Extracting True Implied Probabilities from Vigged Markets: Additive, Multiplicative, Power, and Shin Devigging Compared

Q: How does the Power devigging method guarantee valid probabilities across arbitrary multi-outcome books?

Power devigging solves for a unique scalar exponent k > 1 such that sum_{i=1}^n (pi_i)^k = 1, setting each devigged probability to p_i = (pi_i)^k. Because f(k) = sum (pi_i)^k is strictly decreasing and convex for pi_i in (0, 1) with f(1) = S > 1 and lim_{k -> infinity} f(k) = 0, a unique root k > 1 always exists, converges quadratically under Newton-Raphson, never yields negative tail probabilities, and compresses longshots proportionally more than favorites.

Q: What does Shin's parameter z represent in vigged market quotes?

Derived from Hyun Song Shin's (1991, 1993) microstructure equilibrium, z in (0, 1) estimates the share of market volume originating from insiders who possess superior information about the realized state versus (1 - z) noise liquidity. Higher z implies steeper adverse-selection protection on low-probability outcomes, where a small informed flow inflicts disproportionate dealer payout liability.

SECTION 01

Overround Mechanics & the Microstructure of Favorite-Longshot Bias

Consider a partition of $n \ge 2$ mutually exclusive and exhaustive event outcomes $i \in \{1, \dots, n\}$. Whether quoted as decimal odds $O_i > 1$ by a sharp syndicate market maker or as best executable ask prices $\pi_i \in (0, 1)$ across binary event brackets on a central limit order book (CLOB), raw implied probabilities $\pi_i = 1 / O_i$ satisfy:

$$S = \sum_{i=1}^{n} \pi_i > 1, \qquad V = S - 1 > 0$$

where $S$ is the book sum and $V$ is the dealer overround (vigorish or half-spread load). Comparing an internal model forecast $\hat{p}_i$ directly against a raw quote $\pi_i$ conflates genuine informational alpha with liquidity cost, while using an improperly devigged benchmark manufactures phantom positive-EV signals in the tails.

Crucially, market makers do not distribute $V$ uniformly across outcomes. Across equities options, macro prediction brackets, and sharp sports books, empirical settlement frequencies exhibit favorite-longshot bias (FLB): low-probability longshots carry dramatically higher percentage margin than high-probability favorites. Two microstructure forces drive this asymmetry:

First, convex dealer liability under informed flow: if an informed participant discovers private signal that a $5\%$ tail contract has shifted to $12\%$, buying a modest notional inflicts a $19\times$ payout liability on the liquidity provider, whereas a $75\% \to 82\%$ shift on a favorite alters liability marginally. Second, discrete tick granularity on whole-cent venues imposes a much larger relative spread on a $\$0.04$ contract ($25\%$ of price per cent) than on a $\$0.75$ contract ($1.33\%$ per cent).

SECTION 02

Why Additive & Multiplicative Normalization Fail in Production

2.1 Multiplicative (Proportional) Devigging

The textbook proportional normalization divides every raw implied quote by the book sum $S$:

$$p_i^{\text{mult}} = \frac{\pi_i}{\sum_{j=1}^{n} \pi_j} = \frac{\pi_i}{S}$$

Notice that $p_i^{\text{mult}} / p_j^{\text{mult}} = \pi_i / \pi_j$: Multiplicative devigging assumes the market maker shaded every outcome by the exact same proportional factor $S$. In a multi-bracket macro event with $S = 1.08$, a longshot priced at $\pi_i = 0.040$ (whose true probability may be $0.024$) is devigged only to $0.040 / 1.08 = 0.0370$. Any consensus agent comparing a sharp prior against Multiplicative probabilities will systematically misjudge tail fair value by 50–150 basis points.

2.2 Additive (Equal-Margin) Devigging

Additive devigging deducts an equal share of the overround from every outcome:

$$p_i^{\text{add}} = \pi_i - \frac{S - 1}{n}$$

While deducting constant probability mass compresses small $\pi_i$ more heavily in relative terms than large $\pi_i$, Additive devigging violates probability axioms whenever a deep tail outcome satisfies $\pi_i \le (S - 1)/n$, yielding $p_i^{\text{add}} \le 0$. In a 10-bucket CPI or Fed rate distribution with $8\%$ cumulative ask overround, every tail bracket quoted at $\$0.008$ or below maps to zero or negative probability.

