A Shannon Entropy Framework for Prediction Market Alpha

Beyond Calibration

Most prediction market analysis focuses on calibration. Did 70% contracts resolve at 70%? Useful, but incomplete.

A market can be well-calibrated and still leak money to informed players. Calibration tells you the market is not systematically biased. It does not tell you whether the market is resolving the uncertainty it should.

Brier Score Decomposition

The Brier score is a proper scoring rule for probabilistic predictions:

BS = (1/N) × Σ(fᵢ - oᵢ)²

Where fᵢ is the forecast probability and oᵢ is the outcome (0 or 1).

Murphy (1973) showed the Brier score decomposes into three additive components:

BS = Uncertainty - Resolution + Reliability

Uncertainty (U): The inherent variance in outcomes. For binary events, U = p̄(1 - p̄), where p̄ is the base rate. This is the entropy of the climatology. Maximum at p̄ = 0.5, zero when outcomes are deterministic.

Resolution (RES): How much forecasts deviate from base rates. Higher resolution means forecasts contain more information. Resolution measures the variance of conditional probabilities around the base rate.

Reliability (REL): Calibration error. Zero when forecasts match observed frequencies. A 70% forecast should resolve 70% of the time.

The decomposition separates what a forecaster can control (resolution, reliability) from what they cannot (uncertainty).

Entropy Efficiency

I track a ratio:

R = Resolution / Uncertainty

Call it entropy efficiency. It measures the fraction of baseline entropy resolved by the market's forecasts.

Why This Ratio?

Absolute resolution is hard to compare across markets. A market resolving 0.05 bits against U = 0.25 is working harder than one resolving 0.05 bits against U = 0.10.

Entropy efficiency normalizes across different base rates. It answers: given the inherent uncertainty, how much is the market resolving?

Interpretation

Diagnosing Market Inefficiency

Low R with External Signals: Underconfidence

Low R in a genuinely low-information environment is rational. Low R when public data, models, or domain expertise support sharper estimates is a dispersion inefficiency.

The market hugs base rates not because information is absent but because participants are unwilling or unable to aggregate it.

This creates opportunity. Your edge exists when:

1. R is low
2. You have information the market has not incorporated
3. You can estimate the true probability more precisely

High R with Poor Reliability: Overconfidence

When R is high but reliability is poor, the market is overconfident. It pushes to 90%+ without the informational basis for that precision.

Overconfident markets price certainty they do not have. Fading extreme prices becomes profitable.

Connection to Expected Growth

Proper Scoring Rules

Under proper scoring rules like log score, expected score improvement is proportional to KL divergence between your distribution and the market's:

E[Score Improvement] ∝ D_KL(P_true || P_market)

Kelly Criterion

For a bettor sizing via Kelly criterion, expected log wealth growth scales with the same quantity:

E[log(W_t+1 / W_t)] = D_KL(P_true || P_market)

The entropy gap is not a metaphor. It maps directly to expected growth rate when your model is better calibrated than the market.

This is Kelly's original insight from 1956: the optimal bet size maximizes expected logarithmic utility, and the growth rate equals the information advantage measured in bits.

Worked Example

Scenario: Market prices a regulatory approval at 52%.

Analysis:

• Base rate for similar approvals: 50%
• Market price: 52%
• Resolution: (0.52 - 0.50)² = 0.0004
• Uncertainty: 0.50 × 0.50 = 0.25
• R = 0.0004 / 0.25 = 0.0016 (≈ 0.2%)

The market resolves 0.2% of available uncertainty despite:

• Public FDA filing data
• Phase trial results
• Committee composition

This is a dispersion inefficiency.

Your model: Based on comparable approvals, committee history, and filing completeness, you estimate 71%.

Entropy calculation:

• Entropy at market price: H(0.52) = -0.52×log₂(0.52) - 0.48×log₂(0.48) = 0.999 bits
• Entropy at model price: H(0.71) = -0.71×log₂(0.71) - 0.29×log₂(0.29) = 0.871 bits

Gap: 0.128 bits

Under Kelly sizing, 0.128 bits is the expected log growth per bet. Over many similar opportunities, this compounds.

Multi-Outcome Markets

For markets with n outcomes, entropy generalizes:

H(p) = -Σᵢ pᵢ × log₂(pᵢ)

Entropy is tail-sensitive. Consider two distributions:

Both have the same favorite at 70%. But the tail mass differs. Entropy captures this.

Most participants fixate on the top outcome while misallocating tail mass. Entropy calculations surface these errors because they weight the full distribution, not just the mode.

Practical Application

Screening for Opportunities

1. Compute base rate entropy U for each market category
2. Estimate market resolution from price variance
3. Calculate R = RES / U
4. Flag markets where R < threshold and external information exists

Position Sizing

When your model differs from market:

1. Compute KL divergence: D_KL(P_model || P_market)
2. This is your edge in bits
3. Kelly fraction: f* = edge / odds (adjusted for multi-outcome)
4. Expected growth: D_KL per bet

Tracking Performance

Track realized vs. predicted entropy reduction:

• If your model reduces entropy more than the market, you have persistent edge
• If not, reassess your information sources

Summary

Calibration tells you if a market's probabilities are accurate. Entropy efficiency tells you if the market is resolving the uncertainty it should. The gap between the two is where alpha lives.