Information Design in Smooth Games

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Information Design in Smooth Games

# Information Design in Smooth Games

**Authors:** Alex Smolin; Takuro Yamashita

**Manuscript date:** 2026-03-11

#### Abstract

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.

[^0]Keywords: Bayesian persuasion, information design, dual certification, firstorder approach, linear-quadratic games, targeted disclosure, Gaussian coupling, linear disclosure.


Information Design in Smooth Games

## 1 Introduction

As advances in IT infrastructure and artificial intelligence expand our capacity to collect, process, and generate data, an increasing number of firms, regulators, and individuals must decide which information to supply to strategically interacting agents. Which information should investors receive to allocate capital most efficiently? Which demand signals should be disclosed to improve market outcomes? What are the limits of Bayesian polarization? We show that these and related questions can be tractably addressed within a single framework, and that the optimal policies are simple, intuitive, and robust.

Our main analysis focuses on concave games of incomplete information, i.e., games in which each player's action lies in a convex set and each player's payoff is weakly concave in that action, but our methods extend to any smooth game. Concave games are a staple of applied economic modeling with fixed information structures, because equilibria can be tractably characterized by first-order conditions. We show that the same considerations render tractable the task of designing the information structure itself.

To solve the information-design problem, we adopt a duality-based approach. The dual can be viewed as an adversarial contracting problem between a principal and an omniscient agent who both observes the state and controls all players' actions. The dual-certification theorem (Theorem 1) states that if a state-action distribution emerges in equilibrium under both an information structure (in the information-design problem) and a contract (in the adversarial-contracting problem), then that information structure and that contract are optimal for their respective problems. The contract thus serves as an optimality certificate. Furthermore, it is capable of certifying any optimal information structure (Proposition 1), which enables us to establish whether the optimal information structure is unique or, when several are optimal, to identify their common features.

We show that (certifiably) optimal information structures are prior-robust: once optimal under one prior, they remain optimal under any other prior as long as the implemented allocation rule's support is contained within the original support in every state (Proposition 2). This has two consequences. First, if an optimal information
structure is fully informative about the state, it remains optimal under all priors. Second, unless the problem is trivial, an optimal information structure cannot induce a fullsupport action distribution in every state, which argues against adding full-support, extraneous, independent noise to individual signals.

We apply the solution method in two broad, symmetric settings with linear-quadratic payoffs for both the players and the designer. In each case, the certifying contract is symmetric across players and affine in actions, yet it yields qualitatively different optimal information policies.

In the first setting, the state is one-dimensional (Section 5). We derive the parameters of the certifying contract and use them to establish the optimality of a novel information structure we term targeted disclosure: it fully reveals the state to a subset of players while leaving the rest completely uninformed. This structure is remarkably simple and distributionally robust-it remains optimal under any prior. In addition, when the state is normally distributed, Gaussian coupling, which adds normal noise to each player's signal that cancels out in aggregate, is optimal as well.

In the second setting, the state is multidimensional with jointly normal components (Section 6). We show that a certifying contract can be found within the class of symmetric affine contracts by solving a single-variable minimization problem. This contract certifies the unique optimality of linear disclosure: an information structure that gives each player some linear statistic of the state. The resulting information structure is noise-free and symmetric ex ante.

In Section 8, we illustrate these results in three concrete applications. In Section 8.1, we analyze optimal capital fundraising when several investors independently choose how much to invest in a project of uncertain quality. We show that the information structure that maximizes the project's expected return is exclusive disclosure, a targeted disclosure in which only one investor is informed. Relative to full or no disclosure, exclusive disclosure prevents the dissipation of returns as the investor pool grows, offering a possible rationale for a common venture-capital practice.

In Section 8.2, we examine the limits of Bayesian polarization when each player
forms a forecast from private information. We show that the information structure that maximizes a natural polarization index is a targeted disclosure that informs exactly half of the players. Hence, maximal polarization occurs when the population splits into two cohorts that are internally uniform but sharply divergent from each other. This finding underscores how media segregation can intensify societal polarization.

In Section 8.3, we study optimal price recommendations in a differentiated-product duopoly with linear demand and stochastic demand shocks. We characterize the information structures that maximize any weighted average of consumer and producer surplus. The resulting recommended prices are linear functions of the demand shocks and exhibit discontinuous regime changes with respect to the weight on consumer surplus. This observation underscores the risk that algorithms generating price recommendations-able to shift objectives and adapt policies far faster than human decision-makers-can destabilize markets.


Information Design in Smooth Games

Related Literature Our paper contributes to the recent and flourishing literature on information design. Much of this literature focuses on information design with a single player, called Bayesian persuasion (Rayo and Segal (2010), Kamenica and Gentzkow (2011)). Popular solution methods are belief based, i.e., they operate within the space of the receiver's belief distributions. ${ }^{1}$ A natural continuation of this research agenda is a study of information design in multiplayer games (Bergemann and Morris (2016), Taneva (2019)). In games, players' beliefs constitute infinite hierarchies, which renders the beliefbased approach less tractable (Mathevet, Perego, and Taneva (2020)). Instead, in games, an action-based approach rooted in the revelation principle is promising, as it frames

[^1]the design problem as a linear program and enables the use of duality machinery. ${ }^{2,3}$
Galperti and Perego (2018) and Galperti, Levkun, and Perego (2024) employ this action-based approach to study information design in games with finitely many actions, imposing minimal structure on payoffs. They develop an economic interpretation of the Lagrange multipliers associated with Bayes' plausibility as the value of data records and propose the idea of pooling externalities across records; all these observations apply to our setting.

In contrast, we study games with infinitely many actions and impose a concavity structure on payoffs, thus enabling us to rely on a first-order approach for incentives (Holmström (1979), Mirrlees (1999)) that leads to more succinct and tractable primal and dual problems. This approach was introduced in a Bayesian persuasion setting by Kolotilin $(2012,2018)$ and further refined by Kolotilin, Corrao, and Wolitzky (2025). These papers study an information-design problem with a single player and a onedimensional state, identifying when censorship or, respectively, assortative disclosures are optimal. We deepen and extend this approach, adapting it to settings with multiple players and a multidimensional state.

Most of our current understanding of optimal information in games is drawn from the study of Gaussian signals in games with quadratic payoffs and a normally distributed state (e.g., Angeletos and Pavan $(2007,2009)$, Bergemann and Morris (2013), Bergemann, Heumann, and Morris (2015, 2021), Ui (2020)). In that literature, as well as in a vast body of work in macroeconomics and finance, the Gaussian form of players' signals is imposed ad hoc for analytical convenience.

Recent arguments suggest that Gaussian signals are not only convenient but often optimal among all information structures. For example, Tamura $(2012,2018)$ estab-

[^2]lished the optimality of a Gaussian signal in a setting with a single player by building on the statistical properties of a covariance matrix of posterior expectations. Bergemann, Heumann, and Morris (2017) and Miyashita and Ui (2023) extend this argument to games. ${ }^{4}$ Our results align with these findings and show that in many scenarios the optimal Gaussian structures are symmetric and noise-free; moreover, the certification method allows direct identification of the optimal informational parameters. Furthermore, we provide tools that establish when Gaussian information structures are uniquely optimal.

At the same time, we show that in many settings targeted disclosure is optimal. These information structures are simple to implement and distributionally robust, offering a strong positive message for the information design literature, where existing solutions often lack these properties.


Information Design in Smooth Games

## 2 Design Problem

We study a standard information design problem as presented by Bergemann and Morris (2016), extended to accommodate a continuum of players' actions.

Payoffs There are $N$ players indexed by $i, 1 \leq N<\infty$, and an information designer. Each player chooses an action $a_{i} \in A_{i}=\mathbb{R}$. We denote an action profile by $a \in A=\times_{i} A_{i}$ and write $\left(a_{i}, a_{-i}\right)$ when highlighting player $i$ 's action.

A state $\omega$ is distributed over a Polish set $\Omega$, according to a full-support prior $\mu_{0} \in$ $\Delta(\Omega)$. The action profile and the state jointly determine payoffs through

$$
\begin{aligned}
& u_{i}: A \times \Omega \rightarrow \mathbb{R}, \\
& v: A \times \Omega \rightarrow \mathbb{R},
\end{aligned}
$$

for each player $i$ and for the designer, respectively. The tuple $\left(\left(A_{i}, u_{i}\right)_{i=1}^{N}, \mu_{0}\right)$ constitutes

[^3]the basic game.

Information The players and the designer start with a commonly known prior belief about the state $\omega$ that coincides with the prior $\mu_{0}$. The designer can provide additional information to players by choosing an information structure $\mathcal{I}=(S, \pi)$ that consists of a measurable signal set $S=\times_{i} S_{i}$ and a likelihood function $\pi \in \Delta(S \times \Omega)$ that has $\mu_{0}$ as its state marginal distribution. This information structure determines the sets of private signals the players can observe and, through the likelihood function, their informational content.

