Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 2026
From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
Grossman and Stiglitz (1980) proved that informationally efficient markets are impossible because efficiency destroys the incentive to produce information. Their proof assumes that trading profit is the sole return to information production. This paper relaxes that assumption. When agents accumulate track record capital - a portable asset built by publishing structured judgments verified against outcomes - information production is decoupled from trading, and the paradox dissolves. A CARA-normal general equilibrium model with a judgment market operating alongside the asset market formalizes the resolution. This paper derives the independence of judgment market returns from price efficiency by decomposing the principal's willingness to pay into a consensus premium that vanishes as markets become efficient and a constraint premium that remains permanently bounded above zero. Incentive compatibility is established within the Hurwicz-Myerson framework (Myerson, 1979, 1981): under the Brier scoring rule with coverage weight \(\alpha < 4/5\), honest comprehensive disclosure strictly dominates strategic selection for all agents regardless of calibration quality.
Keywords: information economics, Grossman-Stiglitz paradox, track records, disclosure, mechanism design, incentive compatibility, proper scoring rules, calibration, judgment market
JEL Codes: D82, D83, D86, G14, C72, D47
The impossibility result that Grossman and Stiglitz published in 1980 remains among the most consequential in financial economics. If prices already reflect available information, paying to acquire that information yields no return. But if nobody acquires information, there is nothing for prices to reflect. No equilibrium exists in which prices are both fully efficient and supported by costly information production.
What followed was a large literature exploring how markets manage this tension without resolving it. Three strands are relevant. The first addressed the microstructure of how informed trading transmits information into prices: Kyle (1985) modeled strategic order sizing, Admati and Pfleiderer (1988) formalized intraday clustering, and Black's (1986) noise trading became the standard mechanism for preserving informed traders' incentive (see Vives, 2008, for a comprehensive treatment). The second strand explored voluntary disclosure: Grossman (1981) and Milgrom (1981) established the unraveling logic, Verrecchia (1983) and Dye (1985) characterized partial disclosure, and Diamond and Verrecchia (1991) linked disclosure to financing costs. The third strand examined career concerns and non-trading incentives: Hong and Kubik (2003) found that accurate analysts receive better career outcomes, while Scharfstein and Stein (1990) showed that career concerns produce herding and Ottaviani and Sorensen (2006) formalized how reputational incentives distort forecasts. Among partial decouplings from trading, Admati and Pfleiderer (1986) modeled information selling (though their seller degrades signal quality to prevent free-riding), and Garcia and Sangiorgi (2011) analyzed markets where information sales and strategic trading coexist, though the information sold remains a raw signal.
One assumption has survived every proposed resolution. Grossman and Stiglitz built it into the model's structure rather than stating it, and no subsequent work has extracted it for examination: that the return to information production about asset values is captured exclusively through trading on those assets. This paper identifies and relaxes that assumption by introducing a second incentive channel. Under the Disclosure Incentive, the return to information production accrues through track record capital - a portable, compounding professional asset built by publishing structured judgments that are verified against outcomes over time. Because the track record measures judgment quality rather than trading timing, its value does not depend on whether the price has already adjusted.
Two formal questions arise immediately. If honest disclosure is the mechanism's incentive basis, is truth-telling actually incentive-compatible in the multi-agent, multi-period, strategic environment the mechanism requires? And is the judgment market's return to calibration quality genuinely independent of price efficiency - or does assuming this independence amount to assuming the paradox away?
The structural case for a supplementary channel rests on a result derived in a companion paper (Beier, 2026c). The price system's scalar compression introduces excess distortion on constraint dimensions that is strictly positive whenever the compression rate exceeds a threshold determined by the source covariance structure, strictly increasing in the number of active constraint dimensions, and invariant to market efficiency. No improvement to the price system - no increase in liquidity, no reduction in noise, no advance in market microstructure - can deliver constraint information to decision-makers through the price channel. A supplementary channel is necessary. This paper shows that such a channel, based on track record capital, is incentive-compatible and resolves the Grossman-Stiglitz paradox.
This paper addresses both questions. Sections 2 through 4 state the paradox, introduce the mechanism, and prove the resolution in a CARA-normal general equilibrium framework - including a formal welfare result showing that structured disclosure strictly dominates a price signal of equal precision for multi-dimensional decisions. Section 5 - the paper's technical core - micro-founds the judgment market by decomposing the principal's willingness to pay for calibration quality into a consensus premium that vanishes as markets become efficient and a constraint premium that remains permanently bounded above zero, with an explicit perturbation bound quantifying robustness when the benchmark assumptions are relaxed. Section 6 establishes incentive compatibility within the Hurwicz-Myerson mechanism design framework and shows that for the Brier score with reasonable parameter choices, the calibration threshold for honest disclosure vanishes entirely. Section 7 addresses the transmission mechanism. Testable predictions, limitations, and conclusions follow in Sections 8 through 10.
Consider a market with two types of agents: informed agents who pay cost \(c\) to observe a signal \(s\) about the asset's fundamental value \(v\), and uninformed agents who observe only the market price \(p\). If the price is fully revealing, informed agents cannot earn returns in excess of \(c\). But if the expected profit is zero, no rational agent will pay \(c\) to become informed. And if no agent becomes informed, the price carries no information about \(v\).
Grossman and Stiglitz (1980) showed that for any positive cost \(c > 0\), there exists no equilibrium in which prices are fully informationally efficient. The market settles into a noisy rational expectations equilibrium (Radner, 1979; Hellwig, 1980) in which prices are partially but never fully revealing, and a fraction of agents find it profitable to become informed because noise preserves their informational advantage.
The assumption this paper isolates is:
Assumption GS: The return to information production is captured exclusively through trading profit in the asset whose value the information concerns.
This assumption is never stated in the original paper because it is built into the model's structure. Agents choose between paying \(c\) and trading on the signal, or not paying \(c\) and trading on the price. There is no third option. Every subsequent resolution has preserved Assumption GS. Black's (1986) noise trader resolution works because noise traders provide the liquidity against which informed traders extract profit. Kyle's (1985) strategic trading model studies how informed agents optimally exploit their information through trading. Sims's (2003) rational inattention framework allocates cognitive bandwidth based on expected returns - when applied to asset markets, those returns are trading returns. Hirshleifer (1971) showed that the private value of information captured through trading diverges sharply from its social value, since trading profits transfer wealth between participants rather than creating it. The Disclosure Incentive addresses this deeper problem alongside the narrower one.
The claim that the literature has never examined non-trading returns to information production requires careful engagement with several bodies of prior work. Admati and Pfleiderer (1986) modeled agents who profit from selling information, demonstrating that information markets and asset markets can coexist. Their seller, however, maximizes revenue by degrading signal quality to prevent free-riding, so the decoupling from trading is partial. Garcia and Sangiorgi (2011) allow multiple compensation channels but treat the information sold as a raw signal. Hong and Kubik's (2003) career-concerns finding - that forecast accuracy predicts career outcomes - is empirically close to track record capital, but the return channel runs through employer promotion decisions, making it non-portable and institutionally mediated. What distinguishes the present paper from all of these is developed in Section 3, after the formal structure of the Disclosure Incentive has been specified.
Track record capital is the cumulative, publicly observable record of an agent's structured judgments verified against outcomes over time.
An agent \(i\) publishes a sequence of structured judgments \(\{J_{i1}, J_{i2}, \ldots, J_{iT}\}\) at times \(t = 1, 2, \ldots, T\). Each judgment contains: the evidence consulted (Input), the analytical method employed (Method), the conclusion reached (Output), the identity of the agent (Identity), and the agent's assessment of what happens when the relevant conditions change at scale (Scaling). At some subsequent time \(t + k\), the outcome is observable. The agent's calibration - the correspondence between stated confidence levels and observed frequencies (Dawid, 1982) - is measured using a strictly proper scoring rule (Brier, 1950; Gneiting and Raftery, 2007), a scoring function with the property that an agent uniquely maximizes expected score by reporting true beliefs. The track record is the cumulative calibration score across all published judgments.
The five-field structure requires justification. Input, Method, and Output formalize the epistemic content of an inference - evidence, reasoning, and conclusion. Removing any of the three leaves a disclosure that cannot be meaningfully evaluated. Identity enables verification and track record accumulation. Scaling forces the discloser to articulate interaction effects across agents and across time - the dimension most likely to contain information that scalar prices compress away. A fund manager's assessment of a single firm's water usage is Input-Method-Output. The same manager's assessment of what happens to aquifer levels when five operators in the same watershed simultaneously increase extraction at the rates their current trajectories imply is Scaling. It captures the constraint reasoning that the price system structurally cannot carry, because prices aggregate individual transactions without preserving the interaction structure among them.
