Source-linked AI summary
Information Design in Smooth Games
Alex Smolin, Takuro Yamashita
TL;DR
The paper shows that information-design questions in games can be tractably addressed. The paper uses a duality-based principal-agent certification approach and first-order conditions to characterize optimal information structures in symmetric linear-quadratic games. Certifiably optimal information structures are prior-robust, remaining optimal under any prior.
Problem
The paper shows that information-design questions in games can be tractably addressed.
Method
The paper uses a duality-based principal-agent certification approach and first-order conditions to characterize optimal information structures in symmetric linear-quadratic games.
Results
Certifiably optimal information structures are prior-robust, remaining optimal under any prior.
Takeaways & Limitations
Optimal information policies can be simple, intuitive, and robust, with applications to venture capital, Bayesian polarization, and price competition.
Takeaways & Limitations
The investor application assumes congestion parameter r > 0 and opportunity cost c > 0, and its first-best allocation cannot be implemented through pure information.
Abstract
from arXiv · showhide
We study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.
1 Introduction
The paper develops a tractable certification approach for information design in smooth games and characterizes robust, often simple optimal structures. In symmetric linear-quadratic applications, targeted and linear disclosure yield distinct optimality results, illustrated in venture capital, polarization, and pricing.
- Certification approach: A dual-certification theorem makes an information structure optimal when its induced state-action distribution is also implemented by an adversarial contract.The contract provides an optimality certificate and can certify uniqueness or common features among optimal structures.
- Linear-quadratic games: Targeted disclosure fully reveals a one-dimensional common value to some players while leaving the rest uninformed, and remains optimal under any prior.The structure is simple and distributionally robust.
- Linear-quadratic games: Gaussian coupling is optimal with normally distributed values because player-specific noise cancels in aggregate, while linear disclosure is uniquely optimal in the multidimensional setting.Linear disclosure gives each player a linear statistic of the state and is noise-free and symmetric ex ante.
- Applications: Applications show exclusive disclosure prevents return dissipation in venture capital, half-population disclosure maximizes polarization, and price recommendations can switch regimes discontinuously.The pricing application considers weighted averages of consumer and producer surplus.
- Certification approach: The first-order approach exploits concavity and continuously differentiable payoffs to make continuum-action game incentives tractable.The framework extends beyond concave games while focusing its main analysis on concave games with infinitely many actions.
2 Design Problem
The design problem chooses an information structure and equilibrium to maximize the designer’s expected payoff in a Bayesian game. Concavity permits direct action recommendations and first-order obedience constraints, yielding a measure-based optimization problem.
- Concavity and incentives: Players choose real-valued actions, and weak concavity with continuous differentiability in own action makes best responses satisfy first-order conditions.The payoff maximum is attained at a finite action value.
- Information structure: The designer chooses a signal structure, the state and signals are realized, and each player privately observes a signal before choosing an action.Full disclosure gives every player the state, whereas no disclosure gives each player a single signal.
- Equilibrium and objective: An information structure and strategy profile induce a state-contingent allocation rule, with the designer selecting the equilibrium yielding the highest expected payoff.If multiple equilibria exist, the designer may choose her preferred one; without an equilibrium, the value is undefined.
- Direct formulation: By the revelation principle, the problem can restrict attention to direct structures recommending actions that all players willingly obey.These structures correspond to measures over action profiles and states.
- Direct formulation: The resulting constrained maximization imposes obedience conditions for every player and Bayes’ plausibility for the state marginal.For continuum recommendations, the obedience constraints require the relevant marginal weighted by marginal utilities to equal zero measure.
3 Solution Method
The paper converts information design into an adversarial contracting dual, where a shared implementation certifies optimality. This framework also establishes when certificates exist and constrains all optimal measures they certify.
- Contracting interpretation: The dual problem has a contracting interpretation: a principal chooses action-dependent incentive functions, then a fully informed agent selects the action profile.The principal minimizes the agent’s expected payoff, and implementability means the induced best responses generate the target action-state measure.
- Dual certification: An allocation measure is optimal when it is implementable both by information and by incentives under a contract.The primal and dual problems then have equal values.
