Introduction

From Inference to Prediction

In Part I of this series , we established that financial markets function as distributed Bayesian inference engines, aggregating heterogeneous private information into equilibrium prices through a process formally equivalent to posterior computation in probabilistic graphical models. The present paper takes the next logical step: if markets perform inference, what exactly do they infer? And can we build a complete theory of markets by taking the predictive function as the foundational primitive?

The classical answer—that markets estimate fundamental value—is unsatisfying for several reasons. “Fundamental value” is itself a derived concept, defined as the expected discounted sum of future cashflows. But expected with respect to which probability measure? Discounted at which rate? And who decides? These circularities suggest that the conventional approach has the logical order backwards: it attempts to define value first and derive prediction as a byproduct, when in fact prediction is the more primitive operation.

We propose a radical inversion. In Predictive Market Theory (PMT), the fundamental output of a market is not a price but a prediction—specifically, a probability measure over the space of possible future states of the world. Price is then derived from this predictive measure through a pricing functional that reflects agents’ risk preferences. This inversion is not merely philosophical: it leads to different mathematics, different theorems, and different empirical predictions.

Historical and Intellectual Context

The idea that markets are prediction machines has deep roots. The earliest formal prediction markets date to the Iowa Electronic Markets established in 1988 for forecasting election outcomes. Arrow recognized that competitive equilibria in complete markets are isomorphic to probability measures over states of nature. De Finetti’s foundational work on subjective probability established that coherent betting behavior implies—and is implied by—a probability measure, a result that applies directly to market prices of contingent claims.

More recently, the literature on prediction markets has explicitly studied markets designed for forecasting. The key insight from this literature is that market prices for binary contracts (“Will event \(E\) occur?”) can be interpreted directly as probability estimates. What has been missing is a unified theoretical framework that:

1. Extends the prediction market insight from binary contracts to the full complexity of financial markets.

2. Provides axiomatic foundations for the market’s predictive function.

3. Derives standard asset pricing results as consequences of the predictive framework.

4. Quantifies the precision and limitations of market prediction.

5. Connects to the thermodynamic and information-theoretic structures identified in Part I.

Predictive Market Theory accomplishes all five.

Overview of Contributions

The principal contributions of this paper are:

1. Axiomatic foundations (Section 2): Five axioms from which the entire theory follows.

2. The Predictive Measure \(\mathcal{P}\) (Section 3): A new probability measure on outcome space, distinct from both \(\mathbb{P}\) and \(\mathbb{Q}\), encoding the market’s collective forecast.

3. Variational Prediction Principle (Section 4): The predictive measure minimizes expected predictive free energy.

4. Price as a Functional of Prediction (Section 5): Standard pricing results derived from the predictive framework.

5. The Fundamental Theorem of Predictive Markets (Section 6): A market-completeness result for predictive measures.

6. Predictive Resolution and Uncertainty Relations (Section 7): Fundamental limits on market forecasting precision.

7. Resolution of Classical Puzzles (Section 8): New explanations for the equity premium puzzle and excess volatility.

8. Connections to Kelly Theory and de Finetti (Section 9): Unifying threads with optimal betting and operational subjective probability.

9. Predictive Dynamics (Section 10): The stochastic evolution of the predictive measure.

10. Empirical predictions and experimental design (Section 13).

The Axioms of Predictive Market Theory

We now state the five axioms on which Predictive Market Theory is erected. Throughout, let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space where \(\Omega\) is the space of possible future states of the world, \(\mathcal{F}\) is a \(\sigma\)-algebra of events, and \(\mathbb{P}\) is the physical (objective) probability measure. Let \(\{P_t^{(k)}\}_{k \in \mathcal{K}}\) denote the collection of prices of all traded assets at time \(t\), where \(\mathcal{K}\) is the asset index set. Let \(\mathcal{F}_t\) denote the market filtration at time \(t\)—the information available from observing market activity up to time \(t\).

Discussion of the Axioms

Axiom 1: Predictive Completeness

This is the predictive analogue of market completeness in the Arrow–Debreu sense . In the Arrow–Debreu framework, a complete market has a traded security for every state of nature. Predictive Completeness is weaker: it requires only that a portfolio (possibly synthetic) exists whose price encodes the probability of any event. This is achievable whenever the market has a sufficient variety of derivative instruments, particularly options at different strike prices.