The formal model in Section 4 represents the disclosed signal as a scalar \(s_i\) for tractability. The five-field structure is an implementation specification for the disclosure format, not a feature of the formal model. The resolution and incentive compatibility results hold for any disclosure that carries a probability assessment scorable by a strictly proper scoring rule; the welfare dominance result (Proposition 3) additionally requires the disclosure to carry precision about multiple eigenbasis dimensions. The five fields are argued to be jointly sufficient for the mechanism to operate and individually necessary for the track record to be meaningful, but they are not derived as uniquely optimal from the formal apparatus - they are a design choice motivated by the epistemic requirements of structured judgment.
The track record is portable - it belongs to the agent rather than to any employer - and it compounds, since each verified judgment adds precision to the calibration estimate. Tetlock (2005) and Tetlock and Gardner (2015) established that expert judgment quality varies enormously and is measurable through track records, but did not formalize the economic incentive structure that would cause agents to voluntarily build such records. This paper provides that formalization.
Against the prior work surveyed in Section 2, the Disclosure Incentive differs in that the disclosure carries the full five-field structure rather than being compressed into a price target or letter grade, and scoring is proper in the technical sense - the two features that distinguish it from Admati-Pfleiderer information selling and Hong-Kubik career rewards. The record belongs to the agent across institutional affiliations and the ontology is open (the discloser defines the domain), which together make it portable in a way that employer-mediated reputation is not. Most importantly, the disclosure discipline improves judgment quality over time rather than merely making existing quality visible - a property no prior model contains. These properties produce a full decoupling of information production from both trading profit and institutional affiliation.
Track record capital accumulation, rather than trading profit, is the return channel that defines the Disclosure Incentive.
An agent who observes signal \(s\) about the fundamental value of some economic variable faces a choice under Assumption GS: trade on \(s\) or forgo the return. The Disclosure Incentive opens a second path: the agent publishes a structured judgment based on \(s\), waits for the outcome, and allows the track record to update. The return accrues in the labor market, the consulting market, the capital allocation advisory market - anywhere demonstrated judgment quality commands a premium. Because the track record measures judgment quality rather than trading timing, the return does not depend on the asset price. A well-calibrated judgment published after the price has moved is worth exactly as much to the track record as one published before, so long as it was timestamped before the outcome was known.
The crowding-out concern raised by Benabou and Tirole (2006) deserves engagement. Calibration quality is not a proxy for the intrinsic objective of good analytical reasoning; it is that objective, directly measured. And because the reward accrues through a public record rather than a private bonus, it engages reputational motivation, which their analysis identifies as less susceptible to crowding-out than purely pecuniary incentives. The concern nonetheless has force at the margin: agents might optimize for narrow calibration metrics while neglecting broader analytical depth. The five-field format limits this by scoring the reasoning path, not just the conclusion.
Track record capital differs from the closest existing concepts in economics. Human capital (Becker, 1964) accumulates through training and experience; track record capital does not, since two analysts with identical credentials can have dramatically different records. Signaling (Spence, 1973) communicates an unobservable type indirectly through costly credentials, whereas a track record is a direct measurement of calibration quality, verified against reality rather than an institutional standard. The subtlest distinction is from reputation in the career concerns literature (Holmstrom, 1999; Chevalier and Ellison, 1999), where reputation inferred from noisy behavioral signals creates incentives to conform (Scharfstein and Stein, 1990; Trueman, 1994) rather than to be well-calibrated. The proper scoring rule in the Disclosure Incentive breaks this herding: it rewards calibration rather than consensus, with foundational results tracing to Good (1952) and Savage (1971). The formal proof of incentive compatibility occupies Section 6.
Track record capital has a property that no prior model of reputation or information selling contains: the act of structured disclosure improves judgment quality over time, not merely reveals it. The discipline of articulation, verification, and iterative learning generates better quality as a consequence of the disclosure process itself.
The improvement claim rests on conditions identified by Kahneman and Klein (2009): feedback improves judgment reliably only in high-validity environments where the relationship between cues and outcomes is regular enough for learning. The Disclosure Incentive's structured timestamping, proper scoring, and explicit confidence levels are designed to engineer such an environment in domains where it is naturally absent - consistent with Einhorn's (1980) finding that unstructured experience does not improve judgment. Lichtenstein and Fischhoff (1980) demonstrated experimentally that explicit calibration feedback produces measurable improvement, and Ericsson, Krampe, and Tesch-Römer's (1993) deliberate practice framework identifies the conditions: structured task, unambiguous feedback, sufficient iteration. The five-field format, proper scoring, and rolling disclosure cadence map onto each condition.
The improvement mechanism is the paper's most distinctive informal contribution. The formal model in Section 4 is static and sufficient to resolve the paradox without it. A dynamic extension in which \(\tau_\varepsilon\) improves as a function of disclosure history is specified in Section 8 as a testable prediction. The static resolution stands on its own; the dynamic claim strengthens it but is not formalized here.
Consider an economy with a single risky asset paying terminal value \(v\) at date 2, where \(v \sim N(\bar{v}, \tau_v^{-1})\), with \(\tau_v\) denoting the precision of the prior. A riskless asset has zero net return. The total supply of the risky asset is normalized to zero. A continuum of agents indexed on \([0,1]\) have CARA utility with common risk aversion coefficient \(\gamma\). Noise traders submit random demand \(u \sim N(0, \tau_u^{-1})\), independent of \(v\) and all other random variables.
At date 0, each agent \(i\) chooses one of three types:
Uninformed (U): pays no cost, observes no signal.
Trader (T): pays cost \(c > 0\), privately observes signal \(s_i = v + \varepsilon_i\), where \(\varepsilon_i \sim N(0, \tau_\varepsilon^{-1})\), independent across agents. Trades on private information.
Discloser (D): pays cost \(c > 0\), observes signal \(s_i = v + \varepsilon_i\), publishes \(s_i\) immediately, and earns track record return from the judgment market. The discloser's signal becomes public upon publication.
The clean partition into types is a modeling simplification. In practice, an agent could trade on a signal and then disclose it. For the paradox resolution (Proposition 1), the partition is without loss of generality: if an agent who trades and then discloses earns \(\pi_T + r_D\), the trading component \(\pi_T \to 0\) as prices become efficient while \(r_D\) remains positive (by J4), so the trade-then-disclose agent's incentive reduces to the pure discloser's in the limit where the paradox bites. The resolution holds a fortiori when agents can access both channels.
The equilibrium comparison results (Propositions 2, 4) and the no-paradox result (Proposition 5) rely on the partition at interior parameter values, not just in the limit. If agents can trade and disclose simultaneously, the equilibrium fractions \((\lambda_T^*, \lambda_D^*)\) change, altering \(\tau_p\) and potentially breaking the informational equivalence of Proposition 2. The sequential determination (Condition 1 pins down \(\lambda_D^*\) independently, then Condition 2 pins down \(\lambda_T^*\)) depends on the partition. These results hold under the partition as specified; a richer model admitting mixed types would require a simultaneous equilibrium characterization. Section 7 discusses the rolling timing architecture that accommodates mixed strategies in practice.
Let \(\lambda_T\) and \(\lambda_D\) denote the fractions choosing T and D respectively, with \(\lambda_T, \lambda_D \ge 0\) and \(\lambda_T + \lambda_D \le 1\).
The timing is: Date 0, agents choose type. Date 1, disclosers simultaneously publish their signals; all agents observe the aggregate disclosure; the asset market clears and price \(p\) is determined; disclosers receive their track record return. Date 2, the asset pays \(v\).
When \(\lambda_D > 0\), the published signals \(\{s_j : j \in D\}\) are publicly observable. In the continuum, the aggregate disclosure statistic
\[\bar{d} = \frac{1}{\lambda_D} \int_{j \in D} s_j \, dj = v + \eta\]
where \(\eta \sim N(0, (\lambda_D \cdot \tau_\varepsilon)^{-1})\), provides a sufficient statistic for the disclosed signals with precision \(\tau_D = \lambda_D \cdot \tau_\varepsilon\). (The exact aggregation invokes the law of large numbers over a continuum of independent random variables, following the standard convention in the noisy rational expectations literature. The measure-theoretic difficulty was identified by Judd, 1985; the positive resolution was provided by Sun, 2006.)
After disclosure, the public precision (prior plus disclosure) is:
\[\tau_{\text{pub}} = \tau_v + \tau_D = \tau_v + \lambda_D \cdot \tau_\varepsilon\]
This is the precision available to every agent before the price is observed. It exceeds the prior precision \(\tau_v\) whenever \(\lambda_D > 0\).
Given the public information, the economy reduces to a standard Grossman-Stiglitz problem with effective prior precision \(\tau_{\text{pub}}\) and fraction \(\lambda_T\) of informed traders.
