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.

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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.

Scope of Application¶

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 (Ai, ui){i \in I}, \Theta, \psi \rangle , where Ai is the set of possible actions (with A = \prod{i \in I}.
Preliminaries. An information structure is defined as a tuple S = \langle (Ti){i \in I}, \pi \rangle , where Ti is a set of possible signals (or types) each player can receive (with.
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.

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.

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.

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.

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 (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.

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