The Breeden–Litzenberger result provides the classical construction: the price of a European call option \(C(K)\) with strike \(K\) encodes the tail probability of the underlying exceeding \(K\) at expiration. Formally: \[\begin{equation} \label{eq:breeden_litz} \mathcal{P}_t(S_T > K) = -e^{r(T-t)} \frac{\partial C(K)}{\partial K} \end{equation}\] and the full predictive density is: \[\begin{equation} \label{eq:risk_neutral_density} \frac{d\mathcal{P}_t}{dK}(K) = e^{r(T-t)} \frac{\partial^2 C(K)}{\partial K^2} \end{equation}\]

We emphasize that our \(\mathcal{P}\) is not the risk-neutral measure \(\mathbb{Q}\). The relationship between \(\mathcal{P}\), \(\mathbb{Q}\), and \(\mathbb{P}\) is developed in Section 3.

Axiom 2: Bayesian Coherence

This axiom asserts that the market, viewed as a collective prediction device, obeys the laws of probability. It is the market-level analogue of individual rationality: just as a rational agent’s beliefs satisfy the axioms of probability, so too do the market’s collective beliefs. The key empirical content is the dynamic updating rule—the market does not simply assign static probabilities but revises them in response to new information in a manner consistent with Bayes’ theorem.

This axiom rules out certain pathological market behaviors:

• Dutch books: If \(\mathcal{P}_t\) were not a proper probability measure, an arbitrageur could construct a Dutch book—a combination of trades that guarantees a profit regardless of the outcome .

• Non-Bayesian updating: If the market updated in a manner inconsistent with Bayes’ rule (e.g., overreacting systematically to new information), the resulting \(\mathcal{P}\) would fail the coherence axiom.

Axiom 3: Informational Irreversibility

This axiom encodes a form of the second law of thermodynamics for markets. As information is incorporated, the predictive distribution sharpens—uncertainty decreases, and this process is irreversible. The market cannot “unlearn” information that has been publicly reflected in prices.

Note that this does not preclude increased price volatility. Volatility can increase even as entropy decreases: a bimodal distribution (e.g., “the company will either succeed spectacularly or fail completely”) can have lower entropy than a diffuse unimodal distribution, yet exhibit higher variance. The axiom constrains entropy, not variance.

Axiom 4: Predictive Sufficiency of Price

This is the strongest axiom and the one most subject to empirical challenge. It asserts that prices contain all the information needed to reconstruct the predictive measure. In its strong form, this is equivalent to the strong-form Efficient Market Hypothesis. We will later weaken it to an approximate version, introducing the concept of predictive residuals.

Axiom 5: Thermodynamic Dissipation

This axiom connects the predictive theory to the thermodynamic framework of Part I. The predictive free energy combines two terms: the expected surprise of outcomes under the predictive measure (an accuracy term) and the KL divergence from the predictive measure to the physical measure (a complexity or confidence penalty). The axiom states that the market’s prediction improves over time in the free-energy sense.

The Predictive Measure

Three Measures in Finance

Standard mathematical finance operates with two probability measures:

1. The physical measure \(\mathbb{P}\): the objective probability distribution governing actual outcomes. Under \(\mathbb{P}\), stocks have positive expected returns (the equity premium), and the expected return on any asset reflects its systematic risk.

2. The risk-neutral measure \(\mathbb{Q}\): a mathematical construct under which all assets have the same expected return (the risk-free rate). The risk-neutral measure is not a genuine probability forecast—it is a pricing convenience that encodes risk preferences alongside probability assessments.

We introduce a third measure:

Relationship Between \(\mathbb{P}\), \(\mathbb{Q}\), and \(\mathcal{P}\)

The three measures are related through Radon–Nikodym derivatives. Under regularity conditions (absolute continuity), we can write: \[\begin{equation} \label{eq:radon_nikodym_chain} \frac{d\mathbb{Q}}{d\mathbb{P}} = \frac{d\mathbb{Q}}{d\mathcal{P}} \cdot \frac{d\mathcal{P}}{d\mathbb{P}} \end{equation}\]

The Radon–Nikodym derivative \(d\mathbb{Q}/d\mathbb{P}\) is the classical pricing kernel or stochastic discount factor \(M\): \[\begin{equation} \label{eq:pricing_kernel} M(\omega) = \frac{d\mathbb{Q}}{d\mathbb{P}}(\omega) \end{equation}\]

We decompose this into two components:

The predictive accuracy kernel \(A(\omega)\) measures how well the market’s collective forecast matches the true probabilities: if \(A(\omega) = 1\) for all \(\omega\), the market is a perfect forecaster. The risk distortion kernel \(R(\omega)\) captures pure risk preferences: even a perfectly forecasting market will price risky assets differently from risk-neutral valuation because agents demand compensation for bearing risk.