Each trader's private information, conditional on the public disclosure, has precision \(\tau_\varepsilon\). Disclosers have no residual private information for trading purposes. Following Grossman and Stiglitz (1980) and Hellwig (1980), the price signal provides uninformed agents with precision \(\tau_p(\lambda_T)\) about \(v\) beyond what \(\tau_{\text{pub}}\) already reveals.
The posterior precision for each agent type is:
Trader: \(\tau_T = \tau_{\text{pub}} + \tau_p(\lambda_T) + \tau_\varepsilon\)
Uninformed/Discloser: \(\tau_U = \tau_{\text{pub}} + \tau_p(\lambda_T)\)
The trader indifference condition is:
\[\Phi(\lambda_T, \tau_{\text{pub}}) = \exp(2\gamma c)\]
where \(\Phi(\lambda_T, \tau_{\text{pub}}) = \frac{\tau_{\text{pub}} + \tau_p(\lambda_T) + \tau_\varepsilon}{\tau_{\text{pub}} + \tau_p(\lambda_T)}\)
This is the Grossman-Stiglitz indifference condition, modified only in that \(\tau_{\text{pub}}\) replaces \(\tau_v\). The function \(\Phi\) is strictly decreasing in \(\lambda_T\) and strictly decreasing in \(\tau_{\text{pub}}\).
The track record return captures the market value of demonstrated judgment quality. A judgment market operates alongside the asset market: principals (employers, capital allocators, institutional clients) hire or allocate capital to agents based on demonstrated calibration.
Let \(\bar{r}_D(\lambda_D)\) denote the expected track record return per discloser when the fraction of disclosers is \(\lambda_D\). The following conditions are derived from the principal's optimization problem in Section 5:
(J1) \(\bar{r}_D\) is continuous and strictly decreasing on \([0,1]\).
(J2) \(\bar{r}_D(0) > c\): when no agent is disclosing, the first entrant earns a return exceeding information cost.
(J3) \(\bar{r}_D(1) < c\): if all agents disclose, the judgment market premium is competed below cost.
(J4) \(\bar{r}_D(\lambda_D)\) is independent of price efficiency: the judgment market compensates calibration quality, not trading timing.
Section 5 derives these conditions - particularly J4 - from a micro-founded model of the principal's portfolio problem. J4 is a result (Proposition 7), not a primitive assumption.
By J1-J3 and the intermediate value theorem, there exists a unique \(\lambda_D^* \in (0,1)\) such that \(\bar{r}_D(\lambda_D^*) = c\). This is the judgment market equilibrium.
The equilibrium of the augmented economy is a pair \((\lambda_T^*, \lambda_D^*)\) satisfying:
Condition 1 (Judgment market clearing): \(\bar{r}_D(\lambda_D^*) = c\)
Condition 2 (Trader indifference): \(\Phi(\lambda_T^*, \tau_v + \lambda_D^* \cdot \tau_\varepsilon) = \exp(2\gamma c)\)
The equilibrium is determined sequentially. Condition 1 pins down \(\lambda_D^*\) independently of the asset market (by J4). Given \(\lambda_D^*\), Condition 2 determines \(\lambda_T^*\) as in the standard framework, but with the augmented public precision \(\tau_{\text{pub}} = \tau_v + \lambda_D^* \cdot \tau_\varepsilon\).
Proposition 1 (Resolution). In the augmented economy, information production is bounded below by \(\lambda_D^* > 0\) regardless of noise trading intensity. The Grossman-Stiglitz impossibility does not hold.
Proof. In the pure-trading economy (\(\lambda_D = 0\)), the trader indifference condition determines \(\lambda_T^*\). As \(\tau_u\) increases (noise variance decreases), the price becomes more informative for any given \(\lambda_T\), reducing the trading advantage. In the limit, \(\lambda_T^* \to 0\) and information production collapses. In the augmented economy, \(\lambda_D^*\) is determined by Condition 1 independently of \(\tau_u\). As \(\tau_u\) increases, the same logic pushes \(\lambda_T^* \to 0\), but \(\lambda_D^*\) is unaffected. Total information production \(\lambda_T^* + \lambda_D^* \ge \lambda_D^* > 0\) for all values of \(\tau_u\). The paradox required that the sole incentive for information production vanishes as prices become efficient. With a second incentive that is independent of price efficiency (J4, proved in Proposition 7), the sole-incentive premise fails. QED.
Proposition 2 (Informational equivalence and compositional improvement). The augmented equilibrium has the same total information available to uninformed agents as the standard Grossman-Stiglitz equilibrium, but the composition of information differs.
Proof. In both equilibria, the trader indifference condition equates the same ratio. In the standard economy: \(\Phi(\lambda_{GS}^*, \tau_v) = \exp(2\gamma c)\). In the augmented economy: \(\Phi(\lambda_T^*, \tau_v + \lambda_D^* \cdot \tau_\varepsilon) = \exp(2\gamma c)\). Both require:
\[\frac{\tau_\varepsilon}{\tau_{\text{pub}} + \tau_p} = \exp(2\gamma c) - 1\]
where the right side is the same constant. Therefore \(\tau_v + \tau_p(\lambda_{GS}^*) = \tau_v + \lambda_D^* \cdot \tau_\varepsilon + \tau_p(\lambda_T^*)\). Total uninformed precision is the same. The trader indifference condition endogenously adjusts \(\lambda_T^*\) downward to maintain the marginal trader's informational advantage.
The compositional difference is: in the augmented economy, a portion \(\lambda_D^* \cdot \tau_\varepsilon\) of the total information enters the public domain directly as structured reasoning, while in the standard economy all information enters through the noisy price channel. The disclosed information is observable in its full dimensionality - Input, Method, Output, Identity, Scaling - while the price signal is a scalar compression. QED.
The potential objection rooted in Morris and Shin (2002) - that increasing public signal precision can reduce welfare when agents have coordination motives - does not apply. In the CARA-normal framework, agents are price-takers with no coordination motive. The Morris-Shin pathology requires a beauty-contest payoff structure that is absent from the Grossman-Stiglitz setting. The five-field disclosure structure further works against overweighting by preserving the analytical reasoning behind the conclusion, enabling agents to evaluate the public signal rather than treating it as a focal point.
The informational equivalence of Proposition 2 is a floor on the welfare comparison, not the whole story. When agents face multi-dimensional decision problems, the structure of information matters beyond its scalar precision.
Proposition 3 (Strict welfare gain from compositional improvement). Consider an uninformed agent who must choose an action vector \(a = (a_1, \ldots, a_K)\) to minimize expected loss \(E[l(a, V)]\) where \(l(a, V) = \sum_k (a_k - V_k)^2\) and \(V = (V_1, \ldots, V_K)\) are the eigenbasis fundamentals. Assume that constraint dimensions have weakly less prior precision than the consensus dimension: \(\tau_{V_k} \le \tau_v\) for \(k \ge 2\). Compare two information regimes of equal total scalar precision \(\tau_{\text{total}}\): Regime P (price only), carrying precision \(\tau_{\text{total}}\) about \(V_1\) and zero about \(V_2, \ldots, V_K\); and Regime D (disclosure), carrying precision \(\tau_{D_k}\) about each dimension with \(\sum_k \tau_{D_k} = \tau_{\text{total}}\) and \(\tau_{D_k} > 0\) for at least two dimensions. Then \(E[l^D] < E[l^P]\): structured disclosure of equal total precision strictly dominates the price signal.
Proof. Under Regime P, the expected loss is \(E[l^P] = (\tau_v + \tau_{\text{total}})^{-1} + \sum_{k=2}^K \tau_{V_k}^{-1}\): the first dimension benefits from the price precision while constraint dimensions retain their prior variance. Under Regime D, \(E[l^D] = \sum_k (\tau_{V_k} + \tau_{D_k})^{-1}\). Because \(f(\tau) = 1/\tau\) is convex, concentrating a fixed precision budget on one dimension minimizes the total variance reduction across dimensions. Distributing precision to dimensions with lower prior precision produces larger reductions per unit allocated, since the marginal value \(\partial f / \partial \tau = -1/\tau^2\) is larger when \(\tau\) is smaller. Assume that \(\tau_{V_k} \le \tau_v\) for \(k \ge 2\) - that is, the prior is weakly less precise about constraint dimensions than about the consensus dimension. This holds when \(V_1\) is the dimension about which the market has aggregated the most prior information - a condition the eigenbasis construction in Section 5.2 is designed to capture. Under this condition, any interior allocation with \(\tau_{D_k} > 0\) for some \(k \ge 2\) strictly reduces total expected loss relative to the concentrated allocation. QED.
Proposition 3 means the compositional improvement identified in Proposition 2 is not merely aesthetic. Equal total precision delivered as a scalar (the price) versus as a structured vector (the disclosure) produces strictly different welfare when agents face multi-dimensional decisions. The price allocates information where it is least needed; structured disclosure allocates it where it is most needed. The marginal dominance of disclosure bits over price-precision bits is established formally in the companion paper (Beier, 2026c, Proposition D).
