Prior Disclosure Games

• Prior Disclosure Games are asymmetric information models where an informed sender strategically reveals signals to shape receiver beliefs and actions.
• Key methodologies include static and dynamic formulations, verifiable and large-evidence disclosures, and equilibrium analyses ensuring consistency and optimal outcomes.
• These models have practical applications in Bayesian persuasion, dynamic information design, and algorithmic game search, impacting policy, welfare, and online learning.

Searching arXiv for recent and foundational papers on prior disclosure and related disclosure-game frameworks. arxiv_search(query="prior disclosure games information disclosure sender receiver Bayesian persuasion disclosure game", max_results=10) arxiv_search(query="\"prior disclosure\" disclosure games Bayesian persuasion information disclosure", max_results=10) arxiv_search(query="Commitment games conditional information disclosure disclosure games large evidence spaces From Design to Disclosure Withholding Verifiable Information", max_results=10) Prior disclosure games study strategic environments in which an informed party decides whether, how, or when to reveal information before another party acts. The disclosed object may be a closed-set message containing the true type, a truncation of hard evidence, a publicly announced experiment, a public signal, or even a prior itself. The literature spans static voluntary disclosure, commitment and conditional disclosure, dynamic information design on Markovian systems, repeated zero-sum environments with information spillover, and algorithmic variants in imperfect-information search. This suggests that “prior disclosure games” is best understood as a family of asymmetric-information models organized around disclosure technologies, belief updating, and equilibrium selection rather than as a single canonical game (Ali et al., 2024, Jiang, 2019, Tang et al., 2024, Vangala, 21 Aug 2025).

1. Core formalization and disclosure technologies

A common static formulation begins with a sender of type $\theta\in\Theta$, drawn from a prior $F$, who can send any message $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$. In this formulation, both $\Theta$ (“no-reveal”) and $\{\theta\}$ (“full-reveal”) are feasible. After observing a message $m$, the receiver updates beliefs by Bayes’ rule whenever possible, uses off-path beliefs concentrated on a “worst-case type,” and chooses an action in $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$; a Perfect Bayesian Equilibrium requires sender optimality, receiver optimality, consistency of beliefs, and the specified off-path beliefs (Ali et al., 2024).

A closely related verifiable-disclosure model specializes the state space to $\Theta=[0,1]$, restricts the receiver to a finite ordered action set $A=\{a_0,\dots,a_{n-1}\}$, and again permits only messages that are closed sets containing the true state: $$M(\omega)=\{m\subseteq[0,1]: m \text{ closed}, \omega\in m\}$$. Here the sender’s payoff is state independent and depends only on the induced action, while the receiver’s best response is pinned down by the posterior mean relative to cut-points $F$0 (Zhang, 2022).

In large-evidence disclosure games, the message technology is encoded by a preorder $F$1 on the evidence space $F$2: if $F$3, a sender with evidence $F$4 can imitate evidence $F$5. In the left-censoring case,

$F$6

for some truncation length $F$7, so disclosure consists of revealing a left truncation of a signal sequence (Jiang, 2019).

Dynamic formulations replace one-shot messages with publicly observable experiments. In one such model, the principal publicly announces an experiment kernel $F$8, nature draws a message according to $F$9, both players observe the realized message, the receiver acts, and the state evolves through a Markov kernel $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$0. Under a truthful disclosure rule, the chosen experiment and its realized outcome must be announced honestly (Tang et al., 2024).

Model family	Disclosure object	Representative equilibrium notion
Voluntary disclosure with verifiable sets	Closed message set containing the true type	Perfect Bayesian Equilibrium
Large-evidence disclosure	Evidence report $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$1 under a preorder	Truth-leaning Perfect Bayesian Equilibrium
Dynamic truthful disclosure	Public experiment and realized message	Canonical Belief-Based Perfect Bayesian Equilibrium

These variants share a common structure: disclosure is constrained by a technology, receiver behavior is belief-based, and equilibrium analysis asks which posteriors, actions, and payoffs can be sustained under those constraints.

2. Static equilibrium structure and voluntary withholding

In the large-evidence framework, the focal refinement is the truth-leaning Perfect Bayesian Equilibrium. The receiver uses the unique best response $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$2, the sender’s support at evidence $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$3 is contained in $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$4, and if truthful disclosure itself is optimal then the sender discloses truthfully. Off-path messages are interpreted at face value, so an unused message $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$5 is believed to reflect truthful evidence $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$6 (Jiang, 2019).

A central characterization uses the sender’s face-value payoff

$$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$7

and equilibrium value function $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$8. The sender’s equilibrium value function is unique; in the binary-signal left-censoring example it is summarized by the maximal-difference truncation $$\mathcal M(\theta):=\{m\subseteq\Theta \text{ closed}: \theta\in m\}$$9. The equilibrium disclosure rule then has a distinctive feature: seemingly sub-optimal truncations are disclosed, and the disclosure contains the longest truncation that yields the maximal difference between the number of good and bad signals. The receiver treats all on-path reports with the same $\Theta$0 identically, and posterior beliefs after on-path reports depend only on $\Theta$1 rather than the full signal pattern (Jiang, 2019).