- Dual certification: A certificate constrains every optimal measure, not merely the information structure used to construct it.All optimal measures must be best responses to the same dual contract.
- Dual certification: Certification is possible exactly when primal and dual solutions exist and the duality gap is zero.A constructed certificate itself proves zero duality gap and existence of both solutions.
- Existence and scope: Under compact action and state spaces with suitable regularity, standard duality arguments provide a benchmark strong-duality result.For noncompact action spaces, the paper instead verifies strong duality constructively in applications.
4 Robustness of Optimal Information Structures
The paper shows that certified optimality can persist across priors when the induced allocation rule remains implementable with no larger statewise action support. This yields robust conclusions for full-support noise and full state information.
- Prior robustness: A certifiably optimal allocation remains certifiably optimal under another prior when the new implementable rule has no larger support in every state.The result depends on the common certificate and applies within the paper’s concave-game setting.
- Scope: The robustness result is specific to continuum-action games and does not generally extend to games with finitely many actions.The paper links this scope condition to players’ more flexible best responses.
- Full-support noise: If an optimal measure has full action support in every state, any information structure is certifiably optimal.With full support, the support restriction becomes vacuous.
- Full-support noise: Full-support extraneous noise cannot be certifiably optimal except when the designer’s expected payoff is invariant to information.This conclusion concerns concave games with finitely many players.
- Full state information: A fully informative optimal information structure remains optimal under all priors.Full state information means each player can deduce the state with certainty from her private signal.
5 Linear-Quadratic Games: Common Value
In symmetric linear-quadratic games, the paper uses affine certificates and linear allocation rules to characterize optimal disclosure. Targeted disclosure is optimal with a common state, while Gaussian coupling supplies an alternative under normally distributed states.
- Targeted disclosure: Targeted disclosure fully reveals the state to selected players and provides no information to the remainder; zero and N targeting equal no and full disclosure.The policy is implemented through an allocation rule that can also be induced by an affine contract.
- Targeted disclosure: Targeted disclosure is optimal in common-state symmetric linear-quadratic games: an interior k∗ selects k∗ informed players, while boundary cases favor no or full disclosure.If k∗ is noninteger and interior, ceiling-targeted disclosure is asymptotically optimal as the number of players grows.
- Targeted disclosure: The same targeted-disclosure policy is optimal for all priors, despite being asymmetric when only some players receive information.Which players are informed can change surplus distribution without changing aggregate equilibrium outcomes.
- Gaussian coupling: Gaussian coupling adds independent Gaussian noise to private signals while coupling noises so aggregate signals remain a deterministic function of the state.Each individual signal is imperfect, but observing all signals reveals the state when β ≠ 0.
- Gaussian coupling: With normally distributed states, optimal Gaussian coupling implements the target linear allocation through an equilibrium in which each player chooses a_i ≡ s_i.The paper presents this as an alternative optimal information provision method.
6 Linear-Quadratic Games: Interdependent Values
In symmetric linear-quadratic games with interdependent, normally distributed values, the certification approach yields an optimal linear disclosure, uniquely determining recommendations under generic conditions.
- Theorem 4: An R∗-linear disclosure is optimal in normalized symmetric linear-quadratic Gaussian games with multidimensional states.The allocation rule is implemented by a symmetric affine contract, certifying optimality in both the dual and primal problems.
- Certification: Theorem 4 reduces the search for optimal linear disclosure parameters to a one-dimensional minimization problem.An optimal structure can be found within symmetric noise-free Gaussian information structures.
- Uniqueness: The optimal information structure is unique in the interdependent-value setting because richer state uncertainty yields a single best response.By contrast, the common-state setting permits multiple optimal action profiles under the certificate.
7 Extensions
The paper extends its certification approach beyond the baseline finite, concave setting, while identifying boundaries from bounded actions, general smooth payoffs, and infinite-player economies.
- Bounded Action Spaces: For bounded action spaces, boundary first-order conditions become inequalities, and the dual adds sign constraints on contract functions at boundary actions.The primal problem otherwise remains the same.
