Bayes Correlated Equilibrium¶

In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information.

Version: v1 · 2026-09-28 · History
Domain-specific #: 8139
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomains: Game Theory, Information Design → Economics & Finance

Core Idea¶

Bayes Correlated Equilibrium is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information.

In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. It is both a generalization of the correlated equilibrium perfect-information solution concept to bayesian games, and also a broader solution concept than the usual Bayesian Nash equilibrium thereof. Additionally, it can be seen as a generalized multi-player solution of the Bayesian persuasion information design problem.

Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have. It was first proposed by Dirk Bergemann and Stephen Morris. A game is defined as a tuple G = \langle (A_i, u_i){i \in I}, \Theta, \psi \rangle , where A_i is the set of possible actions (with A = \prod (\Theta) is a full support common prior over the states of the world.} A_i ) and u_i : A\times \Theta \rightarrow \mathbb{R} is the utility function for each player, and \psi \in \Delta_{++

For Bayes Correlated Equilibrium, the abstraction is narrower than the article's general subject matter: a positive case must preserve In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics, logic, and statistics, which is why this identity is domain-specific rather than prime.

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Follow the Helper's Whisper

Imagine a game where each player knows some secret things and not others. A helper who knows the secret situation whispers a suggested move to each player. If, no matter what each player secretly knows, nobody wants to ignore the helper's whisper, the plan is a Bayes correlated equilibrium.

Advice Nobody Wants to Ignore

In game theory, players choose actions and get rewards, and sometimes they don't know everything, like the true state of the world. A Bayes correlated equilibrium is a way to describe stable plans in such games. Think of a helper who can send each player a private suggestion, and the suggestions can be linked to each other and to the hidden state. The plan counts as an equilibrium if, for every possible piece of private knowledge a player might have, following the suggestion is the player's best choice. This idea includes more possible outcomes than older ideas where players act without such linked suggestions. It was introduced by Dirk Bergemann and Stephen Morris.

Obedient Recommendations Under Uncertainty

A Bayes correlated equilibrium is a solution concept for static games of incomplete information, where players don't know the state of the world. A game is described by each player's actions and payoffs, which depend on the chosen actions and the state, plus a common prior over states. A Bayes correlated equilibrium is a way of recommending actions to players, possibly depending on the state and on their private information, so that no player wants to deviate from their recommendation for any type they might have. It extends the ordinary correlated equilibrium to games with uncertainty, and it includes more outcomes than the usual Bayesian Nash equilibrium. It can also be seen as a multi-player version of Bayesian persuasion, where a designer chooses what information to give. It was first proposed by Dirk Bergemann and Stephen Morris.

A Bayes correlated equilibrium is a solution concept for static games of incomplete information, introduced by Dirk Bergemann and Stephen Morris. A basic game is a tuple G = ⟨(A_i, u_i)_{i∈I}, Θ, ψ⟩, where A_i is player i's action set with A = ∏A_i, u_i: A × Θ → ℝ is i's utility, and ψ is a full-support common prior over states of the world Θ. The equilibrium object is a decision rule that assigns a joint distribution over action profiles to each state (and, with an information structure, to players' types), interpreted as a mediator's private recommendations. It is an equilibrium when obedience holds: for every player and every type or recommendation they may receive, following the recommended action is optimal given their posterior beliefs. The concept generalizes correlated equilibrium from complete-information games to Bayesian games and is weaker, hence broader, than Bayesian Nash equilibrium. It also serves as a multi-player generalization of the Bayesian persuasion information-design problem, since the set of Bayes correlated equilibria characterizes outcomes achievable by some information structure.