First, the designer chooses an information structure $\mathcal{I}$. Second, the state $\omega$ and the signal profile $s$ are realized according to $\mathcal{I}$. Finally, each player privately observes his signal $s_{i}$ and chooses an action $a_{i}$.

We will regularly refer to two benchmark information structures: full disclosure, with $S_{i}=\Omega$ and $s_{i} \equiv \omega$ for all $i$; and no disclosure, with $\left|S_{i}\right|=1$ for all $i$.

Equilibrium The basic game together with the information structure chosen by the designer determine a Bayesian game of incomplete information. In that game, each player's behavior is described by a strategy that maps any received signal to a possibly random action, $\sigma_{i}: S_{i} \rightarrow \Delta\left(A_{i}\right)$, and we consider as an equilibrium concept a Bayes Nash equilibrium:

Definition 1. (Bayes Nash Equilibrium) For a given information structure $\mathcal{I}$, a strategy profile $\sigma=\left(\sigma_{1}, \ldots, \sigma_{N}\right)$ constitutes a Bayes Nash equilibrium if

$$
\mathbb{E}_{\mathcal{I}, \sigma_{i}, \sigma_{-i}}\left[u_{i}\left(a_{i}, a_{-i}, \omega\right)\right] \geq \mathbb{E}_{\mathcal{I}, \sigma_{i}^{\prime}, \sigma_{-i}}\left[u_{i}\left(a_{i}^{\prime}, a_{-i}, \omega\right)\right]
$$

for all $i$ and $\sigma_{i}^{\prime}: S_{i} \rightarrow \Delta\left(A_{i}\right) .{ }^{5}$
An information structure and a strategy profile determine a distribution over the action profiles in each state $\alpha: \Omega \rightarrow \Delta(A)$, which we call an allocation rule, and the corre-

[^4]sponding designer's expected payoff $\mathbb{E}_{\mathcal{I}, \sigma}[v(a, \omega)]$. The value of an information structure is defined as the maximal designer's expected payoff that can arise in equilibrium of the induced game: if the game has multiple equilibria, the designer can choose the one she prefers, whereas if no equilibrium exists, the value is undefined. An information-design problem consists of finding an information structure with a maximal value without placing any additional restrictions on the sets of signals or the likelihood function, apart from a mild "admissibility" condition. (This condition, and other omitted formal details, are deferred to Appendix A.)

Definition 2. (Optimal Information Structure) An information structure is optimal if there does not exist an information structure with a strictly higher value.

The search for an optimal information structure is complicated by the scale of the basic game: multiple players, after receiving private signals, choose actions from a continuum while anticipating one another's behavior. Our central simplifying assumption is:

Assumption 1. (Concave Payoffs) For all $i=1, \ldots, N, \omega \in \Omega$, and $a_{-i} \in A_{-i}$, $u_{i}\left(a_{i}, a_{-i}, \omega\right)$ is continuously differentiable in $a_{i}$, weakly concave in $a_{i}$, and obtains its maximum at some finite value.

We call a basic game in which Assumption 1 is satisfied a concave game. ${ }^{6}$ In a concave game, for any player $i$ and equilibrium belief $\mu \in \Delta\left(A_{-i} \times \Omega\right)$ about the others' actions and the state, a best response $a_{i}^{*}$ exists and satisfies the first-order condition:

$$
\left.\frac{\partial}{\partial a_{i}} \mathbb{E}_{\mu}\left[u_{i}\left(a_{i}, a_{-i}, \omega\right)\right]\right|_{a_{i}=a_{i}^{*}}=\left.\mathbb{E}_{\mu}\left[\frac{\partial}{\partial a_{i}} u_{i}\left(a_{i}, a_{-i}, \omega\right)\right]\right|_{a_{i}=a_{i}^{*}} \triangleq \mathbb{E}_{\mu}\left[\dot{u}_{i}\left(a_{i}^{*}, a_{-i}, \omega\right)\right]=0,
$$

where we denote the player's marginal payoff function by

$$
\dot{u}_{i}(a, \omega) \triangleq \frac{\partial u_{i}(a, \omega)}{\partial a_{i}} .
$$

[^5]An equilibrium under any given information structure is characterized by a system of such conditions, one for each player's signal.

The information-design problem can be simplified by appealing to the revelation principle (Myerson (1982)): it is without loss of generality to focus on direct information structures that inform each player about a recommended action $S=A$ and are such that all players are obedient, i.e., are willing to follow the recommendations. Each direct information structure corresponds to a measure $\pi \in \Delta(A \times \Omega)$, and the informationdesign problem can be formulated as a constrained maximization over these measures:


Information Design in Smooth Games

$$
\begin{aligned}
V^{P} \triangleq & \sup _{\pi \in \Delta(A \times \Omega)} \int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi \\
& \quad \text { s.t. } \int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) \mathrm{d} \pi=0 \quad \forall i=1, \ldots, N, \text { measurable } A_{i}^{\prime} \subseteq A_{i}, \\
& \int_{A \times \Omega^{\prime}} \mathrm{d} \pi=\int_{\Omega^{\prime}} \mathrm{d} \mu_{0} \quad \forall \text { measurable } \Omega^{\prime} \subseteq \Omega .
\end{aligned}
$$

Constraints (6) are a proper formulation of first-order conditions (4) in light of a continuum of recommended actions. These constraints capture players' obedience and effectively require that for each player $i$, the marginal of $\pi$ on $A_{i}$ weighted by the marginal utilities equals zero measure. Constraints (7) capture Bayes' plausibility and, likewise, require that the marginal of $\pi$ on $\Omega$ equals the prior $\mu_{0}$.


Information Design in Smooth Games

## 3 Solution Method

Problem (5-7) is linear in $\pi$, yet unwieldy to solve directly. In the spirit of linear programming, we view it as a primal problem and call any $\pi \in \Delta(A \times \Omega)$ a primal measure. If a primal measure satisfies the constraints of the primal problem, then we call that measure implementable by information. We can construct a dual problem as
follows (e.g., Anderson and Nash (1987)):

$$
\begin{aligned}
& V^{D} \triangleq \inf _{\lambda \in \times_{i} \mathcal{M}\left(A_{i}\right), \gamma \in \mathcal{M}(\Omega)} \int_{\Omega} \gamma(\omega) \mathrm{d} \mu_{0} \\
& \quad \text { s.t. } \sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega)+\gamma(\omega) \geq v(a, \omega) \forall a \in A, \omega \in \Omega,
\end{aligned}
$$

where $\mathcal{M}(X)$ denotes the space of measurable real-valued functions on $X$. The minimization arguments, the dual variables $(\lambda, \gamma)$, represent the Lagrange multipliers associated with the primal incentive constraints (6) and the feasibility constraints (7), respectively.

The problem (8) is a generalization of the dual problem of Kolotilin (2018) and Kolotilin et al. (2025) to multiple players. By the arguments analogous to those of Galperti et al. (2024), the optimal $\gamma^{*}(\omega)$ measures the marginal benefit of increasing the frequency of state $\omega$. Importantly, the optimal $\gamma^{*}(\omega)$ can be solved away: the objective in (8) is additively separable in $\gamma(\omega)$, and the constraints at different states $\omega$ are linked only through the variables $\lambda$. Hence, for any $\lambda$ and $\omega$, the optimal choice of $\gamma(\omega)$ is the smallest value consistent with the dual constraints:

$$
\gamma^{*}(\omega ; \lambda)=\sup _{a \in A} w(a, \omega, \lambda),
$$

where the dual payoff function $w$ is defined as

$$
w(a, \omega, \lambda) \triangleq v(a, \omega)-\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) .
$$

As a result, the problem (8) can be restated as follows:

$$
V^{D}=\inf _{\lambda \in \times_{i} \mathcal{M}\left(A_{i}\right)} \mathbb{E}\left[\sup _{a \in A} w(a, \omega, \lambda)\right] .
$$

Problem (10) admits a simple economic interpretation as a contracting problem between a dual principal and a dual agent. First, the principal chooses an incentive contract $\lambda$ that consists of $N$ functions $\lambda_{i}\left(a_{i}\right)$ and determines the agent's payoff according to (9). Second, the state $\omega$ is realized. Finally, the agent perfectly observes the state and chooses
the whole action profile $a \in A$ to maximize his payoff. The contracting is adversarial in that the principal aims to minimize the agent's expected payoff.

If the best responses exist at all states and induce the joint action-state measure $\pi(a, \omega)$, then we say that $\lambda$ implements $\pi$ by incentives and that $\pi$ is implementable by incentives, by contract $\lambda$.

Theorem 1. (Weak Duality. Dual Certification) If $\pi \in \Delta(A \times \Omega)$ is implementable by information and is implementable by incentives by contract $\lambda$, then (i) $\pi$ solves the information-design problem, (ii) $\lambda$ solves the adversarial-contracting problem, and (iii) $V^{P}=V^{D}$.