- Infinite Economies: The analysis does not directly cover games with a continuum of players.The paper instead studies finite-player games and examines limits as the number of players approaches infinity.
- Infinite Economies: With normally distributed states, Gaussian information structures are certifiably optimal and optimal aggregate behavior is deterministic in the state.Finite economies require no extraneous noise or carefully coupled noise, whereas infinite economies can rely on independent noise and the law of large numbers.
- Infinite Economies: Even in infinite economies, targeted disclosure may also be optimal and robustly so.
- General Smooth Games: In general smooth games, first-order conditions may select a suboptimal local maximum or even a minimum because payoffs need not be concave.The approach therefore requires a final verification that players obey the recommended actions.
- General Smooth Games: When recommendations are obeyed after verification, the information structure solving the relaxed first-order-condition problem also solves the original design problem.The relaxed primal problem is solved using the certification approach.
8 Applications
The applications show how optimal information design manages investment congestion, expectation polarization, and competitive pricing through targeted or linear disclosure structures.
- 8.1 Persuading Investors: For any number of investors, one-targeted disclosure is optimal in the investment application.The same structure remains optimal across congestion and cost parameters and project quality.
- 8.1 Persuading Investors: Optimal information design avoids rent dissipation as the investor population grows, keeping total project profit above V[ω]/4 and converging to that level as N →∞.The limiting payoff increases with project-quality variance.
- 8.2 Polarizing Predictions: For an even number of players, N/2-targeted disclosure is optimal and achieves expectation polarization irrespective of the prior.The result extends the insight that informing one of two players can be optimal.
- 8.2 Polarizing Predictions: Belief polarization and expectation polarization differ beyond binary states, yet both share the same optimal information structure.
- 8.3 Informing Competitive Pricing: In competitive pricing, the optimal designer’s choices favor consumers at δ = 1, producers at δ = 0, and social efficiency at δ = 1/2.As δ varies over [0, 1], the corresponding solutions span the relevant welfare trade-off.
- 8.3 Informing Competitive Pricing: The uniquely optimal pricing disclosure is symmetric, linear, and personalized, preventing firms from identifying whether recommendations reflect own or rival demand conditions.Generically, full disclosure is suboptimal and restricting disclosure to public information may entail losses.
- 8.3 Informing Competitive Pricing: A small change in the consumer welfare weight can discontinuously change information provision, equilibrium price volatility, correlations, and market behavior.Volatility and correlations can diagnose which side of the market the structure favors.
9 Conclusion
The paper develops a certification approach for information-design problems in games and applies it to symmetric linear-quadratic settings. It derives theoretical support for targeted disclosure, Gaussian coupling, and linear disclosure, with applications to investment, market control, and Bayesian polarization.
- 9 Conclusion: The certification approach solves information-design problems in games by connecting information implementability with incentive implementability.The framework is presented as effective and tractable in symmetric linear-quadratic settings.
- 9 Conclusion: The analysis provides theoretical justification for targeted disclosure, Gaussian coupling, and linear disclosure.
- 9 Conclusion: The findings clarify disclosure practices guiding investment, socially efficient information control in markets, and the limits of Bayesian polarization.
- 9 Conclusion: The framework lays groundwork and offers tools for studying information design in general smooth games.
- 9 Conclusion: The approach could extend to contests, public-good provision, labor or financial markets, information elicitation, spillovers, and dynamic interaction.The authors identify these as promising directions and expect the extensions to be feasible within their framework.
A Appendix
The appendix defines admissible equilibria and establishes a first-order approach under conditions ensuring well-behaved payoff differences. These conditions support the required interchange of operators and cover the equilibria characterized in the paper.
- A Appendix: The admissibility definition requires best responses to satisfy the paper’s condition (4) under equilibrium beliefs.
- A Appendix: Admissibility requires players’ payoff differences to be locally well behaved at best responses.It ensures interchangeability of operators in condition (4).
- A Appendix: Admissibility holds for all equilibria characterized in the paper and is automatic on compact action and state spaces with continuously differentiable payoffs.
- A Appendix: For general action and state spaces, polynomial payoffs provide one sufficient condition for admissibility.