Structural Signature¶

Sig role-phrases:

Defining carrier — That is, every player obtains a higher expected payoff by following the recommendation from the decision rule than by deviating to any other possible action.
Constitutive relation — Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have.
Operating condition — It was first proposed by Dirk Bergemann and Stephen Morris.
Recognition evidence — Let I be a set of players, and \Theta a set of possible states of the world.
Admissible variation — By joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game.
Characteristic consequence — A decision rule for the incomplete information game \Gamma = (G, S) is a mapping \sigma: T \times \Theta \rightarrow \Delta (A) .
Failure boundary — A Bayes correlated equilibrium (BCE) is defined to be a decision rule \sigma which is obedient: that is, one where no player has an incentive to unilaterally deviate from the recommended joint strategy, for any possible type they may be.

What It Is Not¶

Not the whole field of mathematics, logic, and statistics. The node requires the specific identity stated by In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information.
Not an over-broad reading. Let I be a set of players, and \Theta a set of possible states of the world.
Not an over-broad reading. By joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game.
Not an over-broad reading. A decision rule for the incomplete information game \Gamma = (G, S) is a mapping \sigma: T \times \Theta \rightarrow \Delta (A) .
Not automatically Bayesian Nash Equilibrium. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Bayes Correlated Equilibrium applies literally inside mathematics, logic, and statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Bayesian persuasion. More specifically, let v : A \times \Theta \rightarrow \mathbb R be the information designer's objective function.
Preliminaries. A game is defined as a tuple G = \langle (A_i, u_i){i \in I}, \Theta, \psi \rangle , where A_i is the set of possible actions (with A = \prod (\Theta) is a full support common prior over the states of the world.} A_i ) and u_i : A\times \Theta \rightarrow \mathbb{R} is the utility function for each player, and \psi \in \Delta_{++
Preliminaries. An information structure is defined as a tuple S = \langle (T_i){i \in I}, \pi \rangle , where T_i is a set of possible signals (or types) each player can receive (with T = \prod T_i ), and \pi : \Theta \rightarrow \Delta (T) is a signal distribution function, informing the probability \pi (t | \theta) of observing the joint signal t \in T when the state of the world is \theta \in \Theta .
Documented setting. Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have.
Preliminaries. Let I be a set of players, and \Theta a set of possible states of the world.
Preliminaries. By joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game.

Outside mathematics, logic, and statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Theory or should be marked as analogy.

Clarity¶

A clear use of Bayes Correlated Equilibrium names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. The strongest recognition evidence in the frozen account is: Let I be a set of players, and \Theta a set of possible states of the world. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Let I be a set of players, and \Theta a set of possible states of the world. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Bayes Correlated Equilibrium compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have.—and the practical consequence—a decision rule for the incomplete information game \Gamma = (G, S) is a mapping \sigma: T \times \Theta \rightarrow \Delta (A) . This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information.
Check operation and conditions. It was first proposed by Dirk Bergemann and Stephen Morris.
Demand recognition evidence. Let I be a set of players, and \Theta a set of possible states of the world.
Test variation. Change an implementation or setting while preserving by joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Theory.

Knowledge Transfer¶

Within the home domain. Knowledge about Bayes Correlated Equilibrium transfers literally when a new case preserves the same carrier type, relation, and recognition test. More specifically, let v : A \times \Theta \rightarrow \mathbb R be the information designer's objective function. A game is defined as a tuple G = \langle (A_i, u_i){i \in I}, \Theta, \psi \rangle , where A_i is the set of possible actions (with A = \prod (\Theta) is a full support common prior over the states of the world.} A_i ) and u_i : A\times \Theta \rightarrow \mathbb{R} is the utility function for each player, and \psi \in \Delta_{++

Beyond the home domain. No canonical parent is asserted for Bayes Correlated Equilibrium. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

If there is no uncertainty about the state of the world (e.g., if \Theta is a singleton), then the definition collapses to Aumann's correlated equilibrium solution. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information; recognition evidence → Let I be a set of players, and \Theta a set of possible states of the world

Applied / In Practice¶

In this case, \sigma \in \Delta (A) is a BCE if, for every i \in I , we have. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Correlated equilibrium; invariant → In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information; boundary → the case exits the class when let I be a set of players, and \Theta a set of possible states of the world

