Quantitative Forecasting & Prediction Markets
Shin Method vs Multiplicative Vig-Removal Formula
Strip bookmaker overround S = 1/O_1 + 1/O_2 - 1 via Hyun Song Shin’s (1993) insider-trading equilibrium to correct Favorite-Longshot Bias in implied probabilities.
Why proportional vig removal overestimates longshots — and how Hyun Song Shin’s (1993) model fixes it!
When converting decimal market odds (O_1, O_2 > 1) into true forecasting probabilities, raw inverse odds q_1 = 1 / O_1 and q_2 = 1 / O_2 sum to S = q_1 + q_2 >= 1 (Overround / Vigorish = S - 1). Naive Multiplicative Normalization divides equally by S (p_mult_i = q_i / S), assuming dealers shade favorites and longshots by the same proportion! Hyun Song Shin (1992, 1993) proved that when dealers protect against informed order flow share z in [0, 1), longshots carry far larger percentage markups because a single informed bet on a 4.00 longshot inflicts 3x the dealer loss of a 1.25 favorite! Solving Shin’s exact 2-way equilibrium gives p_Shin_i = (SQRT(z^2 + 4*(1 - z)*(q_i^2 / S)) - z) / (2*(1 - z)) with exact closed-form insider share z* = ((q_1^2 / S) - p_1^2) / (p_1 * (1 - p_1)) where p_1 = q_1 - (S - 1) / 2!
Verify on Favorite_Odds_O1 = 1.25 (B2 -> q_1 = 0.8000) and Longshot_Odds_O2 = 4.00 (B3 -> q_2 = 0.2500): Overround is 1.0500 - 1 = 0.0500 (5.00% vig). Multiplicative normalization assigns Longshot 0.25 / 1.05 = 0.2381 (23.81%), whereas Shin’s model recovers Implied Insider Share z* = 0.0510 (5.10%) and lowers Longshot True Probability to 0.2250 (22.50%, raising Favorite to 0.7750)!
| Implied Probability Metric (O_1 = 1.25, O_2 = 4.00) | Formula | Verified Output |
|---|---|---|
| Bookmaker Overround / Vig (S - 1) | (1 / O_1) + (1 / O_2) - 1 | 0.0500 (5.00% Overround) |
| Multiplicative Longshot Prob | q_2 / S | 0.2381 (23.81% — Overstated!) |
| Shin Implied Insider Share (z*) | ((q_1^2 / S) - p_1^2) / (p_1 * (1 - p_1)) | 0.0510 (5.10% Informed Flow) |
| Shin Fair Longshot Probability | (SQRT(z*^2 + 4*(1-z*)*q_2^2/S) - z*)/(2*(1-z*)) | 0.2250 (22.50% True Prob!) |
Shin Implied Insider-Trading Fraction z* Formula
Validate decimal odds O_1, O_2 > 1 with non-negative overround S = 1/O_1 + 1/O_2 >= 1 (and interior probabilities p_i > 0).
=LET(o_1, B2, o_2, B3, IF(OR(o_1 <= 1, o_2 <= 1), "Require Decimal Odds > 1", LET(q_1, 1 / o_1, q_2, 1 / o_2, s_sm, q_1 + q_2, IF(s_sm < 1, "Arbitrage Market (Sum < 1)", IF(s_sm = 1, 0, LET(p_1, q_1 - 0.5 * (s_sm - 1), IF(OR(p_1 <= 0, p_1 >= 1), "Invalid Extreme Overround", ROUND(((q_1^2 / s_sm) - (p_1^2)) / (p_1 * (1 - p_1)), 4))))))))Exact Shin Vig-Removed Longshot Fair Probability Formula
Feed z* directly into Shin’s general N-outcome square-root formula (SQRT(z^2 + 4*(1 - z)*(q_2^2 / S)) - z) / (2*(1 - z))!
=LET(o_1, B2, o_2, B3, IF(OR(o_1 <= 1, o_2 <= 1), "Require Decimal Odds > 1", LET(q_1, 1 / o_1, q_2, 1 / o_2, s_sm, q_1 + q_2, IF(s_sm < 1, "Arbitrage Market (Sum < 1)", LET(p_1, q_1 - 0.5 * (s_sm - 1), IF(OR(p_1 <= 0, p_1 >= 1), "Invalid Extreme Overround", LET(z_sh, IF(s_sm = 1, 0, ((q_1^2 / s_sm) - (p_1^2)) / (p_1 * (1 - p_1))), ROUND((SQRT((z_sh^2) + 4 * (1 - z_sh) * ((q_2^2) / s_sm)) - z_sh) / (2 * (1 - z_sh)), 4))))))))Try the example
Use the original values or change them to test your understanding.