Theorem 1 offers a solution method based on optimality certification. When the conditions of the theorem hold, we say that $\lambda$ is a (dual) certificate of $\pi$, that $\lambda$ certifies the optimality of $\pi$, and that $\pi$ is certifiably optimal. Similarly, if an information structure induces a certifiably optimal measure, then we call that information structure certifiably optimal.

We highlight two general properties of certificates. First, every certificate $\lambda^{*}$ has a clear economic meaning: it represents Lagrange multipliers associated with the obedience constraints (6). Accordingly, $\lambda_{i}^{*}\left(a_{i}\right)$ measures the marginal value for the information designer from perturbing the obedience constraint of action $a_{i}$. Second, each certificate is "universal":

Proposition 1. (Universality) If $\lambda$ certifies the optimality of measure $\pi$, and $\pi^{\prime}$ is another optimal measure, then $\lambda$ also certifies the optimality of measure $\pi^{\prime}$.

By Proposition 1, a certificate does more than certify a single information structure- it constrains the entire set of optima. Every optimal measure must be certified by the same certificate; equivalently, it must be the dual agent's best response to the same contract in the dual problem. We exploit this property in Section 5, where multiple optima exist, to isolate their common features, and in Section 6 to show that the optimal measure is unique.

The difference $V^{D}-V^{P}$ between the optimal values of primal and dual problems is nonnegative and constitutes a duality gap. The solution to the information-design problem can be certified if and only if (i) solutions to both primal and dual problems exist and (ii) the duality gap is equal to zero, $V^{D}=V^{P}$. Thus, either all optimal information structures can be certified or none of them can.

Any constructed certificate, by its very existence, proves that the duality gap is zero and that both primal and dual solutions exist. However, it may be useful to know $a$ priori whether one can expect to find such a certificate in a given problem.

Claim 1. (Strong Duality) If each $A_{i}$ and $\Omega$ are compact, and $v$ and each $\dot{u}_{i}$ are continuous on $\Omega \times A$, then $V^{D}=V^{P}$.

Claim 1 follows from standard Fenchel-Rockafellar arguments. We record the result mainly as a benchmark, suggesting that strong duality is to be expected in regular environments. Many applications below feature noncompact action spaces. In those environments, a general strong-duality theorem would require additional coercivity and integrability assumptions to rule out mass escaping to infinity and to ensure that the dual certificates remain well behaved. Rather than impose such conditions abstractly, in the applications below we verify strong duality constructively by exhibiting explicit certificates.


Information Design in Smooth Games

## 4 Robustness of Optimal Information Structures

By Theorem 1, an allocation rule induced by a certifiably optimal information structure must be chosen freely by a fully informed dual agent. This observation has two consequences. First, the prior is irrelevant for the implementability of an allocation rule by incentives since the prescribed action profiles must be optimal state-by-state. Second, if the optimal allocation rule randomizes over several action profiles at a given state, the dual agent must be indifferent among those profiles and could therefore randomize over them with any probabilities. Hence, only the support of the action profiles matters for implementability by incentives:

Proposition 2. (Robustness) Let two concave information-design problems differ only in their priors, if at all. Let $\pi_{1} \in \Delta(A \times \Omega)$ be certifiably optimal in the first problem. If $\pi_{2} \in \Delta(A \times \Omega)$ is implementable by information in the second problem and $\operatorname{supp} \pi_{2}(\cdot \mid$ $\omega) \subseteq \operatorname{supp} \pi_{1}(\cdot \mid \omega)$ for all $\omega \in \Omega$, then $\pi_{2}$ is certifiably optimal in the second problem.

Proposition 2 shows that certifiably optimal information structures are, to some extent, prior-robust: once optimal under one prior, an information structure remains optimal under any other prior, provided it still implements an allocation rule whose support is no larger in every state. It is specific to our setting, which features a continuum of actions and thus more flexible players' best responses, and does not hold in generic games with finitely many actions (cf. Kamenica and Gentzkow (2011)).

Generally, the larger the support is, the easier it is to construct multiple information structures that implement allocation rules within that support. In the extreme case, if the action support covers the whole action space, then the support condition of Proposition 2 has no bite, and any information structure can be certified to be optimal.

Corollary 1. (Full-Support Noise) If measure $\pi \in \Delta(A \times \Omega)$ is certifiably optimal and $\operatorname{supp} \pi(\cdot \mid \omega)=A$ for all $\omega \in \Omega$, then any information structure is certifiably optimal.

Corollary 1 presents a case against using extraneous noises that induce full-support action profiles in concave games with finitely many players. The information structures that employ such noises can never be certifiably optimal, except in trivial cases in which the designer's expected payoff is invariant to the information provided. This finding resonates with the analysis of Taneva (2019), who studied a two-player binary setting and showed that sending conditionally independent signals is never strictly optimal, as well as with the analysis of Candogan and Strack (2023), who established optimality of partitional signals in a class of games. However, such extraneous independent noises may optimally appear in the limit information structure as the number of players grows to infinity, as we discuss in Section 7.

Furthermore, Proposition 2 enables us to assess the optimality of full state transparency. We call an information structure fully informative about the state if each player
can deduce the state with certainty from her private signal. There could be many such structures, each differing in the state-by-state coordination of players' actions, including full disclosure. However, any such structure can implement the same allocation rule under all priors.

Corollary 2. (Full State Information) If a certifiably optimal information structure is fully informative about the state, then it is certifiably optimal under all priors.

Corollary 2 shows that, once full disclosure is optimal, it is extremely robust: it remains optimal under any prior (cf. Jehiel (2015)).


Information Design in Smooth Games

## 5 Linear-Quadratic Games: Common Value

The certification approach developed in the previous section applies to any concave game (see Online Appendix C) and, more generally, to any smooth game (Section 7). In this section, we demonstrate the approach in symmetric linear-quadratic games with a common state. We first develop general results and then illustrate them through two applications: capital fundraising and expectation polarization.

We define a linear quadratic symmetric game with a common state as a game with $\Omega \subseteq \mathbb{R}$, and the following payoff structure:

$$
\begin{aligned}
u_{i}(a, \omega) & =\left(\omega+b_{1}\right) a_{i}+\frac{1}{2}\left(q^{o}-\frac{q^{c}}{N}\right) a_{i}^{2}+q^{c} a_{i} \bar{a}+f_{1}\left(a_{-i}, \omega\right), \\
v(a, \omega) & =\left(h \omega+b_{2}\right) \bar{a}+p^{o} \check{a}+p^{c} \bar{a}^{2}+f_{2}(\omega),
\end{aligned}
$$

where $\bar{a} \triangleq \sum_{j} a_{j} / N, \check{a} \triangleq \sum_{j} a_{j}^{2} / N, f_{1}$ and $f_{2}$ are arbitrary functions of the indicated arguments, $h \geq 0$, and $q^{o}, q^{c}, p^{o}, p^{c}$ satisfy the concavity conditions:

$$
\begin{gathered}
q^{o}<0, q^{o}+q^{c}<0 \\
q^{o} p^{c}>p^{o} q^{c}
\end{gathered}
$$

Condition (13) ensures that $u_{i}$ is strictly concave in $a_{i}$ for all $N \geq 1$, whereas condition
(14) ensures concavity of the dual payoff under conjectured certificates. This game is strategically equivalent to, and thus admits the same solution as, a simpler normalized game, in which $\mathbb{E}[\omega]=0$,

$$
\begin{aligned}
u_{i}(a, \omega) & =\omega a_{i}+\frac{1}{2}\left(q^{o}-\frac{q^{c}}{N}\right) a_{i}^{2}+q^{c} a_{i} \bar{a}, \\
v(a, \omega) & =(h \omega+b) \bar{a}+p^{o} \check{a}+p^{c} \bar{a}^{2},
\end{aligned}
$$

where $h, q^{o}, q^{c}, p^{o}, p^{c}$ are as in the original game and $b=h \mathbb{E}[\omega]+b_{2}-2\left(\mathbb{E}[\omega]+b_{1}\right)\left(p^{o}+\right.$ $\left.p^{c}\right) /\left(q^{o}+q^{c}\right)$. Therefore, we can focus on an analysis of the normalized game.