Structural Tensions¶

T1 — Stable identity versus admissible variation. Let I be a set of players, and \Theta a set of possible states of the world. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. By joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. A decision rule for the incomplete information game \Gamma = (G, S) is a mapping \sigma: T \times \Theta \rightarrow \Delta (A) . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. A Bayes correlated equilibrium (BCE) is defined to be a decision rule \sigma which is obedient: that is, one where no player has an incentive to unilaterally deviate from the recommended joint strategy, for any possible type they may be. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. That is, every player obtains a higher expected payoff by following the recommendation from the decision rule than by deviating to any other possible action. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Bayes Correlated Equilibrium literally, co-instantiate Theory, or only resemble it?

T6 — Autonomy versus reduction. Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Bayes Correlated Equilibrium distinguish that the broader parent Theory leaves together?

Structural–Framed Character¶

Bayes Correlated Equilibrium is structural-leaning. Its structural side is the repeatable organization summarized by In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. Its framed side is the mathematics, logic, and statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: It was first proposed by Dirk Bergemann and Stephen Morris. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Theory. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: That is, every player obtains a higher expected payoff by following the recommendation from the decision rule than by deviating to any other possible action. Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in such a way that no player has an incentive to deviate for every possible type they may have. It further constrains recognition and variation through: It was first proposed by Dirk Bergemann and Stephen Morris. Let I be a set of players, and \Theta a set of possible states of the world.

What is domain-bound. mathematics, logic, and statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Bayes Correlated Equilibrium literal. Its documented scope includes the condition that More specifically, let v : A \times \Theta \rightarrow \mathbb R be the information designer's objective function. Another bounded application condition is that A game is defined as a tuple G = \langle (Ai, ui){i \in I}, \Theta, \psi \rangle , where Ai is the set of possible actions (with A = \prod{i \in I} Ai ) and ui : A\times \Theta \rightarrow \mathbb{R} is the utility function for each player, and \psi \in \Delta{++} (\Theta) is a full support common prior over the states of the world. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—By joining those two definitions, one can define \Gamma = (G, S) as an incomplete information game.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Correlated equilibrium.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Bayes Correlated Equilibrium. The reviewed identity is: In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Bayes Correlated Equilibrium Domain-specific

Parents (1) — more general patterns this builds on

Bayes Correlated Equilibrium is a kind of Correlated equilibrium Domain-specific

Bayes Correlated Equilibrium is a strict kind of Correlated equilibrium: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.

Hierarchy path (1) — routes to 1 parentless root

Bayes Correlated Equilibrium → Correlated equilibrium → Equilibrium → Fixed Point

Neighborhood in Abstraction Space¶

Bayes Correlated Equilibrium sits in a crowded region of the domain-specific corpus (23^rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Political & Strategic Game Models (11 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Theory. The parent omits the specialist differentia. Tell: Can the case establish In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information?
Bayesian Nash Equilibrium. The solution concept for games of incomplete information: recast not knowing an opponent's payoffs as Nature drawing each player's private type from a common prior, then solve for a fixed point of type-conditional strategy functions where every type's action is a best response in expectation and the supporting beliefs are Bayes-consistent. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Mixed Strategy Equilibrium. Solve a game with no stable deterministic play by having each player randomize over their actions in exactly the proportions that leave every opponent indifferent, so no one can profitably deviate. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Subgame Perfect Equilibrium. Refine the Nash equilibria of a sequential game by keeping only strategy profiles that prescribe a best response in every subgame, discarding outcomes propped up by threats a player would never actually carry out. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Bayes Correlated Equilibrium remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics, logic, and statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Theory?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Bayes_correlated_equilibrium (revision 1368685853).
Preserved source candidate: https://econtheory.org/ojs/index.php/te/article/view/20160487
Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0899825698907060
Preserved source candidate: https://www.jstor.org/stable/1911154
Preserved source candidate: https://www.jstor.org/stable/26673203
Preserved source candidate: https://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.