Extracting \(\mathcal{P}\) from Market Data

In practice, how do we disentangle the predictive measure \(\mathcal{P}\) from the risk-neutral measure \(\mathbb{Q}\) observed in option prices? We provide two methods.

Method 1: Calibration via Prediction Markets

If a prediction market exists for the same event, the prediction market price directly estimates \(\mathcal{P}(A)\), while the option market provides \(\mathbb{Q}(A)\). The ratio yields the risk distortion kernel evaluated on the event: \[\begin{equation} R(A) = \frac{\mathbb{Q}(A)}{\mathcal{P}(A)} \end{equation}\]

Method 2: Power Utility Inversion

Under the assumption of power utility with relative risk aversion \(\gamma\), the predictive density can be recovered from the risk-neutral density by: \[\begin{equation} \label{eq:power_inversion} \frac{d\mathcal{P}}{dx}(x) \propto \frac{d\mathbb{Q}}{dx}(x) \cdot x^{\gamma} \end{equation}\] where \(x = S_T/S_t\) is the gross return. This follows from the first-order condition of a representative agent with power utility.

The Predictive Measure as a Martingale Measure

A fundamental structural property of the predictive measure is that it generates a martingale—but not on prices. Rather, it is the predictive probabilities themselves that form a martingale:

This is a strong testable prediction: realized frequencies of events should match their market-implied probabilities, on average.

The Variational Prediction Principle

Free Energy of Prediction

We now show that the predictive measure is characterized by a variational principle: it minimizes the expected predictive free energy over all measures consistent with the market’s information.

The predictive free energy has two competing terms:

• The prediction error term penalizes measures that assign low likelihood to the observed data. A measure that “predicted poorly” in light of what has been observed receives a large penalty.

• The complexity penalty (KL divergence from the prior) penalizes measures that deviate far from the prior, implementing a form of Occam’s razor: among measures that explain the data equally well, prefer the simplest (closest to the prior).

The Market as a Variational Inference Machine

Theorem 7 reveals that the market performs variational inference—the same computational framework used in modern machine learning . The market searches over the space of probability measures to find the one that best balances explanatory power (fitting observed data) against complexity (staying close to the prior).

This has profound implications:

1. Bounded rationality as approximate inference: When the market cannot compute the exact posterior (due to bounded rationality, limited information processing, or insufficient liquidity), it effectively performs approximate variational inference, searching over a restricted family of measures \(\mathcal{M}_0 \subset \mathcal{M}(\Omega)\). The quality of the approximation is measured by the variational gap: \[\begin{equation} \label{eq:variational_gap} \Delta \mathcal{F} = \mathcal{F}[\mathcal{P}_t^{\text{approx}}; \mathcal{F}_t] - \mathcal{F}[\mathcal{P}_t^*; \mathcal{F}_t] = \mathrm{KL}(\mathcal{P}_t^{\text{approx}} \| \mathcal{P}_t^*) \geq 0 \end{equation}\]

2. Model selection: The marginal likelihood \(Z_t\) provides an automatic model selection criterion. Markets that can entertain more complex models (richer asset universes, more derivative instruments) achieve lower free energy, corresponding to better predictions. This provides a principled explanation for financial innovation: new instruments are valuable precisely because they expand the space of measures over which the market can optimize.

3. The prior as market structure: The prior \(\pi_0\) encodes structural assumptions about the economy—the base rate expectations, the background model. Changes in the prior (e.g., regime changes, new economic paradigms) correspond to fundamental reconfigurations of the market’s predictive framework.

Connection to the Free Energy Principle

The variational formulation connects directly to Friston’s Free Energy Principle from computational neuroscience, which posits that biological agents minimize variational free energy to maintain a model of their environment. Under PMT, the market is a collective agent that minimizes predictive free energy, treating the economy as its “environment.” The parallel is exact:

This is more than analogy. Both the brain and the market face the same fundamental computational problem—maintaining an accurate internal model of an uncertain, changing environment—and both solve it by minimizing variational free energy.