Proposition 4 (Pareto improvement). The augmented equilibrium weakly Pareto-dominates the Grossman-Stiglitz equilibrium within the CARA-normal model, and strictly dominates for inframarginal disclosers. Under the additional structure of Proposition 3, the dominance is strict for all agents facing multi-dimensional decisions.
Proof. The proof proceeds agent type by agent type.
Uninformed agents. By Proposition 2, uninformed agents achieve the same posterior precision in both equilibria within the CARA-normal scalar comparison. In the single-asset model, where expected utility depends solely on posterior precision and the price distribution, uninformed agents' expected utility is weakly unchanged. The strict improvement requires that uninformed agents face a multi-dimensional decision problem as specified in Proposition 3, which is a richer environment than the single-asset GE model. Under that additional structure, the compositional shift from scalar price signal to structured disclosure produces a strict welfare gain. The Pareto improvement is therefore weak within the GE model as specified and strict under the multi-dimensional extension.
Marginal traders. In both equilibria, the marginal trader earns expected trading profit exactly equal to \(c\). The indifference condition ensures this. Traders' expected utility is the same across equilibria.
Marginal disclosers. By construction, the marginal discloser earns track record return exactly equal to \(c\) (from Condition 1). The marginal discloser's expected utility equals that of the marginal trader.
Inframarginal disclosers. Agents with calibration quality \(q\) above the marginal cutoff earn track record returns strictly exceeding \(c\). In the standard Grossman-Stiglitz equilibrium, these agents would either trade (earning exactly \(c\) in expectation by indifference) or remain uninformed. In the augmented equilibrium, they earn \(\bar{r}_D(q) > c\). Their expected utility is strictly higher.
Inframarginal disclosers are strictly better off; all other agents are weakly no worse off. Under the multi-dimensional extension of Proposition 3, uninformed agents are also strictly better off. QED.
Proposition 5 (No judgment market paradox). The judgment market does not generate a reflexive paradox analogous to Grossman-Stiglitz.
Proof. The Grossman-Stiglitz paradox has a specific reflexive structure: information production increases price efficiency, which reduces the return to information production, which reduces information production. In the judgment market, the analog would require: disclosure increases the supply of calibrated judgment, which reduces the track record return, which reduces disclosure, which reduces the supply. This is a standard supply-demand adjustment with a stable interior equilibrium at \(\lambda_D^*\) where \(\bar{r}_D(\lambda_D^*) = c\) (by J1-J3). The reflexive structure is absent because track record return does not depend on the information content of the price - it depends on the supply and demand for demonstrated calibration. No analog of the efficiency-destroys-incentive loop exists. QED.
The stability of this equilibrium is reinforced by the empirical finding that calibration quality has large and persistent cross-sectional variance (Tetlock, 2005): even when \(\lambda_D\) is large, top-calibrated disclosers retain a scarce skill premium. The judgment market develops a quality gradient, as labor markets generally do.
A decrease in noise trading variance (higher \(\tau_u\)) pushes \(\lambda_T^*\) toward zero without affecting \(\lambda_D^*\). This is the comparative static that most directly illustrates the resolution: the limiting case that destroys the trading channel leaves the disclosure channel intact.
An increase in the track record return function shifts \(\lambda_D^*\) upward, which raises \(\tau_{\text{pub}}\), which lowers \(\lambda_T^*\). More disclosure crowds out some trading, but total information production increases and a larger share enters the public domain.
An increase in information cost \(c\) reduces both \(\lambda_D^*\) and \(\lambda_T^*\). But the relative resilience differs: the trading channel becomes fragile as \(c\) approaches the maximum trading profit, while the disclosure channel survives as long as the judgment market premium exceeds \(c\) for some agents.
This section derives the properties of the judgment market - particularly J4, the independence of judgment market returns from price efficiency - from a model of the principal's portfolio problem. The derivation operates at three levels: an exact decomposition under the CARA-normal benchmark, an analysis of where the decomposition breaks in the non-separable case, and a conditional lower bound that recovers the economically sufficient result.
A principal must make decisions about the constraint environment that require assessments of future states. The principal's expertise is limited. Hiring an agent with demonstrated calibration quality reduces the expected cost of decisions made under uncertainty.
Let \(\theta\) be the uncertain state of the constraint environment (drawn from distribution \(F\)). The principal must choose an action \(a \in A\) that depends on \(\theta\). The principal's loss from action \(a\) when the state is \(\theta\) is \(l(a, \theta)\). Under delegation, the agent reports an assessment \(r\) of the probability distribution over \(\theta\), and the principal chooses \(a(r) = \text{argmin}_a \, E_r[l(a, \theta)]\).
Definition 1. The calibration quality of an agent is \(q\), measuring the correspondence between the agent's reported distributions and realized outcome frequencies (Dawid, 1982; DeGroot and Fienberg, 1983). A perfectly calibrated agent (\(q = 1\)) reports distributions matching outcome frequencies exactly. An uninformed agent (\(q = 0\)) reports the prior \(F\) regardless of any private signal.
Proposition 6 (Value of calibration). The principal's expected loss from delegation is strictly decreasing in the agent's calibration quality \(q\).
Proof. Let \(L(q) = E[l(a(r_q), \theta)]\) be the expected loss when delegating to an agent with calibration quality \(q\). At \(q = 0\), the agent reports the prior and the principal acts on the prior. At \(q = 1\), the agent reports perfectly calibrated probabilities and the principal's action is optimal given the agent's information. For intermediate \(q\), model the agent's report as \(r_q = q \cdot p + (1-q) \cdot F\), a convex combination of the true posterior \(p\) and the prior \(F\). This garbling preserves the mean, adds noise proportional to \((1-q)\), and produces a signal that is Blackwell-dominated by the true posterior for all \(q < 1\). By the Blackwell (1953) theorem, a more informative signal (higher \(q\)) strictly reduces the expected loss for decision problems where the action depends on the state. Since the principal's problem involves choosing actions contingent on the constraint environment, the strict version holds: \(L(q') < L(q)\) for \(q' > q\) wherever the principal's optimal action is not constant in \(\theta\). QED.
The true payoff of the investment depends on a vector of fundamental variables \(X = (X_1, \ldots, X_K)\) (cash flows, regulatory timelines, aquifer depletion rates, supply chain bottlenecks). Let \(V = U'X\) be the eigenbasis projection, where \(V_1\) is the primary principal component (the consensus value dimension) and \(V_2, \ldots, V_K\) are the orthogonal constraint dimensions. The asset price \(p\) carries public information strictly about \(V_1\) with precision \(\tau_p\), remaining structurally blind to the constraint dimensions (formally established as Proposition C in Beier, 2026c, where the excess distortion is derived in closed form within the CARA-normal framework).
This blindness is a modeling assumption, but it also reflects an empirical claim about what prices do: scalar transaction prices compress high-dimensional judgment into a single number, losing the interaction effects and constraint reasoning that structured disclosure preserves. The claim is testable: if prices already carry constraint-dimension information, then constraint-relevant disclosures should produce no abnormal returns when published. If they do produce abnormal returns with a lag - consistent with prices adjusting to information they did not previously contain - the empirical claim is supported.
The principal can hire a judgment agent with calibration quality \(q\) to provide an assessment \(D\). Real agents do not assess eigenvectors; they assess original-basis variables. The agent's signal \(D\) carries information about a subset of the original variables \(X\). Because \(V_1\) is a linear combination of all \(X\), the agent's signal naturally carries some information about \(V_1\) alongside its information about \(V_2, \ldots, V_K\). The agent's total informational contribution is a mixture of consensus value discovery and constraint assessment.
Assume the principal solves an unconstrained CARA-normal portfolio optimization problem, choosing a vector of risky asset allocations \(a\) alongside a risk-free asset to maximize expected utility \(E[-\exp(-\gamma(W_0 + a'V))]\). The risk-free asset acts as a slack variable, so there is no binding budget constraint coupling the risky allocations.
Structural Assumption (Posterior Independence). Assume the prior over \(V\) is jointly Gaussian with diagonal covariance matrix \(\Sigma_V = \text{diag}(\sigma^2_{V_1}, \ldots, \sigma^2_{V_K})\). The agent observes \(D = X_S + \varepsilon\), where \(X_S\) is a subset of the original-basis fundamentals and \(\varepsilon \sim \mathcal{N}(0, \Omega)\). The observation noise \(\varepsilon\) is structured such that when rotated into the eigenbasis via \(U'\), the transformed noise covariance \(U'\Omega U\) is diagonal. Under this condition, the Bayesian update from \(D\) preserves the zero off-diagonal terms of the posterior covariance over \(V\), ensuring posterior factorization across dimensions.