This corrects a common simplification according to which equilibrium disclosure should consist only of the sender’s most persuasive feasible report. In the large-evidence model, “sub-optimal” disclosures with extra bad signals can appear on path because they deter profitable deviations by slightly better types; if only the maximal-difference truncation were used, some types would deviate to truth-telling at face value (Jiang, 2019).

In state-independent sender-payoff models with verifiable evidence, the sender’s preferred equilibrium is linked to an information-design benchmark. Any equilibrium induces a distribution $\Theta$2 over posterior means and sender payoff $\Theta$3, while the commitment problem maximizes $\Theta$4 subject to $\Theta$5. Extreme points of this design problem are bi-pooling. A bi-pooling solution is an equilibrium outcome of the verifiable disclosure game if and only if its canonical representation $\Theta$6 satisfies $\Theta$7. The sender’s equilibrium payoff set is exactly the interval $\Theta$8, where $\Theta$9 is the unraveling payoff and $\{\theta\}$0 is the preferred-equilibrium payoff, and every payoff in that interval is achieved in a pure-strategy, obedient-recommendation equilibrium whose on-path messages are unions of at most $\{\theta\}$1 intervals (Zhang, 2022).

3. Commitment, conditional disclosure, and information-design equivalence

Commitment changes the feasible disclosure logic. In commitment games with conditional information disclosure, agents have a conditional commitment device that can be used to conditionally disclose private information. The framework proves a folk theorem providing sufficient conditions for ex post efficiency, thereby modeling ideal cooperation between agents without a third-party mediator. It also shows that conditional disclosure can achieve full cooperation in cases where unconditional disclosure cannot, and it develops an implementation yielding program $\{\theta\}$2-Bayesian Nash equilibria corresponding to the Bayesian Nash equilibria of the commitment games (DiGiovanni et al., 2022).

A different line of work establishes an explicit equivalence between voluntary disclosure and information design. In the disclosure game of Ali–Kleiner–Zhang, the sender privately observes $\{\theta\}$3 and can send any closed message set containing $\{\theta\}$4; the set of ex ante equilibrium payoffs of the disclosure game is denoted $\{\theta\}$5, while $\{\theta\}$6 denotes the payoff pairs achievable through Bayes-plausible segmentations. Under the assumptions NoRents, WorstCase, and Continuity, the main theorem states $\{\theta\}$7, so every design-achievable payoff can be approximately supported by an equilibrium of the disclosure game. The proof combines a finite-partitional support lemma with an approximation lemma for arbitrary Bayes-plausible segmentations. Applications include monopoly pricing, veto bargaining, and insurance contracting, and in monopoly pricing a truth-leaning refinement selects the efficient frontier $\{\theta\}$8 (Ali et al., 2024).

The verifiable-withholding framework reaches a closely related boundary result from the opposite direction. It identifies conditions under which an information-design outcome is an equilibrium outcome of the verifiable disclosure game and gives simple sufficient conditions under which the sender does not benefit from commitment power. In that sense, the literature isolates two complementary questions: when commitment strictly enlarges what can be sustained, and when voluntary disclosure already attains the commitment benchmark (Zhang, 2022).

A plausible implication is that the distinction between “disclosure” and “design” becomes substantive only when disclosure technologies, off-path beliefs, or incentive constraints block the sender from implementing Bayes-plausible segmentations directly. Where those frictions are weak, disclosure becomes a microfoundation of design rather than a separate object.

4. Dynamic, repeated, and belief-state formulations

Dynamic information disclosure games place disclosure inside a controlled stochastic process. In the Markovian principal–receiver model, the public history of experiments and messages induces a common posterior belief $\{\theta\}$9, and Canonical Belief-Based strategies use this belief as a sufficient statistic. The belief update has the form

$m$0

after the message and

$m$1

after the action and state transition. A CBB-PBE is characterized by coupled dynamic programs: the receiver solves a belief-MDP, while the principal’s value is obtained by a persuasion step $m$2. The paper also provides a backward inductive procedure for solving an equilibrium in CBB strategies and interprets the result as information compression, because both players can use a compressed version of their information rather than full histories (Tang et al., 2024).

Repeated games introduce strategic spillovers across receivers or across simultaneous interactions. In the three-player repeated zero-sum model with one informed player and two uninformed opponents, the informed player may benefit from revealing information in one component game but thereby leak information into the other. Writing $m$3, the equilibrium payoff set for the informed player lies between

$m$4

and $m$5. If each component game admits a non-revealing equilibrium, then every payoff in that interval is attained as a uniform-equilibrium ex ante payoff. A separate result shows that any equilibrium attaining the upper endpoint must make the processes

$m$6

and

$m$7

martingales; a small example shows that the top of the interval may be unattainable, so spillover can be severe (Pahl, 2021).