SECTION 03

The Power Devigging Method & Quadratic Newton-Raphson Solver

Starling Quantitative’s primary benchmark extractor is the Power Devigging Method (also known in actuarial and logarithmic utility literature as constant-elasticity odds scaling). We model true probabilities as a power transformation of raw implied probabilities with scalar exponent $k > 1$:

$$p_i^{\text{power}} = \pi_i^{\,k} = \left(\frac{1}{O_i}\right)^{k} \quad \text{subject to} \quad \Phi(k) = \sum_{i=1}^{n} \pi_i^{\,k} - 1 = 0$$

Because $\pi_i \in (0, 1)$ for every nontrivial contract and $S = \sum_{i=1}^{n} \pi_i > 1$:

$$\Phi(1) = S - 1 > 0, \qquad \lim_{k \to \infty} \Phi(k) = -1, \qquad \Phi'(k) = \sum_{i=1}^{n} \pi_i^{\,k} \ln(\pi_i) < 0, \qquad \Phi''(k) = \sum_{i=1}^{n} \pi_i^{\,k} \big(\ln \pi_i\big)^2 > 0$$

Because $\Phi(k)$ is strictly decreasing and strictly convex on $[1, \infty)$, Intermediate Value Theorem and global convexity guarantee a unique root $k^* > 1$ and monotonic quadratic convergence from initial iterate $k_0 = 1$ under Newton-Raphson:

$$k_{m+1} = k_m - \frac{\sum_{i=1}^{n} \pi_i^{\,k_m} - 1}{\sum_{i=1}^{n} \pi_i^{\,k_m} \ln(\pi_i)}$$

In production Rust and Python 3.11 pricing loops, machine-precision convergence ($|\Phi(k_m)| < 10^{-12}$) completes in 4 to 6 iterations (< 2 microseconds). Note the structural elegance of $\pi_i^{\,k}$ for $k > 1$: the effective margin ratio $\pi_i / \pi_i^{\,k} = \pi_i^{\,1-k}$ grows smoothly as $\pi_i \to 0$, capturing favorite-longshot bias continuously while strictly preserving $p_i^{\text{power}} \in (0, 1)$ across arbitrarily deep multi-outcome books.

SECTION 04

Shin’s Microstructure Model & Informed-Trader Parameter $z$

Whereas Power devigging specifies a functional curve, Hyun Song Shin (1991, 1993) derived equilibrium quotes from first-principles market microstructure. Suppose a competitive, risk-neutral dealer faces unit-normalized order flow in which fraction $z \in (0, 1)$ originates from informed insiders who observe the realized state beforehand, and fraction $1 - z$ originates from uninformed liquidity traders splitting volume proportionally to market weights.

Setting expected dealer profit on each outcome equal to zero yields a quadratic equilibrium relationship between observed raw implied probability $\pi_i$, book sum $S = \sum_{j=1}^n \pi_j$, and true underlying probability $p_i^{\text{shin}}$:

$$p_i^{\text{shin}}(z) = \frac{\sqrt{z^2 + 4(1 - z)\,\dfrac{\pi_i^2}{S}} - z}{2(1 - z)}, \qquad \sum_{i=1}^{n} p_i^{\text{shin}}(z) = 1$$

Two analytical properties guide our multi-agent consensus checks:

1. Two-Outcome Equivalence ($n = 2$): When $n = 2$, algebraic simplification collapses Shin’s equation identically into the Additive formula $p_1 = \pi_1 - (S - 1)/2$. Consequently, on binary head-to-head contracts with deep skew (such as $\$0.94$ vs. $\$0.10$), Power devigging provides superior tail behavior over two-outcome Shin.

2. Multi-Outcome Adverse-Selection Telemetry ($n \ge 3$): Across multi-bracket macro, weather, or tournament markets ($n \ge 3$), solving $\sum_{i=1}^n p_i^{\text{shin}}(z) = 1$ via bracketed bisection or Jullien-Salanié Newton steps yields an explicit real-time estimate of $z$—the market’s implied share of toxic informed flow. Sudden spikes in $z$ across sequence snapshots serve Tier 02 risk agents as an early warning of private news leakage before headlines cross public wires.