Exact diagonalization is a mathematical convenience approximating a common empirical pattern: specialized agents observe localized, narrow-domain constraint variables whose observation errors are largely uncorrelated with the broad macroeconomic factors driving the consensus dimension. An environmental engineer assessing water table depth introduces measurement noise with little correlation to GDP forecast noise. When off-diagonal terms in \(U'\Omega U\) are small relative to the diagonal entries, the decomposition holds approximately, with cross-dimensional leakage entering as a second-order perturbation.
The principal's willingness to pay for the agent's calibration quality \(q\), given price precision \(\tau_p\), is:
\[W(q, \tau_p) = CE(q, \tau_p) - CE(0, \tau_p)\]
Proposition 7 (Decomposition and asymptotic independence of judgment demand). Under the unconstrained CARA-normal portfolio problem with posterior independence, the principal's total willingness to pay \(W(q, \tau_p)\) decomposes into a Consensus Premium and a Constraint Premium:
\[W(q, \tau_p) = W_{\text{consensus}}(q, \tau_p) + W_{\text{constraint}}(q)\]
The total willingness to pay monotonically declines in price efficiency (\(\frac{\partial W}{\partial \tau_p} < 0\)) but asymptotes to a strictly positive floor:
\[\lim_{\tau_p \to \infty} W(q, \tau_p) = W_{\text{constraint}}(q) > 0\]
Proof.
Step 1: Orthogonal Decomposition of the Signal. The agent's original-basis signal \(D\) can be projected onto the eigenbasis: \(D = D_{V_1} + D_{V_{2\ldots K}}\). Because the price provides a signal about \(V_1\) with precision \(\tau_p\), the component \(D_{V_1}\) is partially redundant with the price. The orthogonal component \(D_{V_{2\ldots K}}\) is strictly non-redundant because the price provides zero bits of information about \(V_2, \ldots, V_K\).
Step 2: Separability of the Certainty Equivalent. The dimensions \(V_k\) are independent in the prior. By the Structural Assumption, they remain independent in the posterior after updating on \(D\). Because the unconstrained CARA optimization lacks a binding budget constraint to couple them, the certainty equivalent additively separates:
\[CE(q, \tau_p) = CE_1(\tau_p, q) + \sum_{k=2}^K CE_k(q)\]
where \(CE_1(\tau_p, q)\) depends on both price precision and agent quality, and \(CE_k(q)\) for \(k \ge 2\) depend only on the agent's quality \(q\) and the prior constraint variances.
Step 3: Decomposition of the Willingness to Pay.
\[W(q, \tau_p) = \underbrace{[CE_1(\tau_p, q) - CE_1(\tau_p, 0)]}_{W_{\text{consensus}}(q, \tau_p)} + \underbrace{\sum_{k=2}^K [CE_k(q) - CE_k(0)]}_{W_{\text{constraint}}(q)}\]
The Consensus Premium captures the marginal value of the agent's \(V_1\) information given the price. Under CARA-normal updating, the posterior precision on \(V_1\) given both \(p\) and \(D\) is \(\tau_p + \tau_{D_{V_1}}(q)\), while the precision given \(p\) alone is \(\tau_p\):
\[W_{\text{consensus}}(q, \tau_p) = \frac{1}{2\gamma} \ln\left(\frac{\tau_p + \tau_{D_{V_1}}(q)}{\tau_p}\right)\]
This is strictly positive for finite \(\tau_p\), strictly decreasing in \(\tau_p\), and vanishes in the limit: \(\lim_{\tau_p \to \infty} W_{\text{consensus}}(q, \tau_p) = 0\).
The Constraint Premium is:
\[W_{\text{constraint}}(q) = \sum_{k=2}^K \frac{1}{2\gamma} \ln\left(\frac{\tau_{V_k} + \tau_{D_{V_k}}(q)}{\tau_{V_k}}\right)\]
Each term depends only on \(q\) and the prior constraint variances, not on \(\tau_p\). Therefore \(W_{\text{constraint}}(q) > 0\) for any \(q > 0\), and \(\frac{\partial W_{\text{constraint}}}{\partial \tau_p} = 0\).
The total derivative follows: \(\frac{\partial W}{\partial \tau_p} = \frac{\partial W_{\text{consensus}}}{\partial \tau_p} < 0\), and \(\lim_{\tau_p \to \infty} W(q, \tau_p) = W_{\text{constraint}}(q) > 0\). QED.
Proposition 7 establishes the decomposition under the unconstrained CARA-normal framework with diagonal posterior covariance. In real-world capital allocation, these conditions frequently fail.
When the principal faces a binding budget constraint or when the asset payoff is a nonlinear function of fundamentals (threshold defaults, for instance), the loss function becomes non-separable. The cross-partial derivative \(\frac{\partial^2 L}{\partial \tau_p \partial q}\) is strictly ambiguous.
The ambiguity has a concrete economic source. If the price reveals that \(V_1\) is catastrophic and the project economics are unviable, the principal abandons the investment, rendering constraint information about \(V_2\) worthless - the signals act as substitutes. Yet when the price confirms strong consensus profitability, the principal scales exposure maximally, spiking vulnerability to tail-risk constraints and the value of constraint judgment - the signals are complements. Which regime obtains depends on the realized price, so the cross-partial's sign cannot be determined ex ante.
Because informational independence does not guarantee payoff independence, the exact decomposition of Proposition 7 does not extend beyond the CARA-normal benchmark without additional structure.
When \(U'\Omega U\) has off-diagonal entries, the posterior covariance becomes \(\Sigma_{\text{post}} = \Sigma_{\text{diag}} + \Delta\), where \(\Sigma_{\text{diag}} = \text{diag}(\Sigma_{\text{post}})\) and \(\|\Delta\|_F = \delta\) captures the off-diagonal perturbation.
Proposition 8 (Perturbation bound). Let \(\delta < \lambda_{\min}(\Sigma_{\text{diag}})\). The willingness-to-pay decomposition error satisfies:
\[|W(q, \tau_p) - [W_{\text{consensus}}(q, \tau_p) + W_{\text{constraint}}(q)]| \le \frac{\|\mu\|^2}{2\gamma} \cdot \frac{\delta}{\lambda_{\min}(\Sigma_{\text{diag}})[\lambda_{\min}(\Sigma_{\text{diag}}) - \delta]}\]
where \(\mu = E[V \mid D]\) is the posterior mean under the agent's signal. For \(\delta \ll \lambda_{\min}(\Sigma_{\text{diag}})\), the bound simplifies to \(O(\|\mu\|^2 \delta / (2\gamma \lambda_{\min}^2))\).
Proof. The ex-post certainty equivalent under CARA-normal optimization is \(CE = \frac{1}{2\gamma}\mu'\Sigma_{\text{post}}^{-1}\mu\). Proposition 7 uses the ex-ante expected certainty equivalent, which integrates over the distribution of \(\mu\); because the bound below holds for each realization of \(\mu\), it survives integration. Applying the resolvent identity \((\Sigma_{\text{diag}} + \Delta)^{-1} = \Sigma_{\text{diag}}^{-1} - \Sigma_{\text{diag}}^{-1}\Delta(\Sigma_{\text{diag}} + \Delta)^{-1}\), the CE error is \(|CE(q) - CE_{\text{sep}}(q)| = \frac{1}{2\gamma}|\mu'\Sigma_{\text{diag}}^{-1}\Delta(\Sigma_{\text{diag}} + \Delta)^{-1}\mu|\). Bounding with operator norms and applying Weyl's inequality for the perturbed inverse yields \(\frac{\|\mu\|^2}{2\gamma} \cdot \frac{\delta}{\lambda_{\min}(\lambda_{\min} - \delta)}\). The willingness to pay is \(W = CE(q) - CE(0)\). Because \(CE(0)\) involves no agent signal and hence no \(\Delta\) perturbation, \(CE(0)\) decomposes exactly. The WTP decomposition error therefore equals the \(CE(q)\) error alone, yielding the stated bound without doubling. QED.
The error scales with \(\delta/\lambda_{\min}^2\): the ratio of off-diagonal noise to the square of the smallest posterior variance. When the agent specializes in a narrow domain, \(\delta\) is small relative to \(\lambda_{\min}\) and the decomposition is reliable. For a generalist whose observation errors correlate across dimensions, \(\delta\) is non-negligible and the decomposition is approximate.
