## Elo Ratings: A Foundation for Sports Modelling
The Elo rating system, originally developed for chess, has become one of the most widely used approaches to rating sports teams. Its elegant simplicity makes it powerful: a team's rating updates after every match based on the result and the prior ratings of both teams.
## The Core Update Formula
After a match:
New_Rating_A = Old_Rating_A + K × (Result − Expected_Result)
Where:
- K = update factor (typically 20–40 in sports)
- Result = 1 (win), 0.5 (draw), 0 (loss)
- Expected_Result = 1 / (1 + 10^((Rating_B − Rating_A)/400))
**Example:**
Team A rating: 1600. Team B rating: 1500. Team A wins.
Expected result = 1 / (1 + 10^((1500−1600)/400)) = 1 / (1 + 10^(−0.25)) = 0.640
Actual result = 1 (win)
New rating for A = 1600 + 30 × (1 − 0.640) = 1600 + 10.8 = 1610.8
## Converting Elo to Win Probability
Expected_Result is directly the probability of winning (accounting for draws via the standard split: e.g. 60% expected → 50% win, 17% draw, 33% loss).
For more accurate 1X2 probabilities, combine the Elo win probability with the historical draw rate for similar match quality differentials.
## Advantages of Elo
- Simple to calculate and update
- Self-correcting: overrated teams lose points until their rating reflects their true quality
- Produces match win probabilities directly
## Limitations
- Does not incorporate within-match information (goals scored, chances created)
- K factor choice significantly affects responsiveness vs stability
- Does not model home advantage directly (must be added manually)
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