SECTION 05

Worked Numerical Benchmark Across Four Devigging Models

Consider a three-way event market quoted at decimal odds $O = (1.40,\, 4.50,\, 9.00)$, corresponding to raw implied probabilities $\pi = (0.7143,\, 0.2222,\, 0.1111)$ and book sum $S = 1.0476$ ($4.76\%$ vig). Solving Power devigging yields $k^* = 1.0606$; solving Shin’s system yields implied insider share $z^* = 0.0243$ ($2.43\%$):

OUTCOME	ODDS	RAW $\pi_i$	MULTIPLICATIVE	ADDITIVE	SHIN ($z=0.0243$)	POWER ($k=1.0606$)
Outcome A (Favorite)	1.40	71.43%	68.18%	69.84%	69.42%	69.99%
Outcome B (Mid)	4.50	22.22%	21.21%	20.63%	20.77%	20.29%
Outcome C (Longshot)	9.00	11.11%	10.61%	9.52%	9.81%	9.73%
Total Sum	—	104.76%	100.00%	100.00%	100.00%	100.00%

Observe the 88-basis-point divergence on Outcome C between Multiplicative ($10.61\%$) and Power ($9.73\%$). An uncalibrated model estimating Outcome C at $10.20\%$ would falsely believe it holds positive edge against a Multiplicative benchmark while actually trading into negative expected value once favorite-longshot shading is stripped via Power or Shin.

SECTION 06

Cross-Venue Execution: Asymmetric Tick Rounding

Devigged probabilities must ultimately map back onto discrete exchange price grids. When routing hedged or relative-value orders across whole-cent venues (such as Kalshi, tick size $\Delta = \$0.01$) and sub-cent CLOBs (such as Polymarket, tick size $\Delta = \$0.001$), symmetric nearest-tick rounding (`round()`) silently leaks up to $0.5\text{c}$ of expected edge per leg.

Starling Quantitative’s Tier 03 execution engine enforces strict directional asymmetric rounding net of venue taker/maker fee functions $\tau(p)$:

$$P_{\text{bid}}^{\text{limit}} = \Delta \cdot \left\lfloor \frac{p_{\text{fair}} - e_{\min} - \tau_{\text{buy}}(p_{\text{fair}})}{\Delta} \right\rfloor, \qquad P_{\text{ask}}^{\text{limit}} = \Delta \cdot \left\lceil \frac{p_{\text{fair}} + e_{\min} + \tau_{\text{sell}}(p_{\text{fair}})}{\Delta} \right\rceil$$

Resting bids always round floor and resting asks always round ceil, guaranteeing that discrete tick quantization strictly widens—never narrows—required net edge $e_{\min}$.

SECTION 07

Technical FAQ

Why does Multiplicative devigging misprice longshot contracts?

Multiplicative devigging divides every raw implied probability $\pi_i$ by the book sum $S$, assuming overround is distributed proportionally across all outcomes. Because liquidity providers shade tail contracts far more heavily than favorites to defend against informed adverse selection, proportional division leaves upward bias in longshot probabilities.

How does the Power devigging method guarantee valid probabilities across arbitrary multi-outcome books?

Because $\Phi(k) = \sum_{i=1}^n \pi_i^{\,k} - 1$ is strictly decreasing and strictly convex on $[1, \infty)$ with $\Phi(1) > 0$ and $\lim_{k\to\infty}\Phi(k) = -1$, a unique root $k^* > 1$ always exists, converges quadratically under Newton-Raphson, never produces non-positive probabilities, and compresses tail quotes more strongly than favorites.

What does Shin’s parameter $z$ represent in vigged market quotes?

In Shin’s (1991, 1993) microstructure equilibrium, $z \in (0, 1)$ estimates the proportion of order flow originating from informed insiders relative to uninformed liquidity traders. Monitoring real-time shifts in $z$ across multi-outcome brackets provides an observable toxicity signal prior to order placement.

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Fractional Kelly Sizing Under Parameter Uncertainty & Out-of-Sample Brier Decomposition

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Extracting True Implied Probabilities from Vigged Markets:
Additive, Multiplicative, Power, and Shin Devigging Compared.