PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 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.
In a market where disclosure is voluntary and scored against outcomes, the decision not to disclose carries information. An agent who has operated in a domain for five years without building a track record in constraint assessment reveals something about their calibration quality in that domain - either they lack the expertise to produce assessments worth scoring, or they lack confidence that their assessments would score well, or both. This paper develops the economics of silence in structured judgment markets. The analysis is distinct from Milgrom's (1981) unraveling result, where silence about a specific fact implies bad news about that fact, and from Dye's (1985) model, where non-disclosure is explained by the cost of producing disclosable information. In the Disclosure Incentive framework (Beier, 2026b), silence about an entire domain - not withholding a specific assessment but having no track record at all in a constraint area - implies something about the agent's type that the market can read and price. The companion paper on incentive compatibility (Beier, 2026r) establishes that agents above the calibration threshold q* find honest comprehensive disclosure optimal. Agents who remain silent in a domain where disclosure is feasible and incentivized are thereby revealed to fall below the threshold - not with certainty, but with a probability that principals can estimate from the population distribution of calibration quality and the disclosure participation rate. Silence is informative precisely because disclosure is incentivized: when good agents have reason to speak, the decision to remain quiet is itself a signal.
Keywords: silence, non-disclosure, unraveling, signaling, track records, calibration, adverse selection, judgment markets, voluntary disclosure
JEL Codes: D82, D83, G14, D86
Milgrom (1981) and Grossman (1981) independently established the unraveling result: when disclosure is costless and verifiable, agents with favorable private information disclose voluntarily, because silence is interpreted as concealment of bad news. The logic cascades - once the best type discloses, the second-best type's silence is interpreted as worse news than before, so the second-best type discloses too, and the process continues until only the worst type remains silent. In the limit, full disclosure obtains even without a mandate.
In Verrecchia's (1983) disclosure model, unraveling is incomplete when disclosure is costly: agents disclose only when the benefit of revealing favorable information exceeds the cost of producing and publishing the disclosure. Dye (1985) extended this to settings where agents are uncertain about whether they possess disclosable information, generating equilibria in which non-disclosure is ambiguous - the market cannot tell whether the agent is withholding bad news or simply lacks information to disclose.
The present paper analyzes silence in a setting that differs from all three predecessors. In the Disclosure Incentive framework, the relevant signal is not a specific fact about a specific asset (Milgrom's setting) or a specific piece of proprietary information (Verrecchia's and Dye's setting). It is the agent's calibration quality across a domain of constraint assessment - a property of the agent's type rather than a property of any specific information the agent holds. The disclosure is not a report about a company's earnings or a bond's credit quality. It is a structured judgment about a constraint environment, scored against outcomes over time, accumulated into a track record that measures the agent's calibration.
In this setting, silence has a different informational structure. An agent who does not disclose in a specific domain has no track record in that domain, and the absence of a track record - in a market where building one is incentivized and where agents above the calibration threshold find it optimal to disclose - reveals something about where the agent falls in the calibration distribution.
Section 2 develops the formal structure of silence as signal. Section 3 distinguishes silence in judgment markets from silence in the unraveling literature. Section 4 analyzes how principals read silence. Section 5 examines the dynamics of silence as disclosure norms evolve. Section 6 states limitations. Section 7 concludes.
Consider a population of agents, each of whom may or may not possess expertise in a specific constraint domain (aquifer dynamics, atmospheric carbon, grid resilience). The companion paper (Beier, 2026r, Proposition 7) establishes that agents with calibration quality q above a threshold q* find honest comprehensive disclosure optimal under the coverage-adjusted scoring rule. Agents below q* may find strategic selection profitable, and agents with very low calibration quality prefer silence - no disclosure at all - because disclosing would produce a track record that reveals poor calibration and reduces rather than enhances their professional value.
Let lambda be the fraction of agents who possess expertise in the domain (who have the capacity to produce assessments that could be scored), and let phi(q) be the distribution of calibration quality among those agents. The fraction of agents who disclose is:
F_D = lambda * integral from q* to 1 of phi(q) dq
This is the disclosure participation rate: the fraction of domain-relevant agents whose calibration quality exceeds the threshold. The remaining agents - those below the threshold and those without domain expertise - are silent.
An observer (a principal hiring for a role that requires constraint assessment expertise) observes an agent's track record. If the agent has a track record in the domain, the principal observes its quality directly. If the agent has no track record - silence - the principal must infer the agent's type from the silence.