A quantitative illustration: with \(K = 3\) dimensions, posterior standard deviation of 10% on the least-informed constraint dimension (\(\lambda_{\min} = 0.01\)), expected excess return of 5% (\(\|\mu\| = 0.05\)), risk aversion \(\gamma = 2\), and off-diagonal perturbation one-tenth the smallest diagonal entry (\(\delta = 0.001\)). The bound gives \(\frac{0.0025}{4} \cdot \frac{0.001}{0.01 \times 0.009} \approx 0.0069\) - roughly 7% of a typical 10-cent-per-dollar constraint premium. Reducing \(\delta\) to one-hundredth of the smallest diagonal entry (a highly specialized agent) drops the error to under 1% of the constraint premium. Whether these magnitudes are acceptable depends on the application; the bound allows the reader to evaluate this for specific parameter configurations.
Within the participation regime, the judgment market does not collapse.
Definition 2. The principal is in the participation regime for asset \(j\) when, given the price information \(p\), the optimal allocation satisfies \(\|a^*(p)\| > 0\). If the price reveals that \(V_1\) is catastrophically low, the principal exits entirely and the judgment market for this specific asset naturally collapses.
Claim 9 (Conditional lower bound on judgment demand). Within the participation regime, the principal's willingness to pay for constraint judgment satisfies:
\[W(q, \tau_p) \ge \kappa_{\min}(\tau_p) \cdot \sigma^2 \cdot T \cdot g(q)\]
where \(\kappa_{\min}(\tau_p) > 0\) is the minimum sensitivity of the principal's loss function to constraint variance given price information and conditional on participation; \(\sigma^2 > 0\) is the rate at which the physical environment generates new constraint variance through non-stationary innovations (regime shifts, novel hazards, irreversible threshold crossings); \(T\) is the decision horizon; and \(g(q) = \min_{k \ge 2} \frac{\tau_{D_{V_k}}(q)}{\tau_{V_k} + \tau_{D_{V_k}}(q)}\) is the minimum fractional variance reduction across constraint dimensions, strictly increasing in \(q\).
As long as the principal remains invested, the continuous physical generation of unpriced constraint variance guarantees that calibrated judgment retains strictly positive value. This is labeled a claim rather than a proposition because \(\kappa_{\min}\) is not bounded from below in terms of model primitives - it is asserted to be positive conditional on participation. The claim's content beyond the participation assumption is carried by the \(\sigma^2 \cdot T\) term: the physical environment continuously generates new constraint variance that the price system cannot anticipate. The force is empirical (physical systems do generate non-stationary innovations) rather than purely formal.
The complete J4 result operates at three levels. Under the CARA-normal benchmark (Proposition 7), the willingness to pay for judgment decomposes exactly. The consensus premium monotonically declines as price efficiency improves, reaching zero in the limit. The constraint premium is permanently bounded above zero, independent of price efficiency.
Outside the benchmark, when budget constraints bind or payoffs are nonlinear, the clean decomposition breaks. Consensus and constraint information interact through the loss function's curvature, sometimes as substitutes, sometimes as complements.
The conditional lower bound (Claim 9) recovers a weaker but economically sufficient result: within the participation regime, physical generation of new constraint variance ensures strictly positive demand for calibrated judgment. J4 holds as a derived result, not an assumption.
Definition 3. The judgment market consists of principals who demand calibration quality and agents who supply it through disclosure. The market return per discloser when the fraction of disclosers is \(\lambda_D\) is:
\[\bar{r}_D(\lambda_D) = \int W(q) \, dG(q \mid \lambda_D)\]
where \(W(q) = L(0) - L(q)\) is the principal's willingness to pay for calibration quality \(q\) (from Proposition 6), and \(G(q \mid \lambda_D)\) is the distribution of calibration quality among the \(\lambda_D\) disclosers.
Proposition 10 (Judgment market equilibrium). Under the following conditions, there exists a unique stable interior equilibrium \(\lambda_D^* \in (0, 1)\):
(M1) \(W(q)\) is strictly increasing in \(q\) (from Proposition 6) and \(W(0) = 0\).
(M2) The supply of disclosers at each quality level is increasing in the return, so that as \(\lambda_D\) increases, the average quality of the marginal discloser decreases.
(M3) The total demand for calibrated judgment is finite and grows more slowly than supply as \(\lambda_D\) increases, so that \(\bar{r}_D\) is continuous and strictly decreasing in \(\lambda_D\).
(M4) \(\bar{r}_D(0) > c\): when no agents disclose, the first discloser earns a return exceeding cost.
(M5) \(\bar{r}_D(1) < c\): if all agents disclose, the per-discloser return falls below cost.
These conditions reproduce J1-J4 with the addition that they are now derived from the principal's optimization problem. In particular, J4 follows from Proposition 7 and Claim 9: the constraint premium is independent of price efficiency, ensuring the demand curve for calibrated judgment asymptotes to a strictly positive floor.
Proof. By M3, \(\bar{r}_D\) is continuous and strictly decreasing on \([0,1]\). By M4 and M5, \(\bar{r}_D(0) > c > \bar{r}_D(1)\). By the intermediate value theorem, there exists \(\lambda_D \in (0,1)\) such that \(\bar{r}_D(\lambda_D) = c\). Uniqueness follows from strict monotonicity. Stability follows from the adjustment dynamics: if \(\lambda_D < \lambda_D^*\), the return exceeds cost and entry increases \(\lambda_D\); if \(\lambda_D > \lambda_D^*\), the return falls below cost and exit decreases \(\lambda_D\). QED.
This section establishes that the Disclosure Incentive is incentive-compatible within the Hurwicz-Myerson mechanism design framework (Hurwicz, 1972; Myerson, 1979, 1981). A mechanism in that sense consists of a message space, an outcome function mapping messages to allocations, and a solution concept specifying agent behavior. The Disclosure Incentive's message space is structured judgments in the five-field format; the outcome function has two components (track record score and track record return); and the relevant solution concept is Bayesian incentive compatibility.
The Hurwicz (1972) impossibility - that no mechanism achieves allocative efficiency and incentive compatibility across all preference profiles - does not apply in its full generality because the mechanism operates on a restricted domain. The payoff from disclosure is mediated by a scoring function with known mathematical properties rather than being an arbitrary function of the message.
Let \(\Omega\) be a finite outcome space and \(\Delta(\Omega)\) the set of probability distributions over \(\Omega\). A scoring rule \(S: \Delta(\Omega) \times \Omega \to \mathbb{R}\) assigns a score to a reported distribution \(r\) when outcome \(\omega\) is realized.
Definition 4. A scoring rule \(S\) is strictly proper if for all distributions \(p\) in the interior of \(\Delta(\Omega)\), \(E_p[S(p, \omega)] > E_p[S(r, \omega)]\) for all \(r \ne p\).
Proposition 11 (Classical result; Savage, 1971; Gneiting and Raftery, 2007). Let \(S\) be a strictly proper scoring rule. An agent with true belief \(p\) maximizes expected score by reporting \(r = p\). Truth-telling is strictly dominant in the single-report setting.
Proof. Define \(G(r, p) = E_p[S(r, \omega)]\). By strict properness, \(G(p, p) > G(r, p)\) for all \(r \ne p\). No information about other agents enters the maximization, so truth-telling is dominant. QED.
Definition 5. The cumulative track record score after \(T\) reports is \(R_T = \frac{1}{T} \sum_{t=1}^T S(r_t, \omega_t)\).
Proposition 12 (Sequential truth-telling). If \(S\) is strictly proper and the agent's per-period payoff is a weakly increasing function of \(R_T\), then truth-telling in every period maximizes the expected cumulative payoff.
Proof. Since \(R_T\) is additively separable across periods and future outcomes are independent conditional on the agent's beliefs, the maximization decomposes: the agent maximizes \(E[S(r_t, \omega_t)]\) at each \(t\) independently. By Proposition 11, the maximum at each \(t\) is achieved by \(r_t = p_t\). Since the payoff function is weakly increasing in \(R_T\), maximizing \(R_T\) maximizes the payoff. QED.
A qualification: if the payoff function is strictly concave in \(R_T\), the agent becomes risk-averse over score variance, which could incentivize shading toward consensus to reduce volatility. The weakly increasing assumption rules this out but deserves scrutiny for specific functional forms. Strategic interaction - where the agent's score at time \(t\) affects other agents' behavior feeding back to the agent's future payoff - could also break the decomposition. The commitment mechanism addresses this.
Propositions 11 and 12 establish that truth-telling is dominant given that the agent reports. But the agent also chooses whether to report. An agent who makes ten assessments and publishes only the five that turned out well overstates calibration quality. This selection problem - identified in the peer prediction literature by Witkowski and Parkes (2012) - is the principal threat to incentive compatibility.
The Disclosure Incentive constrains strategic selection through a commitment structure. At the beginning of each period, the agent commits to a disclosure domain \(D_i\) - a specification of which types of assessments the agent will disclose. The commitment is public, timestamped, and scored.