Deriving Price from Prediction

The Pricing Functional

In PMT, price is not a primitive—it is derived from the predictive measure through a pricing functional that encodes aggregate risk preferences.

Equation [eq:pricing_decomposed] makes the decomposition transparent: the asset price equals the discounted predicted expected payoff, adjusted by a risk premium that depends on the covariance of the payoff with the risk distortion kernel.

Recovering Classical Results

We now show that the standard results of asset pricing theory follow as corollaries of the predictive framework.

The PMT perspective reveals that the Black–Scholes model is not merely a pricing formula but a statement about the form of the market’s prediction: it assumes the market predicts log-normal returns. Deviations from Black–Scholes (volatility smiles, skew) are therefore deviations of the predictive measure from log-normality, not just deviations of the pricing kernel.

The Fundamental Theorem of Predictive Markets

We now state and prove the central result of the theory: the Predictive Market Completeness Theorem.

Predictive Completeness

Implications for Market Design

The Fundamental Theorem provides a precise criterion for evaluating market design: a well-designed market should maximize its predictive span. This has concrete implications:

1. Options markets expand predictive completeness: A market with only stock and bond cannot distinguish between symmetric and skewed predictive distributions. Adding options at multiple strikes fills in the full predictive density.

2. Prediction markets are maximally predictive: A prediction market with binary contracts on all events of interest achieves predictive completeness by construction.

3. Financial innovation as predictive expansion: New financial instruments (CDOs, variance swaps, volatility-of-volatility products) expand the predictive span, allowing the market to express forecasts along dimensions that were previously unresolvable.

Predictive Resolution and Uncertainty Relations

Quantifying Forecasting Precision

The Predictive Uncertainty Relation

We now derive a fundamental result: an uncertainty relation that constrains the joint precision of market predictions across time and across states.

Consequences of the Uncertainty Relation

Resolution of Classical Puzzles

The Equity Premium Puzzle

The equity premium puzzle notes that the historical excess return of equities over risk-free assets (\(\sim\)6% per year) is far too large to be explained by standard consumption-based models with plausible risk aversion parameters. In PMT, this puzzle admits a natural resolution.

The key new term is the predictive bias: the systematic difference between the physical expected return and the market’s predicted expected return. If the market systematically underestimates equity returns (i.e., is too pessimistic), this generates an apparent equity premium even with moderate risk aversion.

This is empirically testable: surveys of professional forecasters consistently show that median forecasts for equity returns underperform realized returns, supporting the pessimism hypothesis.

The Excess Volatility Puzzle

Shiller demonstrated that stock prices are far more volatile than can be justified by subsequent changes in dividends. In PMT, excess volatility has a natural interpretation.

In this decomposition, the “predictive updating” term captures genuine information-driven price changes, while the “risk preference shifts” and “interaction” terms capture changes in risk attitudes. The excess volatility puzzle arises when the total variance exceeds what the predictive updating term alone would justify—the excess is due to fluctuations in risk preferences (the risk distortion kernel \(R\)).

The Value of Active Management

A persistent puzzle in finance is why active management persists when index funds consistently outperform. PMT provides a resolution: active managers are part of the market’s predictive infrastructure.

Active managers are, in effect, paid to run the market’s distributed inference algorithm. The fees they charge are the cost of maintaining the market’s predictive accuracy. If all managers switched to passive indexing, the predictive free energy would increase (predictions would worsen), leading to less efficient capital allocation. The equilibrium fee level is set by the marginal value of the prediction improvement they provide.

Connections to Kelly Theory and de Finetti

The Kelly Criterion as Optimal Prediction Exploitation

The Kelly criterion prescribes the bet size that maximizes the expected log growth rate of wealth: \[\begin{equation} \label{eq:kelly} f^* = \mathop{\mathrm{arg\,max}}_f \mathbb{E}_\mathbb{P}[\log(1 + f \cdot R)] \end{equation}\] where \(f\) is the fraction of wealth bet and \(R\) is the return.

In PMT, the Kelly criterion admits a natural interpretation as optimal exploitation of predictive superiority.