To find optimality certificates, observe that the players' marginal payoffs are linear:

$$
\dot{u}_{i}(a, \omega)=\omega+q^{o} a_{i}+q^{c} \bar{a},
$$

and the dual payoff function takes the form

$$
w(a, \omega, \lambda)=(h \omega+b) \bar{a}+p^{o} \check{a}+p^{c} \bar{a}^{2}-\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right)\left(\omega+q^{o} a_{i}+q^{c} \bar{a}\right) .
$$

Given the linear-quadratic nature of the environment, we conjecture that an optimal allocation rule is linear in the state. Because the allocation rule must be a best response given the dual payoff (16), it is natural to posit a certificate that is affine and symmetric across players:

$$
\lambda_{i}\left(a_{i}\right)=-\frac{1}{N}\left(x a_{i}+x_{0}\right) .
$$

With this certificate, the dual payoff function becomes

$$
w\left(a, \omega, x, x_{0}\right)=x_{0} \omega+\left((h+x) \omega+b+x_{0}\left(q^{o}+q^{c}\right)\right) \bar{a}+\left(p^{c}+x q^{c}\right) \bar{a}^{2}+\left(p^{o}+q^{o} x\right) \check{a} .
$$

If the coefficient in front of $\check{a}$ is nonzero, $p^{o}+q^{o} x \neq 0$, then the dual best-response is uniquely defined, linear in state, and symmetric across players; thus, such $x$ could certify only the optimality of full disclosure or no disclosure. To certify the optimality of partial
disclosure, we must have

$$
\begin{gathered}
x^{*}=-\frac{p^{o}}{q^{o}} \\
w\left(a, \omega, x^{*}, x_{0}\right)=x_{0} \omega+\left(\left(h-\frac{p^{o}}{q^{o}}\right) \omega+b+x_{0}\left(q^{o}+q^{c}\right)\right) \bar{a}+\left(p^{c}-\frac{p^{o} q^{c}}{q^{o}}\right) \bar{a}^{2} .
\end{gathered}
$$

With this choice of $x$, the dual payoff function is a function solely of the average action $\bar{a}$. Because of (15) and $\mathbb{E}[\omega]=0$, any incentive compatible expected average action must equal to 0; therefore we must have

$$
\begin{gathered}
x_{0}^{*}=-\frac{b}{q^{o}+q^{c}} \\
\bar{a}^{*}(\omega)=\frac{p^{o}-h q^{o}}{2\left(p^{c} q^{o}-p^{o} q^{c}\right)} \omega
\end{gathered}
$$

Any allocation rule that results in this average action is implementable by incentives by the affine contract (17). By Theorem 1, if the same allocation rule can be implemented by information, then that allocation rule is optimal. ${ }^{7}$ A natural way to implement such a linear average action is to fully inform a subset of players:

Definition 3. (Targeted Disclosure) For $k \in\{0,1, \ldots, N\}$, a $k$-targeted disclosure is an information structure that fully reveals the state to $k$ players while providing no information to others.

A 0-targeted disclosure is no disclosure and $N$-targeted disclosure is full disclosure. Under a $k$-targeted disclosure, there exists an equilibrium where each uninformed player plays $a_{i}^{N D} \equiv 0$, whereas each informed player plays

$$
a_{i}^{I}(\omega)=-\frac{N}{q^{c} k+q^{o} N} \omega,
$$

[^6]resulting in the average action
$$
\bar{a}(\omega)=\frac{k}{N} a_{i}^{I}(\omega)=-\frac{k}{q^{c} k+q^{o} N} \omega .
$$

Theorem 2. (Optimality of Targeted Disclosure) In a normalized linear-quadratic symmetric game with a common state, if

$$
k^{*} \triangleq \frac{q^{o}\left(h q^{o}-p^{o}\right)}{q^{o}\left(2 p^{c}-h q^{c}\right)-p^{o} q^{c}} N
$$

is in $\{0,1, \ldots, N\}$, then $k^{*}$-targeted disclosure is optimal. If $k^{*} \notin[0, N],{ }^{8}$ then either (i) $p^{o}<h q^{o}$, in which case no disclosure is optimal, or (ii) $\left(h q^{o}-p^{o}\right)\left(q^{o}+q^{c}\right)>2\left(p^{c} q^{o}-p^{o} q^{c}\right)$, in which case full disclosure is optimal.

Proof. If $k$ is in $\{0,1, \ldots, N\}$, then by Theorem 1, comparing (18) and (19), a $k$-targeted disclosure is optimal if

$$
\begin{aligned}
& -\frac{k}{q^{c} k+q^{o} N}=\frac{p^{o}-h q^{o}}{2\left(p^{c} q^{o}-p^{o} q^{c}\right)} \\
& \Leftrightarrow k=\frac{q^{o}\left(h q^{o}-p^{o}\right)}{q^{o}\left(2 p^{c}-h q^{c}\right)-p^{o} q^{c}} N .
\end{aligned}
$$

If $k \notin[0, N]$, then given the conditions (13), either (i) or (ii) holds. ${ }^{9}$ In case of (i), the optimality of no disclosure can be certified by an affine contract with $\left(x, x_{0}\right)=$ $\left(-h,-b /\left(q^{o}+q^{c}\right)\right)$. In case of (ii), the optimality of full disclosure can be certified by an affine contract with $\left(x, x_{0}\right)=\left(h-2\left(p^{o}+p^{c}\right) /\left(q^{o}+q^{c}\right),-b /\left(q^{o}+q^{c}\right)\right)$. (See Appendix A. 3 for details.) $\square$


Information Design in Smooth Games

Theorem 2 highlights the importance of targeted disclosure in linear-quadratic games with a common state. First, if $k^{*} \notin(0, N)$, then either 0-targeted disclosure or $N$ targeted disclosure is optimal. Second, if $k^{*}$ is in $\{0,1, \ldots, N\}$, then $k^{*}$-targeted disclosure is exactly optimal, and the parameter space that yields it spans a wide range of

[^7]economically relevant settings. Third, if $k^{*} \in(0, N)$ but is not an integer, a $\left\lceil k^{*}\right\rceil$-targeted disclosure is asymptotically optimal. Indeed, the affine contract above provides, by weak duality, an upper bound on the designer's value, and under any obedient allocation the contract term has expectation zero. Under this contract, the dual payoff depends only on the average action and is maximized at (18). Because the average action induced by $k$-targeted disclosure depends continuously on $k / N$, and $\left\lceil k^{*}\right\rceil / N \rightarrow k^{*} / N$, the payoff under $\left\lceil k^{*}\right\rceil$-targeted disclosure converges to the optimum as $N$ grows.

The optimality of targeted disclosure is notable for three reasons. First, when $k^{*} \in$ $[1, N-1]$, the policy is asymmetric even though the underlying environment is symmetric. Second, because the argument never relies on the prior, the same targeted-disclosure policy is optimal for all priors (cf. Proposition 2). Third, the policy is remarkably simple to implement. The latter two properties highlight the exceptional practical applicability of targeted disclosure.

At the same time, optimal information structures need not be unique. When targeted disclosure is optimal, the designer is free to choose which players receive information. That choice leaves aggregate equilibrium outcomes unchanged but shifts surplus among players. For example, an ex-ante symmetric variant of $k^{*}$-targeted disclosure first publicly draws a random subset of $k^{*}$ players and then applies $k^{*}$-targeted disclosure to that subset. Furthermore, qualitatively distinct information structures can also be optimal, specifically when the state is normally distributed.

For the rest of this section, assume $N \geq 2$.

Definition 4. (Gaussian Coupling) For $\beta \in \mathbb{R}$ and $\sigma^{2}>0, a\left(\beta, \sigma^{2}\right)$-Gaussian coupling is an information structure such that for all $i$ and $\omega$,

$$
s_{i}=\beta \omega+\varepsilon_{i}-\frac{1}{N-1} \sum_{j \neq i} \varepsilon_{j},
$$

where each noise term $\varepsilon_{i} \sim N\left(0, \sigma^{2}\right)$ is independent from $\omega$ and $\varepsilon_{j}$ for $j \neq i$.
A Gaussian coupling distorts the state by adding Gaussian noise to each player's signal, with the individual noises coupled in such a way that they vanish upon aggregation-
so that the sum of all signals is a deterministic function of the state $\omega$. When $\beta \neq 0$, knowledge of all signals perfectly reveals the state. However, observing only $s_{i}$ provides imperfect information about the state; in this sense, a Gaussian coupling splits the complete state information across players. When the state is normally distributed, a Gaussian coupling corresponds to a Gaussian information structure-that is, the signals and the state are jointly normally distributed.

Theorem 3. (Optimality of Gaussian Coupling) In a normalized linear-quadratic symmetric game with a common state, if $\omega \sim N\left(0, \sigma_{\omega}^{2}\right)$ and $k^{*}$ as defined in (20) lies in $(0, N),{ }^{10}$ then a $\left(\beta, \sigma^{2}\right)$-Gaussian coupling is optimal, where

$$
\begin{aligned}
\beta & =\frac{p^{o}-h q^{o}}{2\left(p^{c} q^{o}-p^{o} q^{c}\right)} \\
\sigma^{2} & =\frac{N-1}{N} \frac{\beta\left(1+\left(q^{c}+q^{o}\right) \beta\right)}{-q^{o}} \sigma_{\omega}^{2}
\end{aligned}
$$

In the optimal equilibrium, $a_{i} \equiv s_{i}$.
Proof. Consider $\left(\beta, \sigma^{2}\right)$-Gaussian coupling with $\left(\beta, \sigma^{2}\right)$ as defined by (21) and (22). If each player plays $a_{i} \equiv s_{i}$, then $\bar{a}(\omega)=\beta \omega=\bar{a}^{*}(\omega)$, as defined in (18). By Theorem 1, if $a_{i} \equiv s_{i}$ is incentive-compatible and $\sigma^{2}>0$, then the $\left(\beta, \sigma^{2}\right)$-Gaussian coupling is optimal.