The inference is Bayesian. The prior probability that a silent agent is below the calibration threshold (given that they have domain expertise) is:
P(q < q* | silence, expertise) = integral from 0 to q* of phi(q) dq / integral from 0 to q* of phi(q) dq = 1
conditional on the agent having expertise. If the agent has expertise and is above q*, they would disclose (Proposition 7). Therefore silence from a domain expert is a certain signal of below-threshold calibration.
In practice, the inference is softened by uncertainty about whether the agent has domain expertise at all. A silent agent may be below the threshold, or may simply lack expertise in that specific domain - a fund manager with excellent calibration on technology stocks may have no track record in aquifer assessment because aquifers are outside their domain, not because their aquifer calibration is poor. The principal's inference from silence depends on how likely the agent is to have domain expertise, which depends on the agent's background, the domain's breadth, and the disclosure participation rate in the domain.
The information content of silence is therefore a function of the disclosure participation rate. When F_D is high (most domain-relevant agents disclose), silence strongly implies below-threshold calibration, because the prior probability that a silent agent lacks domain expertise is low relative to the probability that they have expertise but poor calibration. When F_D is low (few agents disclose), silence is more ambiguous, because most silent agents may simply not yet have joined the disclosure ecosystem for reasons unrelated to their calibration quality.
The unraveling result (Milgrom, 1981; Grossman, 1981) operates through a specific mechanism: an agent who possesses favorable information and does not disclose it is inferred to possess unfavorable information, because favorable information would have been disclosed voluntarily. The inference is about the content of information the agent holds - the specific fact being withheld.
Silence in the judgment market operates through a different mechanism. The inference is not about a specific fact the agent is withholding but about the agent's type - their calibration quality in a domain. An agent who has no track record in aquifer assessment is not withholding a specific aquifer forecast; they are revealing, through the absence of any scored history, that they have not subjected their aquifer judgment to the verification process that the disclosure framework provides.
The distinction has practical consequences. In the unraveling framework, silence is always bad news - the agent is concealing something unfavorable about a specific fact. In the judgment market framework, silence is informative but not necessarily unfavorable in the same way. A silent agent may have excellent general judgment that has never been applied to the specific domain in question. Their silence reveals nothing about their underlying ability - only that they have not demonstrated calibration in that specific area. Principals can distinguish between "no track record because below threshold" and "no track record because outside domain" by observing the agent's track record in adjacent domains, their professional background, and the recency of the domain's disclosure ecosystem.
Dye's (1985) model introduces uncertainty about information possession: the market cannot tell whether a non-discloser is withholding information or lacks information to disclose. The judgment market has an analogous structure - the principal cannot tell whether a silent agent is below threshold or outside the domain - but the resolution mechanism differs. In Dye's model, the ambiguity persists in equilibrium because there is no mechanism to resolve it. In the judgment market, the ambiguity resolves over time: an agent who enters the disclosure ecosystem and accumulates a track record eliminates the ambiguity, and an agent who remains silent over an extended period during which comparable agents build track records generates increasingly strong inference about their type.
Verrecchia's (1983) disclosure cost model explains silence through the cost of producing disclosable information. In the Disclosure Incentive framework, the relevant cost is the information acquisition cost c (the same cost that appears in the Grossman-Stiglitz model). Agents who find c too high relative to the expected track record return remain silent, and their silence reflects the unfavorable cost-benefit calculation rather than hidden unfavorable information. The Disclosure Incentive is designed so that the track record return exceeds c for agents above q* (the judgment market equilibrium condition in Beier, 2026b, Section 4.4), which means that silence from agents whose cost is at or below c is informative about calibration quality specifically.
A principal evaluating a candidate for a role that requires constraint assessment expertise faces a screening problem with three observable categories: agents with strong track records (demonstrably above q*), agents with weak track records (demonstrably below q* but at least partially revealed through their scoring history), and silent agents (no track record in the relevant domain).
The strong-track-record agents are direct: the scoring data speaks for itself. The weak-track-record agents are also informative, though the information is unfavorable: they attempted to build a track record and the scores reveal poor calibration. The silent agents are the ambiguous category.
Principals resolve the ambiguity through several channels. The first is adjacent-domain inference: an agent with a strong track record in energy-sector constraint assessment and no track record in water constraint assessment is inferred to have domain expertise in energy but not water - the silence in water is not evidence of poor calibration but of non-participation in a different domain. The second is temporal inference: an agent who has been active in a market for ten years while comparable agents built water-domain track records over the same period generates stronger below-threshold inference than an agent who entered the market last year. The third is contextual inference: an agent who previously worked at a firm known for water-sector analysis and has no water-domain track record generates different inference from an agent who has never worked in a water-adjacent role.