De Finetti’s Theorem and Market Coherence

De Finetti’s fundamental insight was that coherent betting behavior—behavior that does not admit a Dutch book—is equivalent to the existence of a probability measure governing the bets. This provides the operational foundation for subjective probability.

PMT extends de Finetti’s theorem from individual agents to markets:

Scoring Rules and the Market as a Mechanism

The theory of proper scoring rules provides mechanisms that incentivize truthful probability reporting. A scoring rule \(S(p, \omega)\) assigns a score to a probabilistic forecast \(p\) when outcome \(\omega\) is realized.

This establishes that the market mechanism itself is a proper scoring rule that incentivizes participants to reveal their best probability estimates through trading. The market doesn’t just aggregate predictions—it actively incentivizes truthful prediction.

Predictive Dynamics: The Stochastic Evolution of \(\mathcal{P}_t\)

The Predictive Process

We now study the dynamics of the predictive measure itself. Rather than analyzing the time series of prices (the traditional approach), we analyze the time series of predictions—the stochastic process \(\{\mathcal{P}_t\}_{t \geq 0}\) valued in the space of probability measures on \((\Omega, \mathcal{F})\).

The Stochastic Differential Equation for \(\mathcal{P}_t\)

In continuous time, the evolution of the predictive measure can be expressed as a stochastic partial differential equation (SPDE) on the space of measures. For the finite-dimensional case (prediction over a finite partition \(\{A_1, \ldots, A_n\}\)), the predictive probabilities \(\pi_i(t) = \mathcal{P}_t(A_i)\) satisfy:

Key properties of the predictive SDE:

1. Drift-free: The predictive SDE has no drift term—the evolution is a pure martingale. This is a direct consequence of Theorem 4.

2. Multiplicative noise: The volatility of each probability is proportional to the probability itself, ensuring that probabilities near zero or one change slowly (strong beliefs are hard to move).

3. Conservation: The probabilities sum to one at all times: \(\sum_i d\pi_i(t) = 0\).

4. Absorption: The states \(\pi_i = 0\) and \(\pi_i = 1\) are absorbing boundaries—once the market is certain, it stays certain (under the idealized model).

Information Arrival Rate and Predictive Volatility

The volatility of the predictive process is directly linked to the rate of information arrival:

This result quantifies the intuition that predictions are most volatile when the market is most uncertain—a probability near 50% can swing dramatically in either direction, while a probability near 0% or 100% is inherently stable.

Quantifying Market Prediction Accuracy

Calibration and Discrimination

Following the scoring rule literature , we decompose predictive accuracy into two components:

The Predictive Efficiency Ratio

Comparing Predictive Efficiency Across Markets

The predictive efficiency framework provides a principled way to compare markets. An empirically testable conjecture:

Predictive Market Microstructure

The Predictive Interpretation of Order Flow

In the PMT framework, every trade carries a predictive signal. When agent \(i\) buys, she is expressing the belief \(\mathcal{P}^{(i)}(V > P) > \mathcal{P}(V > P)\)—her prediction assigns higher probability to favorable states than the market consensus.

The Bid–Ask Spread as Predictive Insurance

The bid–ask spread can be understood as the price the market charges for providing instant predictive feedback:

Market Making as Predictive Mediation

Market makers occupy a unique role in the PMT framework: they are predictive mediators who bridge temporal gaps in the flow of predictions.

Empirical Predictions and Experimental Design

Core Empirical Predictions

1. Calibration Test: Option-implied probabilities, adjusted for risk preferences using the power utility inversion [eq:power_inversion], should be well-calibrated: the fraction of events occurring at each probability level should match the predicted probability. This can be tested using large cross-sections of options data.

2. Predictive Martingale Test: The implied probability \(\mathcal{P}_t(A)\) for any event \(A\) should be a martingale under the physical measure \(\mathbb{P}\). Specifically, \(\mathcal{P}_{t+1}(A) - \mathcal{P}_t(A)\) should be uncorrelated with \(\mathcal{F}_t\). Violations indicate failures of Axiom 2.

3. Free Energy Monotonicity: The predictive free energy \(\mathcal{F}[\mathcal{P}_t]\) should be non-increasing over time within a given prediction episode (e.g., from the start of an earnings season to the announcement). The rate of decrease should correlate with information flow (proxied by trading volume).