To show incentive compatibility, observe that $\omega+q^{o} a_{i}+q^{c} \bar{a}$ and $a_{i}$ are jointly normally distributed. Therefore, the incentive compatibility is equivalent to

$$
\begin{aligned}
\mathbb{E}\left[\left(\omega+q^{o} a_{i}+q^{c} \bar{a}\right) a_{i}\right] & =0, \\
\mathbb{E}\left[\left(\omega+q^{o} a_{i}+q^{c} \beta \omega\right) a_{i}\right] & =0, \\
\left(1+q^{c} \beta\right) \beta \sigma_{\omega}^{2}+q^{o}\left(\beta^{2} \sigma_{\omega}^{2}+\sigma^{2}+\frac{1}{N-1} \sigma^{2}\right) & =0,
\end{aligned}
$$

which, after rearrangement, is equivalent to (22).
Finally, it is straightforward to verify that if $k^{*} \in(0, N)$, then $\sigma^{2}>0$; see Appendix A. 3 for details. $\square$

[^8]Theorem 3 highlights an alternative way of providing information optimally-by carefully coupling the players' individual noises. In the case of a normally distributed state, this information structure comes closest to the existing Gaussian information design literature (e.g., Angeletos and Pavan (2007, 2009), Bergemann and Morris (2013), Bergemann, Heumann, and Morris (2015, 2021), Ui (2020)). However, the optimality of a Gaussian coupling depends crucially on the prior's Gaussian form, making this solution less robust than targeted disclosure. Accordingly, in the applications that are presented in Section 8.1 and Section 8.2, we focus on optimal targeted disclosure. Nonetheless, we encourage the reader to bear in mind the fact that if the state is normally distributed in those applications, then providing information via a Gaussian coupling is also optimal.


Information Design in Smooth Games

## 6 Linear-Quadratic Games: Interdependent Values

In this section, we apply the certification approach to strategic environments with interdependent values and thus with a multidimensional state. For tractability, we focus on jointly normally distributed state components. As in the previous section, we first present general results and then illustrate them with an economic application: informing competitive pricing.

Formally, we study symmetric games with $\omega \sim N\left(\mu \mathbf{1}, \sigma^{2} M(1, \rho)\right)$, where $M(x, y)$ denotes the $N \times N$ matrix whose diagonal entries equal $x$ and off-diagonal entries equal $y$, and $\rho \in(-1 /(N-1), 1)$. The players' payoffs are

$$
\begin{aligned}
u_{i}(a, \omega) & =\left(\omega_{i} p^{o}+\breve{\omega}_{-i} p^{c}+b_{1}\right) a_{i}+\frac{q^{o}}{2} a_{i}^{2}+q^{c} a_{i} \breve{a}_{-i}+f_{1}\left(a_{-i}, \omega\right), \\
v(a, \omega) & =\sum_{i=1}^{N}\left(\omega_{i} \tilde{p}^{o}+b_{2}\right) a_{i}+\left(\sum_{i=1}^{N} \sum_{j \neq i} \omega_{i} a_{j}\right) \tilde{p}^{c}+\left(\sum_{i=1}^{N} a_{i}^{2}\right) \tilde{q}^{o}+\left(\sum_{i=1}^{N} \sum_{j \neq i} a_{i} a_{j}\right) \tilde{q}^{c}+f_{2}(\omega),
\end{aligned}
$$

where $\breve{\omega}_{-i} \triangleq \sum_{j \neq i} \omega_{j}, \breve{a}_{-i} \triangleq \sum_{j \neq i} a_{j}, f_{1}$ and $f_{2}$ are arbitrary functions of the indicated arguments, and $q^{o}, q^{c}, p^{o}, p^{c}, \tilde{q}^{o}, \tilde{q}^{c}, \tilde{p}^{o}, \tilde{p}^{c}$ are commonly known parameters that satisfy
the concavity and "genericity" conditions:

$$
\begin{gathered}
q^{o}-q^{c}<0, q^{o}+(N-1) q^{c}<0, p^{o}>\left|p^{c}\right|, \\
q^{o} \tilde{q}^{c} \neq q^{c} \tilde{q}^{o}, \\
\left(q^{o}-q^{c}\right)\left(\tilde{p}^{o}-\tilde{p}^{c}\right) \neq\left(\tilde{q}^{o}-\tilde{q}^{c}\right)\left(p^{o}-p^{c}\right), \\
\left(q^{o}+(N-1) q^{c}\right)\left(\tilde{p}^{o}+(N-1) \tilde{p}^{c}\right) \neq\left(\tilde{q}^{o}+(N-1) \tilde{q}^{c}\right)\left(p^{o}+(N-1) p^{c}\right) .
\end{gathered}
$$

Condition (25) ensures that $u_{i}$ is strictly concave in $a_{i}$ for all $N \geq 1$, whereas other conditions avoid knife-edge cases (e.g., division by zero).

This game is strategically equivalent to-and thus admits the same solution as-a simpler normalized game, in which $\omega \sim N(0, M(1, \rho))$,

$$
\begin{aligned}
u_{i}(a, \omega) & =\left(\omega_{i} p^{o}+\breve{\omega}_{-i} p^{c}\right) a_{i}+\frac{q^{o}}{2} a_{i}^{2}+q^{c} a_{i} \breve{a}_{-i} \\
v(a, \omega) & =\sum_{i=1}^{N}\left(\omega_{i} \tilde{p}^{o}+b\right) a_{i}+\left(\sum_{i=1}^{N} \sum_{j \neq i} \omega_{i} a_{j}\right) \tilde{p}^{c}+\left(\sum_{i=1}^{N} a_{i}^{2}\right) \tilde{q}^{o}+\left(\sum_{i=1}^{N} \sum_{j \neq i} a_{i} a_{j}\right) \tilde{q}^{c}
\end{aligned}
$$

and where $q^{o}, q^{c}, p^{o}, p^{c}, \tilde{q}^{o}, \tilde{q}^{c}, \tilde{p}^{o}, \tilde{p}^{c}$ are the same as in the original game. Therefore, we can focus on the analysis of the normalized game.

Given the greater complexity of the present setting, we proceed slightly differently from the previous section. We still rely on affine contracts for certification; however, instead of calculating the optimal contract parameters in closed form, we invoke an envelope theorem. We show that an optimal symmetric affine contract implements an obedient allocation rule, thereby certifying both its global optimality in the dual problem and the optimality of the corresponding measure in the primal problem.

Formally, observe that the players' marginal payoffs are linear:

$$
\dot{u}_{i}(a, \omega)=\omega_{i} p^{o}+\breve{\omega}_{-i} p^{c}+q^{o} a_{i}+q^{c} \breve{a}_{-i},
$$

and the dual payoff function is

$$
w(a, \omega, \lambda)=v(a, \omega)-\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right)\left(\omega_{i} p^{o}+\breve{\omega}_{-i} p^{c}+q^{o} a_{i}+q^{c} \breve{a}_{-i}\right) .
$$

Given the linear-quadratic nature of the environment, we conjecture that an optimal allocation rule is linear in the state. As this allocation rule must be a best-response given the dual payoff (30), we posit a certificate that is affine and symmetric across players:

$$
\lambda_{i}\left(a_{i}\right)=-\left(x a_{i}+x_{0}\right) .
$$

With this certificate, the dual payoff function can be written as

$$
\begin{aligned}
w\left(a, \omega, x, x_{0}\right) & \sim \omega^{T} M\left(\tilde{p}^{o}+x p^{o}, \tilde{p}^{c}+x p^{c}\right) a+a^{T} M\left(\tilde{q}^{o}+x q^{o}, \tilde{q}^{c}+x q^{c}\right) a \\
& +\left(b+x_{0}\left(q^{o}+(N-1) q^{c}\right)\right) \breve{a},
\end{aligned}
$$

where we omitted an action-independent term $x_{0}\left(p^{o}+(N-1) p^{c}\right) \sum_{i=1}^{N} \omega_{i}$. Because of (29) and $\mathbb{E}[\omega]=0$, any incentive-compatible expected individual action must equal zero, and therefore we conjecture that the coefficient in front of $\breve{a}$ equals zero and thus

$$
\begin{aligned}
x_{0}^{*} & =-\frac{b}{q^{o}+(N-1) q^{c}}, \\
w\left(a, \omega, x, x_{0}^{*}\right) & \sim \omega^{T} M\left(\tilde{p}^{o}+x p^{o}, \tilde{p}^{c}+x p^{c}\right) a+a^{T} M\left(\tilde{q}^{o}+x q^{o}, \tilde{q}^{c}+x q^{c}\right) a \\
& \sim \omega^{T} M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) a+a^{T} M\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right) a .
\end{aligned}
$$

where we defined

$$
\begin{aligned}
& \bar{q}^{o}(x) \triangleq \tilde{q}^{o}+x q^{o}, \bar{q}^{c}(x) \triangleq \tilde{q}^{c}+x q^{c} \\
& \bar{p}^{o}(x) \triangleq \tilde{p}^{o}+x p^{o}, \bar{p}^{c}(x) \triangleq \tilde{p}^{c}+x p^{c}
\end{aligned}
$$

For the optimal allocation rule $a^{*}(\omega)$ to maximize (32), $M\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right)$ must be negative semi-definite; otherwise, the value of the dual payoff diverges to $+\infty$.

