Coordination Mechanisms with Partially Specified Probabilities

Abstract: We study which outcomes are implementable by disclosing coarse statistics of a data-generating process rather than its full distribution. Players observe data whose joint distribution is only partially known: they know the expectations of finitely many random variables and form beliefs by maximum-entropy inference. We obtain two characterizations. When message spaces are unrestricted, implementable outcomes coincide with jointly coherent outcomes, expanding the set of correlated equilibria. With canonical mechanisms, implementability reduces to a single cross-entropy condition: the target outcome must lie on the cross-entropy level set of some correlated equilibrium that passes through that equilibrium itself. Examples and several classes of games illustrate the reach of the framework.

• The paper proposes a mechanism design framework where partial probability information is completed via a maximum-entropy heuristic to enable implementation of joint outcomes.
• It characterizes implementable outcomes for both arbitrary and canonical message spaces using support inclusion and cross-entropy constraints.
• Examples in strategic games demonstrate that maximum-entropy induced belief formation can expand or shift traditional correlated equilibrium outcomes.

Coordination Mechanisms with Partially Specified Probabilities

Motivation and Framework

The paper develops a formal theory for mechanism design under partial information disclosure, extending classical correlated equilibrium analysis. It formalizes situations where agents receive data (e.g., recommendations, signals) whose joint distribution is only partially revealed via coarse statistics—specifically, expectations of finitely many random variables—rather than a full probabilistic specification. Belief formation is governed by a maximum-entropy heuristic, where agents complete the missing information to maximize uncertainty subject to known constraints.

This setting models prevalent real-world informational constraints, such as traders relying on correlated but only partially understood analyst forecasts, social media users exposed to algorithmic feeds, or committee members interpreting overlapping reports. The central technical question is: given partial information, which outcome distributions over action profiles can be implemented in equilibrium?

Main Results

Characterization of Implementable Outcomes

Two principal characterizations are established:

1. Arbitrary Message Spaces: When the designer can freely tailor the message space, every jointly coherent outcome (randomization over strategy profiles in the support of some correlated equilibrium) is implementable. This strictly extends the set of implementable outcomes beyond correlated equilibrium outcomes as originally defined by Aumann (1974, 1987).
2. Canonical Mechanisms: If the message set is restricted to action recommendations and players' strategies are obedient, implementability is characterized via a cross-entropy constraint: the target outcome must reside on a cross-entropy level set for some correlated equilibrium, necessitating that the cross-entropy of the target with respect to the equilibrium is equal to the entropy of the equilibrium itself. The support inclusion condition ($\mathrm{supp}(p) \subset \mathrm{supp}(q)$) is required, with $p$ the target outcome and $q$ the correlated equilibrium.

Numerical and geometric insights: Direct implementability is shown to reduce, in games with unique correlated equilibrium, to the intersection of a support inclusion condition and a single linear constraint in the simplex, yielding a convex polytope of implementable outcomes.

Mechanism Design with Maximum-Entropy Belief Formation

The use of Shannon entropy as a prior selection mechanism leads to critical behavioral implications, prominently "correlation neglect": when agents know only marginal distributions, they model data as independent, even if the true process is correlated. This mechanism allows the designer to induce systematic correlated or anti-correlated behavior using only coarse statistical disclosure, effectively expanding the set of possible equilibrium outcomes.

Examples and Special Cases

Examples are provided across canonical games:

• In the Chicken game, the principle of insufficient reason (maximum entropy matching known support) enables implementation of degenerate outcomes unreachable by correlated equilibrium devices.
• In coordination games, disclosure of marginals in an anti-correlated true process produces anti-coordination outcomes not supported as correlated equilibria but nonetheless jointly coherent.
• General games demonstrate construction of partially specified data-generating processes with message sets and feedback structures tailored to induce desired beliefs, even when the target outcome is not a correlated equilibrium.

These examples highlight that implementable outcomes under maximum-entropy completion can both dominate and fall outside the convex polytope of correlated equilibria.

Implications and Theoretical Insights

Expansion of Mechanism Design Paradigm

The results are significant for mechanism design and information structure design. They show that designers may coordinate agent behavior beyond the scope allowed by correlated equilibrium via controlled coarse disclosure. This is achieved through moment constraint specification and exploitation of maximum-entropy reasoning, a robust heuristic with deep roots in information theory and statistical inference.

Behavioral and Welfare Consequences

Maximum-entropy completion of information leads to both beneficial and detrimental coordination outcomes, depending on the game's structure and the designer's intent. For example, in coordination settings, neglect of correlation induced by entropy maximization can systematically lead to mis-coordination, as players act as if signals are independent.

Connections to Related Solution Concepts

The paper draws connections to self-confirming equilibrium, ambiguity models, and recent work on mechanism design under information misspecification (Lehrer 2012, Spiegler 2021), but distinguishes its approach by focusing on endogenous belief formation and correlation devices, rather than adversarial or ambiguity-averse completion.

Future Directions

Several extensions are proposed:

• Bayesian games: Incorporate payoff-relevant states and reconcile maximum-entropy inference with prior beliefs about the state, relating to Bayesian persuasion (Kamenica and Gentzkow 2011) and Bayes-correlated equilibria (Bergemann and Morris 2016).
• Dynamic games: Extend the framework to settings with learning and feedback, necessitating dynamically consistent beliefs and updated moment constraints.
• Ambiguity attitudes: Explicitly model players' ambiguity preferences or adversarial heuristics, beyond the maximum-entropy foundation.

Conclusion

This paper rigorously demonstrates that partial disclosure of probabilistic information, completed via maximum-entropy inference, constitutes a powerful coordination device in games. It establishes that the corresponding set of implementable outcomes encompasses all jointly coherent profiles, not just correlated equilibria, with canonical mechanisms further constrained by a cross-entropy criterion. Theoretical and practical implications extend to mechanism design, information economics, and behavioral game theory, with rich avenues for further exploration in Bayesian and dynamic settings.

Reference: "Coordination Mechanisms with Partially Specified Probabilities" (2605.07469)