4. Uncertainty Relation Test: For a given market, plot \(\Delta_T\) (time horizon of profitable prediction, measured by holding period of successful strategies) against \(\Delta_S\) (number of states distinguished, measured by the effective number of scenarios in option-implied distributions). The data should satisfy \(\Delta_T \geq \frac{\log_2 \Delta_S}{\mathcal{C}}\) as predicted by Theorem 17.

5. Predictive Efficiency Comparison: Compare the Brier scores of market-implied probabilities across different asset classes. The ordering should follow the hierarchy predicted by Conjecture 40, with liquid equity markets outperforming illiquid alternatives.

Experimental Design: A Controlled Test of PMT

We propose the following experimental framework for testing PMT:

1. Identify a universe of well-defined, verifiable events (e.g., quarterly earnings beating consensus, central bank decisions, election outcomes).

2. Extract \(\mathbb{Q}\)-probabilities from option prices using the Breeden–Litzenberger formula [eq:breeden_litz].

3. Estimate the risk distortion kernel \(R\) using either Method 1 (prediction market calibration) or Method 2 (power utility inversion).

4. Compute \(\mathcal{P}= \mathbb{Q}/ R\) and evaluate calibration, discrimination, and Brier scores.

5. Compare PMT-derived \(\mathcal{P}\) against: (a) raw option-implied \(\mathbb{Q}\)-probabilities, (b) analyst consensus forecasts, (c) pure prediction market prices.

The hypothesis is that \(\mathcal{P}\) (the risk-adjusted market probability) outperforms \(\mathbb{Q}\) (the unadjusted option-implied probability) in calibration and discrimination, as \(\mathcal{P}\) removes the systematic risk distortion that makes \(\mathbb{Q}\) a biased probability estimator.

Data Requirements and Feasibility

Testing the full PMT framework requires:

• High-frequency option chain data (to extract \(\mathbb{Q}\)).

• Contemporaneous prediction market data (for calibration of \(R\)).

• Realized outcomes for a large sample of events.

• Trading volume and order flow data (for microstructure tests).

The richest testing ground is the period around binary events (earnings announcements, FDA decisions, elections) where both option markets and prediction markets provide probabilities.

Conclusion

We have introduced Predictive Market Theory, an axiomatic framework that reconceptualizes financial markets as prediction engines whose fundamental output is a probability measure over future states of the world. The theory rests on five axioms—Predictive Completeness, Bayesian Coherence, Informational Irreversibility, Predictive Sufficiency of Price, and Thermodynamic Dissipation—from which the standard results of asset pricing theory emerge as corollaries.

The key theoretical innovations are:

1. The Predictive Measure \(\mathcal{P}\): A third probability measure, distinct from both the physical \(\mathbb{P}\) and risk-neutral \(\mathbb{Q}\) measures, encoding the market’s collective forecast purged of risk distortions.

2. The Variational Prediction Principle: The market minimizes predictive free energy, unifying Bayesian inference, information theory, and thermodynamics within a single framework.

3. Price as derived from prediction: Rather than deriving prediction from price, PMT derives price from prediction through the pricing functional \(P = e^{-r\tau}\mathbb{E}_\mathcal{P}[X \cdot R]\).

4. The Predictive Uncertainty Relation: A fundamental limit on the joint precision of temporal and state-space prediction, \(\Delta_T \geq \frac{\log_2 \Delta_S}{\mathcal{C}}\).

5. Resolution of classical puzzles: The equity premium, excess volatility, and persistence of active management receive natural explanations as consequences of the predictive structure.

6. Market coherence as an extension of de Finetti: The market’s predictive function satisfies the same coherence conditions as individual rational probability, elevated to the collective level.

Predictive Market Theory shifts the conceptual center of gravity in financial economics from pricing to prediction. In this framework, the primary function of markets is not to “find the right price” but to “find the right probability.” Prices are instrumental—they are the medium through which the predictive function operates—but the prediction is the substance. This inversion suggests that the true measure of market quality is not price efficiency in the Fama sense but predictive efficiency: how well does the market’s probability measure approximate the true distribution of future outcomes?

We believe this framework opens a productive new chapter in the theory of financial markets, one that more naturally connects to adjacent fields—machine learning, information theory, statistical mechanics, and decision theory—and that provides sharper tools for both theoretical analysis and empirical investigation.

GrokRxiv DOI: 10.48550/GrokRxiv.2603.04.predictive-market-theory 4 March 2026