The market's ability to read silence does not, however, generate a coercive pressure to disclose. The coverage-adjusted scoring mechanism (Beier, 2026r) is designed so that agents below the calibration threshold q* are better off remaining silent than producing low-quality disclosures. Silent agents forgo track record capital - they are unpriced in the judgment market - but they are not penalized. Principals allocate their judgment budget to agents with demonstrated track records, and silent agents simply do not compete for that budget. The baseline payoff for non-disclosure remains what it is in the standard economy: ordinary trading returns or base salary, with zero track record capital. Silence is informative to the principal without being coercive to the agent.
The information content of silence evolves as the disclosure ecosystem matures.
In the early stage, when the disclosure framework is newly introduced and few agents participate, silence is weakly informative. Most agents are silent because the ecosystem is new, not because they assessed their calibration and decided against disclosure. Principals cannot distinguish early non-adopters from below-threshold agents, and the inference from silence is diluted.
In the middle stage, as a critical mass of agents builds track records in major constraint domains, silence becomes increasingly informative. The agents who joined early and built strong records have separated themselves from the silent population. The remaining silent agents are a mixture of late adopters, agents outside the domain, and below-threshold agents - and the mixture shifts toward below-threshold as the domain-relevant agents with above-threshold calibration join the disclosed pool. The principal's Bayesian update from silence grows stronger as the disclosure participation rate rises.
In the mature stage, when disclosure in a domain is standard practice among competent agents, silence is strongly informative. An agent who has been active in the relevant market for years without building a track record in a domain where most peers have done so generates a powerful inference. The silence is not proof of below-threshold calibration - there may be idiosyncratic reasons for non-participation - but the evidentiary weight is substantial.
This dynamic creates a tipping pattern. As more agents disclose, the inference from silence grows stronger, which increases the pressure on silent agents to disclose, which raises the disclosure participation rate further, which strengthens the inference from the remaining silence. The tipping logic is the same as in the unraveling literature but operates at the level of domain-wide participation rather than at the level of specific-fact disclosure.
The paper models silence as arising from below-threshold calibration quality, but other reasons for silence exist that the model does not capture. An agent may be silent because of legal constraints (contractual non-disclosure obligations, regulatory restrictions on forward-looking statements), strategic considerations unrelated to calibration (choosing to build a track record privately before going public), or simple unfamiliarity with the disclosure framework. The Bayesian inference from silence is only as good as the model of why agents are silent, and the model's simplification - that silence among domain experts implies below-threshold calibration - overstates the inference's strength.
The adjacent-domain and temporal inference mechanisms described in Section 4 are informal. A formal model of multi-domain inference - how a principal combines track record data across domains to estimate an agent's calibration quality in a domain where they are silent - would require specifying the correlation structure of calibration quality across domains, which is an empirical question the paper does not address.
The tipping dynamics in Section 5 are qualitative. The speed of the transition from weakly informative silence to strongly informative silence depends on the disclosure adoption rate, which depends on the judgment market return, the information acquisition cost, and the institutional infrastructure supporting the disclosure framework. A quantitative model of the tipping dynamics would require calibrating these parameters, which the paper does not attempt.
The paper does not address the welfare implications of silence being informative. If below-threshold agents are penalized by the market's reading of their silence, and if their below-threshold calibration reflects lack of training or experience rather than inherent inability, then the information content of silence has distributive consequences that may warrant institutional response - mentorship programs, calibration training, or protected entry periods during which silence does not generate unfavorable inference.
In a market where disclosure is voluntary, incentivized, and scored, the decision not to disclose is itself a disclosure. The information content of silence depends on the disclosure participation rate: when most domain-relevant agents disclose, the remaining silence strongly implies below-threshold calibration. The inference is distinct from the unraveling literature's logic - it is about the agent's type rather than about a specific withheld fact - and it strengthens over time as the disclosure ecosystem matures and the population of silent agents shifts toward those who assessed the cost-benefit calculation and found disclosure unprofitable.
The practical consequence is that the Disclosure Incentive generates information even from agents who do not participate in it. The silence of non-disclosers is informative to principals, and this informativeness enriches the principal's decision environment without coercing the silent agent. The judgment market extracts signal from every agent in the population - from disclosers through their scored track records, and from silent agents through the inference their silence permits. Those who speak are scored. Those who are silent are read. But the reading does not compel speech - it simply allows principals to allocate judgment budgets efficiently.
This paper presents results from the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics (Beier, forthcoming 2026). The silence economics developed here extends the voluntary disclosure analysis in the companion papers (Beier, 2026b, 2026r) by showing that the Disclosure Incentive generates information even from agents who do not participate in it. The author invites engagement from the voluntary disclosure, signaling, and labor economics communities.