Statistical
Illustrative logistic Elo expected score and one-game rating update
Calculate Arpad Elo's logistic expected match score E_A = 1 / (1 + 10^((R_B - R_A) / 400)) and post-game rating update R'_A = R_A + K * (S_A - E_A) for chess, esports, and LLM arenas.
Expected score is not necessarily win probability
This is a textbook logistic Elo model, not a reproduction of official FIDE tables, rating-period rules or a particular leaderboard implementation. K=32 is a chosen example parameter.
Expected score equals win probability plus half the draw probability when draws score0.5. Without a separate draw model it is not a win probability. The logistic expected-score odds change by a factor10 per400 rating points, not necessarily the actual win-to-loss odds.
The update uses the unrounded expected score. Results displayed to one decimal are illustrative; use your competition’s rules for official ratings. Ratings must be numeric, K nonnegative, and the game score exactly0,0.5 or1.
| Elo Match Metric | Formula | Underdog R_A = 1600 vs R_B = 1800 (K = 32) | Equal Match R_A = R_B = 1600 (Draw) |
|---|---|---|---|
| Expected Score E_A | 1 / (1 + (10 ^ ((B3 - B2) / 400))) | 24.03% (0.2403) | 50.00% (0.5000) |
| New Rating After Win (S_A = 1) | B2 + B4 * (1 - E_A) | 1600 + 24.3 = 1624.3 | 1600 + 16.0 = 1616.0 |
| New Rating After Draw (S_A = 0.5) | B2 + B4 * (0.5 - E_A) | 1600 + 8.3 = 1608.3 | 1600.0 (Zero change!) |
Elo Logistic Expected Match Score E_A (0 to 1)
Note Opponent rating R_B (B3) minus Player A rating R_A (B2) in the exponent numerator.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B4<0,AND(B5<>0,B5<>0.5,B5<>1)),"Invalid inputs",ROUND(1 / (1 + (10 ^ ((B3 - B2) / 400))), 4)))Updated Post-Match Elo Rating R'_A = R_A + K * (S_A - E_A)
Updates current rating B2 using development factor K (B4) and actual match score B5.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B4<0,AND(B5<>0,B5<>0.5,B5<>1)),"Invalid inputs",ROUND(B2 + B4 * (B5 - (1 / (1 + (10 ^ ((B3 - B2) / 400))))), 1)))Try the example
Use the original values or change them to test your understanding.