Lemma 1. $M\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right)$ is negative semi-definite if and only if

$$
x \geq \underline{x} \triangleq \max \left\{-\frac{\tilde{q}^{o}-\tilde{q}^{c}}{q^{o}-q^{c}},-\frac{\tilde{q}^{o}+(N-1) \tilde{q}^{c}}{q^{o}+(N-1) q^{c}}\right\} .
$$

$M\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right)$ is negative definite if and only if $x>\underline{x}$.
If $x>\underline{x}, M\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right)$ is negative definite, and the optimal best-response is

$$
a^{*}(\omega, x)=-\frac{1}{2} M^{-1}\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right) M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) \omega .
$$

The resulting expected value of the dual payoff as a function of $x$ is


Information Design in Smooth Games

$$
\begin{aligned}
W(x) & \triangleq \mathbb{E}\left[w\left(a^{*}(\omega, x), \omega, x, x_{0}^{*}\right)\right] \\
& =-\frac{1}{4} \mathbb{E}\left[\omega^{T} M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) M^{-1}\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right) M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) \omega\right] .
\end{aligned}
$$

Lemma 2. $\lim _{x \downarrow \underline{x}} W(x)=\lim _{x \uparrow+\infty} W(x)=+\infty$, and $\min _{x \geq \underline{x}} W(x)$ exists.
By Lemma 2, the dual payoff admits a minimum at some $x^{*}>\underline{x}$. We show that this $x^{*}$ certifies an optimal information structure.

Definition 5. (Linear Disclosure) For $R=\left(r_{1}, \ldots, r_{N}\right) \in \mathbb{R}^{N \times N}$, an $R$-linear disclosure is an information structure such that for all $i$ and $\omega$,

$$
s_{i}=r_{i}^{T} \omega=\sum_{j=1}^{N} r_{i, j} \omega_{j} .
$$

A linear disclosure informs player $i$ about a linear statistic of the state with the responsiveness weights $r_{i}$. When $N=1$, a linear disclosure corresponds to no disclosure if $r=0$, and to full disclosure otherwise. When $N \geq 2$ and $R$ has full rank, each signal is imperfectly informative, yet the full signal profile perfectly reveals the state. In that case, like a Gaussian coupling, a linear disclosure spreads complete state information across players. When the state is normally distributed, a linear disclosure is a Gaussian information structure.

Theorem 4. (Optimality of Linear Disclosure) In a normalized symmetric linearquadratic Gaussian game with a multidimensional state, an $R^{*}$-linear disclosure is optimal, where

$$
R^{*}=-\frac{1}{2} M^{-1}\left(\bar{q}^{o}\left(x^{*}\right), \bar{q}^{c}\left(x^{*}\right)\right) M\left(\bar{p}^{o}\left(x^{*}\right), \bar{p}^{c}\left(x^{*}\right)\right),
$$

and $x^{*}$ is any point in $\arg \min _{x>\underline{x}} W(x)$. In the optimal equilibrium, $a_{i} \equiv s_{i}$. The induced measure $\pi \in \Delta(A \times \Omega)$ is uniquely optimal.

Proof. By Lemma 2, the dual payoff admits a minimum at some $x^{*}>\underline{x}$. Consider any such $x^{*}$. By (35), the dual best-response is $a^{*}\left(\omega, x^{*}\right)=R^{*} \omega$; hence, this allocation rule is implementable by incentives, by the contract $\lambda_{i}\left(a_{i}\right)=-\left(x^{*} a_{i}+x_{0}^{*}\right)$. The allocation rule is symmetric in $i$ and linear in $\omega$. Recall that

$$
W(x)=\mathbb{E}\left[v\left(a^{*}(\omega, x), \omega\right)+\sum_{i=1}^{N}\left(x a_{i}^{*}(\omega, x)+x_{0}^{*}\right) \dot{u}_{i}\left(a^{*}(\omega, x), \omega\right)\right] .
$$

Since $a^{*}(\omega, x)$ is an interior maximizer and $x$ is an interior minimizer of the dual payoff, it follows by the Envelope Theorem that

$$
\left.\frac{d W}{d x}\right|_{x=x^{*}}=\mathbb{E}\left[\sum_{i=1}^{N} a_{i}^{*}\left(\omega, x^{*}\right) \dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right)\right]=0 .
$$

By symmetry in $i$, it follows that for all $i$,

$$
\mathbb{E}\left[a_{i}^{*}\left(\omega, x^{*}\right) \dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right)\right]=0 .
$$

Because $a_{i}^{*}\left(\omega, x^{*}\right)$ is linear in $\omega$ and the components of $\omega$ are jointly normally distributed, $a_{i}^{*}\left(\omega, x^{*}\right)$ and $\dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right)$ are also jointly normally distributed. Furthermore, $\mathbb{E}\left[\dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right)\right]=0$, and, by (39), $a_{i}^{*}\left(\omega, x^{*}\right)$ and $\dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right)$ are uncorrelated, and thus independent. Therefore, for all $i$,

$$
\mathbb{E}\left[\dot{u}_{i}\left(a^{*}\left(\omega, x^{*}\right), \omega\right) \mid a_{i}^{*}\left(\omega, x^{*}\right)\right]=0 .
$$

Therefore, the allocation rule $a^{*}\left(\omega, x^{*}\right)$ is implementable by information, by $R^{*}$-linear disclosure. By Theorem 1, the optimality follows.

Finally, as the dual agent's best response to $\lambda\left(x^{*}, x_{0}^{*}\right)$ is unique, it follows from Proposition 1 that the corresponding measure is uniquely optimal. $\square$

Theorem 4 shows that an optimal information structure can be found within a simple class of symmetric noise-free Gaussian information structures (cf. Bergemann et al. (2015)). Moreover, its parameters can be identified by solving a one-dimensional minimization of $W(x)$.

In contrast with the common-state setting of Section 5, the optimal measure is unique here. Formally, in that setting the optimal certificate leaves the dual agent indifferent among many action profiles, whereas in this interdependent-value environment it yields a single best response. Conceptually, having one state component per player introduces richer uncertainty, which in turn fully determines the optimal recommendations.


Information Design in Smooth Games

## 7 Extensions

Bounded Action Spaces We have assumed that action spaces are unbounded, to avoid corner solutions and simplify the exposition. However, the certification approach can be easily extended to games with upper and lower bounds on each player's action space, as demonstrated in Appendix B. The primal problem formulation remains the same, except that the boundary first-order conditions must be inequalities rather than equalities. The dual problem formulation is identical, with the added sign constraints on the contract functions evaluated at the boundary actions.

Infinite Economies Our analysis considers games with a finite number of players. It does not directly cover games with a continuum of players, as considered in the literature on infinite economies, e.g., by Angeletos and Pavan (2007) and Bergemann and Morris (2013), even though we anticipate that an analogous certification approach, suitably extended, would work in those settings as well.

Nonetheless, our analysis enables determining an optimal information structure in a game with a fixed number of players and then examining its limit as the number of players approaches infinity. Our results suggest that in many quadratic economies with a normally distributed state, Gaussian information structures are certifiably optimal and an optimal aggregate behavior is a deterministic function of a state. With a finite number of players, the latter condition necessitates the designer to either not introduce extraneous noise at all or to ensure that the noises are carefully coupled. With an infinite number of players, this condition can be met by adding independent noises and relying on the law of large numbers; the optimal aggregate behavior can be ensured by the large population size rather than by noise correlation. However, our analysis suggests that even in infinite economies, alternative information structures such as targeted disclosure may also be optimal, and robustly so.

General Smooth Games The tractability of the certification approach in concave games hinges on the capacity of first-order conditions to succinctly represent players' incentives. In general smooth games, i.e., games in which each player's payoff is continuously differentiable in their own action, these first-order conditions may be insufficient, as they can select a suboptimal local maximum or even a minimum in a player's payoff. However, they remain necessary, which enables a straightforward extension of our analysis: in a smooth but not necessarily concave information-design problem, one constructs a relaxed primal problem that features only first-order conditions. This problem can be solved using the certification approach. In the final step, one verifies whether players obey the recommendation of the information structure found. If so, this information structure solves the original information-design problem.


