Sports Analytics & Tournament Ranking
Colley Bias-Free Laplace Sports Rating Formula
Calculate Wesley Colley’s Laplace-regularized Strength-of-Schedule team rating r_i = (1 + (W - L)/2 + SUM(r_opp)) / (2 + W + L), Effective Wins W_eff, and Preseason Shrinkage vs Raw Win%.
Rank unbalanced schedules cleanly from Week 0 without margin-of-victory bias!
Raw Winning Percentage W / (W + L) fails two basic tournament tests: before Game 1 (0-0) it divides by zero, after Game 1 a 1-0 team ties a 12-0 champion at 1.000, and beating cupcakes counts identically to beating contenders! Astrophysicist Wesley Colley’s Bias-Free Matrix Method (an official NCAA BCS computer ranking model) starts from Pierre-Simon Laplace’s Rule of Succession r_Laplace = (1 + W) / (2 + T) (which gives every 0-0 team 0.5000 and shrinks 1-0 to 2/3 = 0.6667 vs 13/14 = 0.9286 for 12-0!), and then replaces raw wins W with Strength-of-Schedule Effective Wins W_eff = (W - L) / 2 + SUM(r_opp), yielding Colley’s exact self-consistent rating equation r_Colley = (1 + (W - L) / 2 + SUM(r_opp)) / (2 + W + L)!
Verify on Team_Wins W = 7 (B2), Team_Losses L = 3 (B3 -> Total Games T = 10), and Sum_of_Opponent_Ratings R_opp = 6.50 (average opponent rating 0.6500 across a brutal gauntlet schedule, B4): Raw Win% is 7 / 10 = 0.7000; Unadjusted Laplace Rating is (1 + 7) / (2 + 10) = 0.6667; Colley Effective Wins jump to (7 - 3) / 2 + 6.50 = 8.50 wins (crediting +1.50 wins for facing 0.6500 opponents rather than 0.5000 average teams!); and Colley Rating reaches (1 + 8.50) / 12 = 0.7917! Test Week 0 Preseason (W = 0, L = 0, R_opp = 0): Raw Win% hits a guarded "Preseason (0 Games)" state while Colley Rating cleanly returns 0.5000!
| Tournament Ranking Metric (7-3 Record vs 6.50 Opponent Rating Sum) | Formula | Verified Output |
|---|---|---|
| Raw Win Percentage | W / (W + L) | 0.7000 (#DIV/0! at 0-0) |
| Laplace Unadjusted Rating | (1 + W) / (2 + W + L) | 0.6667 (0.5000 at 0-0) |
| Colley Effective Wins W_eff | (W - L) / 2 + R_opp | 8.5000 (+1.50 SoS Credit) |
| Colley Matrix Rating r_Colley | (1 + (W - L)/2 + R_opp) / (2 + W + L) | 0.7917 |
Colley Strength-of-Schedule Rating Formula
Validate W >= 0, L >= 0, and 0 <= R_opp <= W + L (since each opponent rating lies in [0, 1], the sum of T opponent ratings cannot exceed T = W + L)!
=LET(wn_c, B2, ls_c, B3, rp_s, B4, gm_t, wn_c + ls_c, IF(OR(wn_c < 0, ls_c < 0, rp_s < 0, rp_s > gm_t), "Invalid Schedule Record", ROUND((1 + (wn_c - ls_c) / 2 + rp_s) / (2 + gm_t), 4)))Effective Wins & Cupcake Schedule Penalty Check
Switch R_opp to a weak cupcake schedule R_opp = 3.50 (0.3500 average opponent): a 7-3 team drops to W_eff = 5.50 and r_Colley = 6.50 / 12 = 0.5417!
=IF(ISTEXT(E5), E5, ROUND((B2 - B3) / 2 + B4, 4))Try the example
Use the original values or change them to test your understanding.