Definition 6. The coverage-adjusted score for agent \(i\) with commitment \(D_i\) is:
\[C_i = \alpha R_{D_i} + (1 - \alpha) \frac{|A_i|}{|D_i|}\]
where \(R_{D_i}\) is the average proper scoring rule score on published assessments within \(D_i\), \(|A_i|\) is the number of assessments actually published, \(|D_i|\) is the number of assessable events within the committed domain, and \(\alpha \in (0,1)\) weights accuracy against coverage.
Proposition 13 (Truth-telling under commitment). Let \(S\) be strictly proper. Under the coverage-adjusted scoring rule, for an agent with calibration quality \(q\) above a threshold \(q^*\) determined by \(\alpha\), honest comprehensive disclosure yields higher expected coverage-adjusted score than any strategy involving selection or misreporting.
Proof. Consider three deviations from honest comprehensive disclosure.
Deviation 1: Misreport beliefs on published assessments. Holding the publication set fixed, the expected accuracy component is maximized by truth-telling (Proposition 11). Any misreporting reduces the accuracy component without affecting coverage. This deviation is strictly dominated.
Deviation 2: Withhold unfavorable assessments. Suppose the agent withholds assessment \(t\). The coverage component drops by \((1-\alpha)/|D_i|\). The accuracy component may increase if the withheld assessment would have received a below-average score. For an agent with calibration quality \(q\), the variance of individual scores around the mean \(g(q)\) is \(\text{Var}(q)\), which is decreasing in \(q\). The maximum accuracy gain from withholding the worst assessment is bounded by \(\alpha \, \text{Var}(q)^{1/2}/|D_i|\). This bound uses the standard deviation as a proxy for the maximum single-score deviation from the mean, which is tight for bounded scoring rules but not for unbounded rules such as the logarithmic score, where a single confident wrong prediction can deviate from the mean arbitrarily. The general threshold \(q^*\) therefore applies reliably to bounded rules; Proposition 14 provides the sharpest result for the Brier score specifically. The coverage penalty per withheld assessment is \((1-\alpha)/|D_i|\). The deviation is unprofitable when \(\alpha \, \text{Var}(q)^{1/2} < (1-\alpha)\), which gives \(q^*\) implicitly through \(\text{Var}(q^*) = \left(\frac{1-\alpha}{\alpha}\right)^2\).
Joint deviations combining misreporting and withholding. Any strategy that misreports on some assessments while withholding others faces the accuracy penalty on the misreported assessments (Deviation 1) compounded with the coverage penalty on the withheld ones (Deviation 2). Since Deviation 1 is strictly dominated for any fixed publication set, the joint deviation reduces to a weakly worse version of Deviation 2 alone. The threshold \(q^*\) derived for Deviation 2 therefore covers the joint case.
Deviation 3: Manipulate the commitment. The agent could commit to a narrow domain excluding difficult assessments. This is constrained by market discipline: principals hiring based on track records observe the commitment domain and discount narrow commitments. An agent who commits to "technology sector, large-cap equities only" when applying for a role covering all sectors reveals the limitation. The formal constraint is that the equilibrium return \(\bar{r}_D\) incorporates principals' rational discounting of narrow commitments through \(G(q \mid \lambda_D)\), which assigns lower effective quality to agents whose commitment breadth falls below the relevant threshold.
For agents with \(q > q^*\), all deviations are unprofitable. QED.
Proposition 13 is a threshold result. Agents with calibration quality below \(q^*\) may find strategic selection profitable, and agents with very poor calibration may prefer not to disclose at all. The judgment market is designed to differentiate high-calibration from low-calibration agents, so these outcomes serve the mechanism's purpose. Principals who observe that an agent has been active for five years without accumulating a track record draw the appropriate inference from the absence.
The threshold \(q^*\) can be computed explicitly for specific scoring rules. The Brier score yields a particularly clean result.
Proposition 14 (Calibration threshold under the Brier score). Under the coverage-adjusted scoring rule with the Brier score \(S_B(r, x) = 1 - (r-x)^2\) and binary outcomes, the threshold \(q^* = 0\) in two cases. (i) For any strictly proper scoring rule bounded on \([0, 1]\), the threshold vanishes when \(\alpha < 1/2\). (ii) For the Brier score specifically, conditioning on the truth-telling result of Deviation 1, the threshold vanishes when \(\alpha < 4/5\). In both cases, honest comprehensive disclosure strictly dominates strategic selection for every agent regardless of calibration quality.
Proof. Part (i) is established in the companion paper (Beier, 2026r, Theorem 1). For any \([0, 1]\)-bounded strictly proper scoring rule, the maximum accuracy gain from withholding is bounded above by \(\alpha/(2n)\) per dropped assessment, while the coverage penalty is \((1-\alpha)/n\). The condition \(\alpha/2 < 1 - \alpha\) yields \(\alpha < 2/3\) under the Popoviciu variance bound; the companion paper obtains \(\alpha < 1/2\) by bounding expected score values directly rather than working through variances. Part (ii) follows from Corollary 1 of Beier (2026r). Conditioning on truth-telling, the expected Brier score on dimension \(k\) is \(E_{p_k}[S_B(p_k, \omega_k)] = 1 - p_k(1-p_k)\), minimized at \(p_k = 1/2\) with value \(3/4\). The accuracy gain from dropping the worst dimension is at most \(\alpha \cdot (1 - 3/4)/n = \alpha/(4n)\). The coverage penalty remains \((1-\alpha)/n\). No-withholding requires \(\alpha/4 < 1 - \alpha\), yielding \(\alpha < 4/5\). For all \(\alpha\) below the relevant bound, the coverage penalty from withholding any single assessment strictly exceeds the maximum possible accuracy gain from dropping the worst score. QED.
Remark. For \(\alpha\) between \(1/2\) and \(4/5\), the Brier-specific bound applies but the universal bound does not: the mechanism is selection-proof for the Brier score but not for arbitrary bounded scoring rules. For \(\alpha > 4/5\), a nontrivial calibration threshold exists even under the Brier score. The companion paper (Beier, 2026r) provides the full characterization of the threshold as a function of \(\alpha\) and the scoring rule.
The mechanism designer can choose \(\alpha\). Setting \(\alpha < 4/5\) eliminates the selection problem entirely for the Brier score while allowing accuracy to carry four times the weight of coverage. Setting \(\alpha < 1/2\) eliminates the selection problem for any bounded strictly proper scoring rule. The practical range is wide: \(\alpha = 0.75\) is selection-proof under the Brier score, and \(\alpha = 0.49\) is selection-proof under any bounded rule.
For the logarithmic score \(S_L(r, x) = \log(r(x))\), which is unbounded below, the Popoviciu bound does not apply. The score variance for poorly calibrated agents making confident predictions can be arbitrarily large, producing a nontrivial threshold for any \(\alpha > 0\). This is a feature: the logarithmic score penalizes overconfident miscalibration more severely. For practical implementation, the Brier score with \(\alpha < 4/5\) is the permissive choice (universally selection-proof); the logarithmic score with moderate \(\alpha\) is the discriminating choice (stronger incentives at the cost of a positive threshold).
Crawford and Sobel's (1982) model of strategic communication provides the baseline. In cheap talk, the sender's message is costless and unverifiable. Proper scoring aligns the sender's payoff with honest reporting, removing the dependence on preference alignment. The commitment mechanism and coverage scoring address the strategic selection problem that single-report proper scoring does not.
Bergemann and Morris (2005) introduced robust mechanism design, requiring mechanisms to perform well under uncertainty about agents' higher-order beliefs. The proper scoring rule is strictly proper regardless of the agent's beliefs about other agents, so Proposition 11 holds under any belief specification. The commitment mechanism's effectiveness (Proposition 13) depends on observability of the commitment domain - an institutional assumption rather than a belief assumption.
The relationship to prediction markets (Hanson, 2003; Wolfers and Zitzewitz, 2004) illuminates what the mechanism adds. Prediction markets elicit probability assessments by paying based on accuracy, but the payoff functions as trading profit on a prediction contract. Prediction markets therefore inherit the Grossman-Stiglitz structure for producing underlying reasoning. The Disclosure Incentive differs in that the payoff accrues through cumulative track record capital, breaking the reflexive loop. Hanson's (2003) logarithmic market scoring rule is a proper scoring rule embedded in a market mechanism, and the relationship is closer than a superficial comparison suggests: the Disclosure Incentive adds the commitment mechanism and five-field structure that address strategic selection and preserve reasoning, which the logarithmic market scoring rule does not.
The Bayesian Truth Serum (Prelec, 2004) and peer prediction methods (Miller, Resnick, and Zeckhauser, 2005) solve a version of the same problem - eliciting honest reports when outcomes are unverifiable or delayed. The Disclosure Incentive differs in that outcomes are eventually verifiable (through proper scoring against realized states), which provides a stronger incentive foundation than the prior-based mechanisms these methods employ. For long-horizon assessments where verification is delayed by years, interim scoring based on partial outcomes and prediction skill relative to reference classes partially bridges the gap.