Information Design in Smooth Games

## 8 Applications

In this section, we illustrate the developed machinery with three economic applications. ${ }^{11}$

[^9]


Information Design in Smooth Games

### 8.1 Persuading Investors

An important question in securities regulation is how much disclosure should be required from firms seeking capital (e.g., Carvajal, Rostek, and Sublet (2018)). Although many factors matter, in this section we focus on two. On the benefit side, greater transparency helps investors reduce risk and make better decisions; on the cost side, it intensifies cream-skimming, as investors herd into the same projects and crowd out investment gains. ${ }^{12}$ We characterize the optimal balance between these forces within our framework and show that the resulting solution aligns with key venture-capital practices.

Formally, we consider an investment game in the spirit of Angeletos and Pavan (2007) and Bergemann and Morris (2013). There are $N \geq 2$ investors, who simultaneously decide how much to invest in a project, $A_{i}=\mathbb{R}$. The profitability of the project is uncertain; it depends on the unknown project quality $\theta \in \mathbb{R}$ and on the total amount of investment $\breve{a} \triangleq \sum_{i=1}^{N} a_{i}$. The ex post payoff of player $i$ is

$$
u_{i}(a, \theta)=(\theta-r \breve{a}) a_{i}-c a_{i},
$$

where $r>0$ is the congestion parameter and $c>0$ is the opportunity cost of investment. As $r>0$, the project features decreasing returns to scale, i.e., its average profitability decreases in the total investment. We can conveniently rewrite payoff (41) as

$$
u_{i}(a, \omega)=r(\omega-\breve{a}) a_{i}
$$

where the state $\omega \in \Omega$ is a normalized project quality defined as $\omega \triangleq(\theta-c) / r$.
Given (42), for any belief $\mu \in \Delta\left(A_{-i} \times \Omega\right)$, the player $i$ 's best response can be found via the first-order condition to equal

$$
a_{i}^{*}(\mu)=\mathbb{E}_{\mu}\left[\frac{\omega-\breve{a}_{-i}}{2}\right],
$$

[^10]where $\breve{a}_{-i} \triangleq \sum_{j \neq i} a_{j}$, so that the best response linearly increases in the normalized quality expectation and linearly decreases in the expected amount of total investment made by other players. The players' actions are thus strategic substitutes.

The information designer has full control over the information regarding the project quality and can privately convey it to each player, thereby persuading that player to invest more or less. The designer aims to maximize the total profits generated by the project with her ex post payoff being ${ }^{13}$

$$
v(a, \omega)=\sum_{i=1}^{N}(\omega-\breve{a}) a_{i}=(\omega-\breve{a}) \breve{a}=\omega \breve{a}-\breve{a}^{2} .
$$

Investment Control If the designer could control the individual investment directly, then she would set the total investment to respond to the state as $\breve{a}^{F B}(\omega)=\omega / 2$. As we later show, this first-best allocation rule cannot be implemented via pure information control; however, it can be roughly approximated.

No Disclosure and Full Disclosure Two other natural benchmarks are the cases of no disclosure and full disclosure. In these cases, the designer's payoff can be easily computed as:

$$
\begin{aligned}
v^{N D} & =\frac{N}{(N+1)^{2}} \mathbb{E}^{2}[\omega], \\
v^{F D} & =\frac{N}{(N+1)^{2}}\left(\mathbb{E}^{2}[\omega]+\mathbb{V}[\omega]\right) .
\end{aligned}
$$

Comparing the designer's payoffs across the two benchmarks shows that full disclosure strictly dominates no disclosure. However, both information structures suffer from a scaling problem: as the number of players goes to infinity, the designer's payoff and thus the total project profit converges to zero. In the limit, the individual rent as well as the total profit are dissipated. We now show that this problem can be mitigated by adopting an optimal information structure.

[^11]Optimal Information For any information structure, taking the ex-ante expectation of both sides of (43), applying the law of iterated expectations, and summing across players (cf. Bergemann et al. (2017)), we obtain that the expected total investment remains constant at

$$
\mathbb{E}[\breve{a}]=\frac{N}{N+1} \mathbb{E}[\omega]=\frac{N}{N+1} \frac{\mathbb{E}[\theta]-c}{r}>\frac{1}{2} \mathbb{E}[\omega]=\mathbb{E}\left[\breve{a}^{F B}(\omega)\right] .
$$

Consequently, whenever $\mathbb{E}[\omega]>0$, information control can never achieve the full-control benchmark. However, while the designer cannot affect the ex-ante total amount of investment, she can direct investment toward more productive projects. An optimal way to do so is:

Proposition 3. (Persuading Investment by Exclusivity) For any number of players, a 1-targeted disclosure is optimal.

Proof. The setting is an instance of the general setting (11-12) with $h=N, q^{o}=-1$, $q^{c}=-N, p^{o}=0$, and $p^{c}=-N^{2}$. The formula (20) for an optimal $k^{*}$ becomes:

$$
k^{*}=\frac{q^{o}\left(h q^{o}-p^{o}\right)}{q^{o}\left(2 p^{c}-h q^{c}\right)-p^{o} q^{c}} N=\frac{-1(-N-0)}{-1\left(-2 N^{2}+N^{2}\right)-0} N=\frac{1}{N} N=1,
$$

and thus by Theorem 2, 1-targeted disclosure is optimal. $\square$

Proposition 3 shows that an optimal information structure takes a very simple form: the designer designates a single player and provides him with exclusive and full access to information. This player makes fully informed decisions. All other players invest the same amount regardless of the project quality. The same information structure is optimal for any number of investors, congestion and cost parameters, and project quality distribution.

The optimal designer's payoff equals

$$
v^{*}=\mathbb{E}\left[\left(\omega-\frac{\omega}{2}-\frac{N-1}{2(N+1)} \mathbb{E}[\omega]\right)\left(\frac{\omega}{2}+\frac{N-1}{2(N+1)} \mathbb{E}[\omega]\right)\right]=\frac{N}{(N+1)^{2}} \mathbb{E}^{2}[\omega]+\frac{1}{4} \mathbb{V}[\omega] .
$$

As such, optimal information design avoids rent dissipation as the number of players grows to infinity. The project's total profit always stays above $\mathbb{V}[\omega] / 4$ and converges to this level as $N \rightarrow \infty$. The limiting payoff increases with the variance of project quality: under the optimal information structure, just as under full disclosure, riskier projects generate higher realized profits even when their expected quality is the same. ${ }^{14}$


Information Design in Smooth Games

Interestingly, this asymmetric treatment of investors mirrors common venture-capital practice: firms cultivate close relationships with a small set of venture capitalists who learn more about the firm than the broader pool of potential investors (e.g., Bernstein, Giroud, and Townsend (2016)). Although several explanations could account for this pattern, our results highlight its informational benefit: by creating a sharp information asymmetry among investors, it curbs congestion and raises both investment profits and investor welfare.


Information Design in Smooth Games

### 8.2 Polarizing Predictions

Numerous studies document societal polarization, with individuals reaching sharply divergent views of the same issues (e.g., Alesina, Miano, and Stantcheva (2020)). Conventional explanations stress behavioral biases or naïveté, yet recent evidence shows that most people can reliably distinguish fake news from accurate reporting (Angelucci and Prat (2024)). This tension prompts a sharper question: To what extent can polarization arise among fully rational agents solely because they receive different information, and which information structure generates the greatest polarization?

We address this question in a prediction game, in which multiple players try to predict a common underlying state. The state is one-dimensional $\omega \in \mathbb{R}$ and distributed

[^12]according to the prior $\mu_{0}$. There are $N \geq 2$ players, each choosing a prediction $a_{i} \in$ $A_{i}=\mathbb{R}$. The ex post payoff to player $i$ is
$$
u_{i}(a, \omega)=-\left(a_{i}-\omega\right)^{2} .
$$
Because each player's payoff depends only on his own prediction and the state, for any given belief $\mu \in \Delta\left(A_{-i} \times \Omega\right)$, player $i$ 's best prediction is simply the posterior expectation of the state:
$$
a_{i}^{*}(\mu)=\mathbb{E}_{\mu}[\omega] .
$$

We focus on the question of inducing maximal polarization of the players' predictions as measured by the pairwise squared sum:

$$
v(a, \omega)=\sum_{i, j}\left(a_{i}-a_{j}\right)^{2} .
$$

Given the payoff, the designer benefits from sending private signals: any public information structure, including full disclosure or no disclosure, leads the players to make the same predictions and consequently minimizes the designer's objective. An optimal information structure, on the one hand, must provide some state information to move players' predictions and, on the other hand, should heterogeneously obfuscate the information to counteract truth drifting.

Proposition 4. (Polarizing by Segregation) Let the number of players $N$ be even. Then, an $N / 2$-targeted disclosure is optimal.