Public-signalling models in routing use disclosure not only to shape action but to infer prior beliefs. In the routing framework, a traffic information system commits to a public signalling scheme $m$8, users update to posteriors after observing the signal, and then play a Wardrop equilibrium. Under mild conditions, a signalling scheme that allows exact inference of the prior exists, an iterative algorithm finds such a scheme in a finite number of steps, the resulting schemes are robust to small perturbations, and with multiple priors the mixture weights are identifiable when the relevant matrix has full rank (Verbree et al., 2021).

Together these dynamic and repeated models shift the focus from one-shot persuasion to the evolution of beliefs, continuation values, and cross-interaction spillovers.

5. Transparency, competition, and welfare effects

The welfare implications of disclosure are not monotone across models. In the comparison between overt persuasion and covert signaling, the price of transparency is defined as $m$9, where $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$0 is the sender’s equilibrium payoff under covert signaling and $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$1 is the sender’s payoff under overt persuasion. The paper establishes the tight bound $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$2, so transparent commitment is always weakly better for the provider in that framework. The upper bound is attained for strictly Bayesian-posterior competitive games, including zero-sum games, while in continuous games the lower bound is tight in the sense that the ratio can be arbitrarily close to $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$3 (Li et al., 2023).

Public disclosure may also crowd out privately acquired information. In the linear-quadratic-Gaussian game with endogenous private learning, a policymaker chooses public precision $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$4, agents choose private precisions, and the equilibrium private precision $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$5 satisfies $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$6. The robust disclosure rule is particularly sharp: if $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$7, then the worst-case welfare is strictly increasing and the robust rule is $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$8; if $$\arg\max_a \int_\Theta u_R(a,\theta)\,d\mu(\theta\mid m)$$9, then the robust rule is $\Theta=[0,1]$0. The optimal rule outside the robust benchmark depends on the elasticity of marginal information cost and interpolates between the linear-cost and exogenous-private-information cases (Ui, 2022).

Competition among informed senders changes the logic again. In the two-stage sharing-systems model, sellers first choose disclosure levels and then prices. Full disclosure by all sellers and non-disclosure by all sellers both intensify price competition; the former all-disclosure case is never an equilibrium, while the latter non-disclosure case can be an equilibrium under which all sellers get zero profit. Capacity limits and buyers’ estimation biases mitigate competition and encourage disclosure (Ding et al., 2023).

Sequential contests produce yet another comparative static. When a contest designer chooses when to publicly disclose past efforts, information about earlier efforts increases total effort. In the Tullock-style model, full transparency maximizes total effort and no transparency minimizes it; in the more general public-disclosure architecture, many optimal contests collapse to one of three basic structures: simultaneous, first-mover, or fully sequential (Hinnosaar, 2018, Hinnosaar, 2019).

A recurring misconception is that more disclosure always either helps the sender or improves efficiency. The literature rejects that uniform conclusion. Depending on the model, disclosure can generate ex post efficiency, destroy margins through price competition, crowd out private information acquisition, or raise aggregate effort in contests while lowering welfare.

6. Learning, algorithmic extensions, and conceptual broadening

Recent work treats disclosure as an online learning problem. In the repeated persuasion model with Gaussian prior and quadratic costs, the sender faces an adversarial sequence of receiver types and chooses signalling policies online. The results establish $\Theta=[0,1]$1 regret with full information feedback, extend the same rate to a general convex utility through a new parametrization, obtain $\Theta=[0,1]$2 regret when the sender’s objective includes a strongly convex information-penalty term, and give a sublinear regret bound under partial-information feedback, with an $\Theta=[0,1]$3-type one-point-estimator construction in the detailed development (Velicheti et al., 2024).

In imperfect-information game search, the disclosure choice can be internal to the agent’s own belief construction. The mixture-of-public-and-private-distributions framework proposes a new belief distribution that depends on the amount of private and public information desired, studies its use in PIMC and IS-MCTS, and empirically demonstrates an increase in performance; the results also indicate that the new distribution should be used according to the position in the game (Arjonilla et al., 2024).

A conceptually distinct use of prior disclosure games appears in objective Bayesian decision theory. In Bayesimax theory, nature chooses a prior, data are observed, and the Bayesian agent reports a prior under a strictly proper scoring rule. Under the stated injectivity condition, the unique Bayes rule is truthful prior revelation, $\Theta=[0,1]$4, and the minimum Bayes risk equals the conditional generalized entropy of the parameter given the data. For the logarithmic score, this becomes conditional Shannon entropy, $\Theta=[0,1]$5, so Bayesimax priors maximize conditional entropy and can be interpreted as priors selected by minimizing total information (Vangala, 21 Aug 2025).

These algorithmic and decision-theoretic extensions broaden the scope of the field. A plausible implication is that prior disclosure games now function as a unifying language for strategic revelation in economics, online learning, imperfect-information planning, and Bayesian methodology, with the common thread being the endogenous choice of what information becomes common, public, or strategically usable.