devigging draft

Devigging takes a vigged market — where the implied probabilities sum to more than 1 — and backs out fair probabilities that sum to exactly 1 and represent the book's underlying belief. Several methods exist in the literature (multiplicative, additive, power, probit, Shin). Mimir's canonical method is probit, applied to binary (two-sided) markets only. Multi-way (3+) markets are out of scope for v1; future versions may add them.

Probit treats the vig as a uniform shift on a latent normal scale. Convert each side's implied probability q_i to its z-score z_i = Φ⁻¹(q_i), then find a constant c such that Σ Φ(z_i − c) = 1. The fair probabilities are p_i = Φ(z_i − c). Kairos solves for c numerically via bisection on the latent scale. The method is well-suited to markets where the underlying outcome distribution is plausibly Gaussian-ish (point spreads, totals, moneyline-equivalents) and tends to produce results close to the power method while being more stable at extreme prices.

Worked example

Binary moneyline at +150 / −180:

• qA = 1 / 2.50 = 0.4000, qB = 1 / 1.5556 ≈ 0.6429
• Sum = 1.0429, overround = 0.0429 (4.29%) — vig is built in.
• z_A = Φ⁻¹(0.4000) ≈ −0.2533
• z_B = Φ⁻¹(0.6429) ≈ +0.3661
• Solve for c such that Φ(z_A − c) + Φ(z_B − c) = 1. Numerically, c ≈ +0.0561.
• p_A = Φ(−0.2533 − 0.0561) = Φ(−0.3094) ≈ 0.378
• p_B = Φ(+0.3661 − 0.0561) = Φ(+0.3100) ≈ 0.622

Sum: 0.378 + 0.622 = 1.000. The book's fair belief: Side A has ~37.8% probability, Side B has ~62.2%.

Gotchas

• Probit collapses for inputs at exactly 0 or 1. Φ⁻¹(0) and Φ⁻¹(1) are −∞ / +∞. Guard the boundaries before calling probit — treat exactly-0 and exactly-1 implied probabilities as degenerate (unbettable). Kairos's _valid_prob_pair rejects these inputs upstream.
• Methods agree at zero vig. When implied probabilities already sum to 1.0, probit returns them unchanged (c = 0). Testing devig code on a no-vig market doesn't exercise the method's actual behavior — use a vigged input to verify.
• Devig before any cross-book aggregation. Comparing implied probabilities across books or building a consensus-fair-line only makes sense if every book is devigged first, with the same method. Aggregating raw vigged probabilities baked-in book-level vig into the consensus.
• Devig before any EV calculation. EV against the raw vigged line bakes the book's margin into your "edge" — the number is misleading. EV against the devigged fair line is the meaningful one. See vig-adjusted-ev (forthcoming).
• Bisection has bounds. Kairos's probit solver searches over a bounded latent-scale interval. For pathological inputs (extreme overround, near-degenerate prices) the search can fail and Kairos returns None. Don't assume probit always converges; handle the failure.

Open questions

• The probit method assumes a normal latent scale. For markets where outcomes aren't plausibly Gaussian (heavy-tailed event probabilities, exotic props, multi-leg parlays viewed as composite outcomes), probit may be a worse fit than power or another method. Should Mimir surface guidance on when probit is the wrong choice, or treat probit as universally canonical until concrete counter-evidence appears?

Cross-references

• mimir: american-implied-probability
• mimir: book-vig-overround
• mimir: consensus-fair-line