Elo Ratings Math

Interpretation:

• E_A = 1.0 means player A is expected to win with certainty
• E_A = 0.5 means both players are equally matched (50% win probability)
• E_A = 0.0 means player A is expected to lose with certainty

Example: If player A has rating 1600 and player B has rating 1400:

E_A = 1 / (1 + 10^((1400 - 1600) / 400))
E_A = 1 / (1 + 10^(-200 / 400))
E_A = 1 / (1 + 10^(-0.5))
E_A = 1 / (1 + 0.316)
E_A ≈ 0.76

Player A is expected to score 0.76 (76% chance of winning).

2. Rating Update Formula

After a game, ratings are updated based on the actual outcome compared to the expected outcome.

Formula:

R'_A = R_A + K × (S_A - E_A)

Where:

• R'_A = New rating for player A
• R_A = Old rating for player A
• K = K-factor (determines rating volatility)
• S_A = Actual score (1 for win, 0.5 for draw, 0 for loss)
• E_A = Expected score (from formula above)

The Update Difference:

ΔR_A = K × (S_A - E_A)

This difference represents:

• Positive value: Player performed better than expected (rating increases)
• Negative value: Player performed worse than expected (rating decreases)
• Zero: Player performed exactly as expected (no rating change)

3. The K-Factor

The K-factor controls how much ratings can change after each game.

Common K-factor values:

• K = 32: High volatility, used for beginners or provisional ratings
• K = 24: Medium volatility, used for intermediate players
• K = 16: Low volatility, used for established/expert players
• K = 10: Very stable, used for top-level players

Adaptive K-factor example (FIDE chess system):

K = 40  if games_played < 30
K = 20  if rating < 2400
K = 10  if rating >= 2400

4. Rating Difference and Win Probability

The relationship between rating difference and expected win probability:

Formula for any rating difference:

Win_Probability = 1 / (1 + 10^(-ΔR / 400))

Where ΔR = R_A - R_B

5. Two-Player Zero-Sum Property

In a two-player game, the rating changes are equal and opposite:

ΔR_A = -ΔR_B

This is because:

E_A + E_B = 1
S_A + S_B = 1 (for decisive games)

Therefore:

ΔR_A = K × (S_A - E_A)
ΔR_B = K × (S_B - E_B) = K × ((1 - S_A) - (1 - E_A)) = -K × (S_A - E_A) = -ΔR_A

Comprehensive Example

Scenario: Player A (rating 1800) plays Player B (rating 1700), K = 32

Step 1: Calculate Expected Scores

E_A = 1 / (1 + 10^((1700 - 1800) / 400))
E_A = 1 / (1 + 10^(-0.25))
E_A = 1 / (1 + 0.562)
E_A ≈ 0.64

E_B = 1 - E_A ≈ 0.36

Step 2: Actual Outcome - Player B Wins (upset!)

S_A = 0 (loss)
S_B = 1 (win)

Step 3: Calculate Rating Changes

ΔR_A = 32 × (0 - 0.64) = 32 × (-0.64) = -20.48 ≈ -20
ΔR_B = 32 × (1 - 0.36) = 32 × (0.64) = 20.48 ≈ +20

Step 4: New Ratings

R'_A = 1800 + (-20) = 1780
R'_B = 1700 + 20 = 1720

Player B gained 20 points for the upset victory, while player A lost 20 points.

Multi-Player Extensions

For games with more than two players, the Elo system can be extended:

Pairwise Comparison Method: Each player's rating change is the sum of their changes against all opponents:

ΔR_i = K × Σ(S_ij - E_ij)

Where:

• i = player being rated
• j = each opponent
• S_ij = actual score against opponent j
• E_ij = expected score against opponent j

Mathematical Properties

1. Conservation of Rating Points: In a closed system with only two-player games, the total rating points remain constant.

2. Logistic Distribution: The expected score formula uses a logistic curve, which creates smooth probability transitions.

3. Rating Scale Calibration: The choice of 400 in the formula means a 400-point difference corresponds to a 10:1 odds ratio (91% vs 9% win probability).

4. Convergence: Over many games, ratings converge toward players' true skill levels, with convergence speed determined by K-factor.

Implementation Considerations

When implementing Elo ratings:

1. Initial Ratings: Typically start players at 1200, 1500, or 1600
2. Minimum Ratings: Consider setting a floor (e.g., 100) to prevent negative ratings
3. Rating Inflation/Deflation: Monitor average ratings over time
4. Provisional Periods: Use higher K-factors for new players
5. Inactivity Decay: Consider rating decay for inactive players
6. Draw Handling: Use S = 0.5 for both players in draws

Extensions and Variants

Glicko and Glicko-2: Adds rating deviation (RD) to account for uncertainty:

RD² = rating variance (higher = more uncertain)

TrueSkill: Microsoft's system using Bayesian inference with skill mean (μ) and skill standard deviation (σ).

Elo with Home Advantage: Add a constant to the home player's rating in expected score calculation:

E_home = 1 / (1 + 10^((R_away - (R_home + H)) / 400))

Where H is the home advantage (typically 30-100 points).

References

• Elo, A. E. (1978). The Rating of Chessplayers, Past and Present
• FIDE Handbook: Rating Regulations
• Glickman, M. E. (1999). "Parameter estimation in large dynamic paired comparison experiments"