enter the current Elo ratings of both players and select the outcome of the match from Player 1's perspective. The calculator will then show you the new Elo ratings for both players after the match, assuming a standard K-factor of 32.
Example: If Player 1 (1500 Elo) wins against Player 2 (1650 Elo):