- A Appendix: The appendix states a first-order approach for any player and admissible equilibrium under Assumption 1.
- A Appendix: The proof uses ε-based conditions and dominated convergence to justify the relevant limiting and operator-interchange steps.
A.2 Formalism Omitted in Section 3
The appendix proves the duality and certification results underlying the paper’s information-design analysis, then supplies omitted calculations for the linear-quadratic applications. It also records benchmark disclosure conditions and equilibrium normalizations.
- A.2 Formalism Omitted in Section 3: Weak duality follows by integrating dual constraints against any primal-feasible measure and comparing the resulting primal and dual values.
- A.2 Formalism Omitted in Section 3: An information-implementable measure can be certified by dual variables, which makes the same measure incentive-implementable and proves its optimality.
- A.2 Formalism Omitted in Section 3: For compact action and state spaces, Fenchel-Rockafellar duality identifies the Bayes-plausibility and obedience constraints with the dual conjugate conditions.
- A.2 Formalism Omitted in Section 3: Strong duality establishes equality between the primal information-design value and the dual contracting value.Under compactness and continuity conditions, the continuous dual’s feasible set coincides with the primal feasible set.
- A.3 Formalism Omitted in Section 5: In the normalized game, no disclosure is optimal when po + pc ≤ h(qo + qc).
- A.3 Formalism Omitted in Section 5: Full disclosure is optimal in the alternative parameter case, certified by an affine contract and characterized through a concave quadratic maximization.
- A.4 Formalism Omitted in Section 8.1: The appendix derives benchmark equilibria: no-disclosure actions are state-independent, while full disclosure yields parameters k0 = 0 and k1 = 1 N+1.
- A.5 Formalism Omitted in Section 6: Game normalization imposes zero expected marginal payoffs and determines a common expected action through symmetry.
B Bounded Action Spaces
The bounded-action extension preserves the adversarial-contracting interpretation while imposing boundary sign restrictions. Under compactness and continuity, information-and-incentive implementability again certifies optimality and strong duality.
- B Bounded Action Spaces: Interior best responses satisfy first-order conditions, whereas boundary best responses need only be unimprovable under one-sided local deviations.
- B Bounded Action Spaces: The extension to half-bounded action spaces is straightforward.
- B Bounded Action Spaces: Bounded actions restrict allowed contracts at boundary actions through sign constraints, while the adversarial-contracting interpretation remains intact.
- B Bounded Action Spaces: Weak duality continues to hold for bounded action spaces by integrating dual constraints against primal-feasible measures.The inequality may be strict with bounded actions, unlike the unbounded case.
- B Bounded Action Spaces: Information implementability satisfies the bounded-action primal constraints, while incentive implementability requires a feasible contract and complementarity slackness.
- B Bounded Action Spaces: If a measure is implementable by information and incentives under a feasible contract, it solves the information-design problem and the contract solves the adversarial-contracting problem.The bounded-action certification theorem also equates the corresponding values.
- B Bounded Action Spaces: Under compact states and continuous payoffs, strong duality follows by incorporating boundary sign restrictions into the continuous dual.
- B Bounded Action Spaces: The bounded-action obedience constraints impose zero weighted mass on interior subsets and weak inequalities on sets touching action boundaries.
C Online Appendix
The appendix applies a certification approach to a smooth two-sided search game and shows that the optimal public signal remains optimal among all private information structures. For a prior below the threshold, the signal induces posterior splitting and coordinated search at either zero or the cost-minimizing intensity.
- Application: The game models retailer and consumer search intensities, where higher search raises matching chances and benefits increase with the state and the other player's search.
- Public Information: When µ0 ≥ ˆa2, no disclosure is optimal; when µ0 < ˆa2, the optimal public signal splits posteriors between 0 and ˆa2.
- Public Information: The induced actions are either (0, 0) or (ˆa, ˆa) after a low-state signal, while the high state always induces (ˆa, ˆa).
- Optimal Information: Public information is optimal even within the class of all private information structures.
- Certification: With ˆa = 3/4, the certification verifies that (ˆa, ˆa) is optimal in the high state because 1−ˆa−ˆa2 ≤ 0.