If the discloser publishes rather than trades, how does the disclosed information enter the price?
Not through the discloser's own trading activity. The discloser produces a public good - structured, calibrated reasoning about the constraint environment - and the price moves only when other participants act on it. Traders read the disclosed reasoning, weigh it against their own analysis, and those who find it valuable express that assessment through their order flow. Track record capital compensates the discloser for producing the reasoning; trading profit compensates the traders who carried it into the price.
There is a speed problem worth confronting. If a discloser publishes openly and instantly, algorithmic traders will parse and price it in milliseconds, driving trading profit toward zero before anyone can act conventionally. Under that scenario, the entire cost of information production falls on track record return. Section 4.6 establishes that the paradox resolves even in this limiting case, so long as track record return exceeds \(c\) for a positive measure of agents.
The structured disclosure format accommodates a rolling timing architecture. A discloser selects a cadence and delay. One common structure releases the analytical content (reasoning, confidence level, the constraint being assessed) immediately, while holding back the position size for a specified embargo period. Priority for the track record attaches the moment the timestamped signal is published; the embargo preserves an informational advantage during the intervening window. Market forces rather than regulators determine where the equilibrium embargo length settles.
The framework generates predictions testable with existing or near-term data. Magnitude estimates are prior expectations calibrated to the empirical literature cited.
Prediction 1 (Volume of disclosed judgment). Within five years of a track record verification platform launching in a financial sector, the volume of publicly attributed, structured analytical judgments should increase by at least an order of magnitude relative to the pre-platform baseline. The increase should concentrate in long-horizon, constraint-based analysis rather than short-horizon earnings estimates, because the marginal Disclosure Incentive return is highest where trading profit is weakest. Identification: difference-in-differences comparing sectors with and without platform adoption.
Prediction 2 (Track record premium). Agents with verified track records of above-median calibration quality should command a wage premium of 15-40% in advisory and capital allocation roles, controlling for education, experience, and institutional affiliation. The premium should increase with record length at a diminishing rate, consistent with Bayesian updating. Identification: cross-sectional wage regression with track record scores as independent variable and institutional fixed effects. The range is calibrated to the premium documented by Tetlock and Gardner (2015) for superforecaster-level calibration.
Prediction 3 (Price adjustment speed). In sectors where disclosure ecosystems develop, the half-life of price adjustment to constraint events should decrease by 30-60% relative to sectors without such ecosystems. Identification: event studies comparing cumulative abnormal returns around constraint-relevant announcements.
Prediction 4 (Calibration improvement). Agents who practice structured disclosure for two or more years should show measurable calibration improvement (15-30% reduction in Brier score deviation from perfect calibration) relative to matched agents without disclosure practice. This tests the improvement mechanism of Section 3.4. Identification: panel regression of calibration scores on disclosure history length, with agent fixed effects.
The formal model assumes homogeneous information cost \(c\) and homogeneous risk aversion \(\gamma\); relaxing these would produce richer equilibrium predictions about the composition of discloser and trader populations.
The J4 micro-foundation (Proposition 7) establishes asymptotic independence rather than strict independence: total willingness to pay monotonically declines as price efficiency increases, but asymptotes to the strictly positive constraint premium rather than collapsing to zero. The decomposition depends on the Structural Assumption (Posterior Independence), which requires the agent's observation noise to be approximately diagonal in the eigenbasis. Proposition 8 provides explicit perturbation bounds: the error is first-order in the off-diagonal norm, under 1% of the constraint premium for highly specialized agents, and around 7% for moderately specialized ones. Whether these magnitudes are acceptable depends on the application. The eigenbasis itself shifts as the covariance structure of fundamentals changes over time; error bounds under non-stationary factor structures are deferred to future work.
The conditional lower bound (Claim 9) depends on the participation regime - the condition that the principal remains invested. This is a per-asset result. If increasing price efficiency systematically pushes more principals out of the participation regime (by revealing nonviability more frequently), aggregate demand for judgment across the economy could decline even as per-asset demand floors at a positive value. The formal model does not endogenize the measure of viable investments. This is a genuine gap, not a minor one. The empirical prevalence of long-horizon constraint uncertainty suggests the participation regime is common, but the aggregate question is formally open.
The truth-telling result under the commitment mechanism (Proposition 13) is a threshold result. Proposition 14 shows the threshold vanishes entirely for any bounded scoring rule when \(\alpha < 1/2\), and for the Brier score specifically when \(\alpha < 4/5\), resolving the concern for that scoring rule. For the logarithmic score and other unbounded rules, a nontrivial threshold exists. The proof analyzes three classes of deviation and argues that joint deviations reduce to the worst single deviation; a complete characterization covering correlated multi-agent strategies and dynamic commitment manipulation requires separate treatment.
Generative AI capable of producing plausible structured judgments at near-zero cost poses a threat beyond conventional adversarial attacks: it challenges the cost assumption (\(c > 0\)) directly, potentially flooding the judgment market with uncalibrated but superficially credible disclosures. Scoring mechanisms must then distinguish genuine calibration from sophisticated pattern-matching - a verification problem whose difficulty scales with model capability. If the cost of producing credible-looking disclosures falls to zero while the cost of producing genuinely calibrated judgments remains positive, the mechanism's screening function depends entirely on the scoring rule's ability to separate the two over time. Whether proper scoring achieves this separation against adversarial AI is an open question with near-term practical significance.
The delegation model in Section 5 is stylized. A heterogeneous population of principals with different decision problems, risk preferences, and outside options would produce a richer equilibrium with quality-differentiated pricing. The asymptotic independence of the constraint premium depends on the structure of the delegation problem rather than on any particular principal's preferences, so the core decomposition survives heterogeneity, but the equilibrium characterization is incomplete without it.
The constraint premium ultimately flows through principals' investment returns, which raises the question of whether the market will eventually price constraint information and erode the premium. The non-stationarity of the constraint environment (\(\sigma^2 > 0\) in Claim 9) is the mechanism's defense: as long as the physical environment generates new constraint variance faster than the price system incorporates it, the premium persists. Relatedly, disclosed reasoning is a public good (Section 7), and the mechanism requires principals to pay for future judgments from agents with good past records - a coordination problem this paper does not formalize. The mechanism's viability depends on principals finding it individually rational to hire based on track records, which Proposition 6 establishes, but the equilibrium in which track records are widely produced and widely consulted involves a bootstrapping problem that the static model does not address.
The improvement mechanism (Section 3.4) is the paper's most distinctive informal contribution but has no formal content in the current model. The static framework is sufficient to resolve the paradox without it. A dynamic extension where \(\tau_\varepsilon\) improves as a function of disclosure history would formalize this claim; as it stands, it is an empirically motivated conjecture with experimental support (Lichtenstein and Fischhoff, 1980) but no equilibrium characterization.
The interaction between the Disclosure Incentive and existing regulatory mechanisms (mandatory disclosure requirements, professional licensing, fiduciary obligations) is not modeled. The scoring rule's effectiveness depends on outcome verification, which for long-horizon assessments may require years or decades.
The Grossman-Stiglitz paradox has constrained our understanding of how information enters prices since 1980. The paradox is genuine within its assumptions. What this paper shows is that one of those assumptions - that information production is incentivized exclusively through trading profit - is a modeling choice rather than a necessity, and that relaxing it resolves the contradiction.
The CARA-normal general equilibrium model establishes the resolution formally: the disclosure channel sustains information production even in the limiting case where noise vanishes and the trading channel collapses entirely. Uninformed agents receive the same total information but in a qualitatively richer form that strictly dominates the price signal for multi-dimensional decisions, because disclosure distributes precision to constraint dimensions where prices carry nothing. The augmented equilibrium weakly Pareto-dominates the original within the model, and strictly under the multi-dimensional extension.
The judgment market's independence from price efficiency - the assumption on which the resolution rests - is derived rather than imposed. The principal's willingness to pay decomposes into a consensus premium that vanishes as markets become efficient and a constraint premium that remains permanently bounded above zero; an explicit perturbation bound quantifies the decomposition's robustness when the benchmark assumptions are relaxed. Within the participation regime, the physical generation of new constraint variance ensures strictly positive demand even outside the CARA-normal framework. Incentive compatibility is established within the Hurwicz-Myerson framework: proper scoring rules make truth-telling dominant, the commitment mechanism extends this to the strategic environment the mechanism requires, and for the Brier score with appropriate coverage weighting, the calibration threshold vanishes entirely.
The idea is that structured disclosure, scored against outcomes over time, constitutes a distinct return channel for information production - one whose value derives from calibration quality rather than trading timing, and whose independence from price efficiency breaks the reflexive loop that generated the paradox. Whether the mechanism works as described is an empirical question. The predictions in Section 8 specify what to look for.