Proof. The setting is an instance of the general setting (11-12) with $h=0, q^{o}=-1$, $q^{c}=0, p^{o}=2 N^{2}$, and $p^{c}=-2 N^{2}$. The formula (20) for an optimal $k^{*}$ becomes:

$$
k^{*}=\frac{q^{o}\left(h q^{o}-p^{o}\right)}{q^{o}\left(2 p^{c}-h q^{c}\right)-p^{o} q^{c}} N=\frac{-1\left(-2 N^{2}\right)}{-1\left(-4 N^{2}\right)-0} N=\frac{N}{2},
$$

and thus by Theorem 2, $N / 2$-targeted disclosure is optimal. $\square$

Proposition 4 shows that, irrespective of the prior, expectation polarization can be achieved by a strikingly simple information structure that informs only half of the population.

This finding aligns with earlier work in information design. In a two-player, binarystate model, Arieli et al. (2021) characterized the feasible joint belief distributions and reached a parallel conclusion: to maximize polarization of posterior expectations, it is optimal to inform one of the two players. Proposition 4 extends this insight to any number of players and states. Arieli and Babichenko (2024) continue the study of polarization of posterior beliefs. They show that for any even number of players, informing half of them is optimal, because it achieves appropriate statistical bounds. Interestingly, even though belief polarization and expectation polarization differ once the state space is larger than binary, and the underlying arguments are not interchangeable, both problems share the same optimal information structure.

These results contribute to the debate over the sources of societal polarization. Recent studies show that political news consumption is sharply segregated, either because it flows through endogenous peer networks (Bowen et al. (2023)) or because it is curated by social-media algorithms (Braghieri et al. (2024)). As a result, different groups may learn about different topics. This kind of segregation-learning about different topics, rather than learning different facts about the same topic-can maximally polarize a society.


Information Design in Smooth Games

### 8.3 Informing Competitive Pricing

The information design machinery is useful to understand and guide the design of digital platforms and, more generally, of algorithmic information processing (e.g., Bergemann and Bonatti (2024), Ichihashi and Smolin (2025)). In this section, we view the designer as a platform that knows demand conditions better than firms do, thanks to a larger, more recent sales dataset and superior analytics. The platform can privately convey this information to each firm by granting it access to personalized data analysis or direct price recommendations, thereby persuading the firm regarding its pricing decisions. The
platform aims to maximize a weighted average of consumer and producer surplus. We characterize the information structure such a designer would optimally design and the resulting price behavior.

Formally, we consider a differentiated duopoly game in the spirit of Vives (1984) and Gal-Or (1985). ${ }^{15}$ There are two firms that operate in the market. Each firm sells a single product and competes in price with its opponent, so action $a_{i}$ is the price set by firm $i$. Demand is ex ante symmetric across firms and is linear in prices and demand shocks:

$$
q_{i}(a, \omega)=\omega_{i}-a_{i}+\eta a_{-i},
$$

where $\omega_{i}$ captures an individual demand shock for firm $i$ 's product and $\eta$ is a crossprice sensitivity that satisfies $|\eta| \in(0,1)$, so that $-1<\eta<0$ corresponds to the case of complementary products, whereas $0<\eta<1$ corresponds to the case of substitute products.

The state is two-dimensional $\omega=\left(\omega_{1}, \omega_{2}\right) \in \mathbb{R}^{2}$ and comprises the individual demand shocks. The shocks are independently and identically distributed according to a normal distribution, $\omega_{i} \sim N\left(\omega_{0}, \sigma^{2}\right)$. ${ }^{16}$ The firms do not know the state but can be informed about it by the designer.

The firms have quadratic costs of production so that their profits are

$$
u_{i}(a, \omega)=a_{i} q_{i}(a, \omega)-c q_{i}(a, \omega)^{2} .
$$

The resulting ex post values of consumer surplus and producer surplus are

$$
\begin{aligned}
& C S(a, \omega)=\frac{a_{1}^{2}}{2}+\frac{a_{2}^{2}}{2}-\eta a_{1} a_{2}-a_{1} \omega_{1}-a_{2} \omega_{2}, \\
& P S(a, \omega)=u_{1}(a, \omega)+u_{2}(a, \omega) .
\end{aligned}
$$

[^13]The designer's payoff is a convex combination of consumer and producer surpluses, with $\delta \in[0,1]$ representing the weight assigned to the consumer surplus:

$$
v(a, \omega)=\delta C S(a, \omega)+(1-\delta) P S(a, \omega) .
$$

The optimal designer's choices in the extreme cases $\delta=1$ and $\delta=0$ correspond to consumer-optimal and producer-optimal information structures, respectively, whereas the choice in the case $\delta=1 / 2$ corresponds to the socially efficient information structure. As the welfare weight $\delta$ spans the interval [0,1], the corresponding solutions span the Pareto frontier in the space of consumer and producer surpluses.

Price Control We begin the analysis by studying a hypothetical scenario in which the designer can directly control the prices set by the firms. This scenario constitutes a first-best benchmark; it provides an upper bound on the designer's payoff and illustrates the designer's preferred pricing.

This problem admits a solution only if $\delta$ is not excessively high: there is a threshold value $\bar{\delta} \in(0,1)$, such that if $\delta>\bar{\delta}$, then the designer can arbitrarily increase her payoff by setting arbitrarily large negative prices, because the monetary transfer to consumers outweighs any allocation inefficiency. In contrast, if $\delta<\bar{\delta}$, then the designer's problem is well-behaved; it is concave and admits a unique solution that can be found by first-order conditions to (51):

$$
a_{i}^{F B}(\omega)=r_{o}^{F B} \omega_{i}+r_{c}^{F B} \omega_{-i},
$$

where $r_{o}^{F B}$ and $r_{c}^{F B}$ measure the optimal responsiveness of each firm's price to its own demand shock and to the shock of its competitor (explicit formulas appear in Section A.6).

No Disclosure and Full Disclosure The designer does not control prices directly but rather indirectly through demand information she supplies to firms. Before deriving the generally optimal policy, it is instructive to analyze two information benchmarks:
no disclosure and full disclosure.
Under no disclosure, the firms' beliefs stay at the prior, and the equilibrium prices satisfy the first-order conditions derived from (48). In equilibrium, each firm sets a price

$$
a_{i}^{N D}=\frac{1+2 c}{2(1+c)-\eta(1+2 c)} \omega_{0} .
$$

Lacking demand information, firms fix their prices at a level proportional to the expected demand.

In contrast, under full disclosure, the demand shocks are always commonly known. In equilibrium, each firm responds linearly to the shocks perfectly anticipating the price of its opponent:

$$
a_{i}^{F D}(\omega)=r_{o}^{F D} \omega_{i}+r_{c}^{F D} \omega_{-i}=\frac{2(1+c)(1+2 c)}{4(1+c)^{2}-\eta^{2}(1+2 c)^{2}} \omega_{i}+\frac{\eta(1+2 c)^{2}}{4(1+c)^{2}-\eta^{2}(1+2 c)^{2}} \omega_{-i} .
$$

In a sense, this behavior enriches price-setting under no disclosure. If $\omega_{1}=\omega_{2}=\omega_{0}$, then prices are the same as those under no disclosure, $a_{i}^{F D}\left(\omega_{0}, \omega_{0}\right)=a_{i}^{N D}$. If $\omega_{1} \neq \omega_{2}$, then demand is asymmetric across firms, and prices are adjusted to reflect competitive advantages. However, the prices average to the prices under no disclosure, $\mathbb{E}\left[a_{i}^{F D}(\omega)\right]=$ $a_{i}^{N D}$.

Optimal Information The choice of any of the extreme information structures has drawbacks. Providing no disclosure misses the opportunity to strengthen the link between demand and allocation and thus potentially limits efficiency. Providing full disclosure may exacerbate competition and dissipate firm profits. Instead, and as a direct consequence of Theorem 4, the optimal information structure is partially informative and takes a form of a linear disclosure.

Proposition 5. (Managing Competition by Linear Statistics) There is a parameter $\hat{\delta}(\eta, c)$ such that if $\delta \neq \hat{\delta}(\eta, c)$, an optimal direct information structure exists, is unique,
and recommends for some coefficients $a_{0}^{*}, r_{o}^{*}$ and $r_{c}^{*}$ prices

$$
a_{i}(\omega)=a_{0}^{*}+r_{o}^{*} \omega_{i}+r_{c}^{*} \omega_{-i} .
$$


Information Design in Smooth Games

The uniquely optimal direct information structure treats the firms symmetrically. Under it, each firm can only infer a linear combination of the individual demand shocks, and thus not able to infer whether the recommendation of a given price stems from its own demand conditions or the conditions of its competitor. Generically, $r_{c}^{*} \neq r_{c}^{F D}$ and $r_{o}^{*} \neq r_{o}^{F D}$, so providing full disclosure is suboptimal; instead, each firm receives personalized information that differs from its competitor's. This finding suggests that restricting disclosure to be public, as often done in the oligopoly literature on information sharing (Vives (1990, 1999)), though natural in some contexts, may entail losses.

Proposition 5 enables the calculation of optimal information structures. It can be shown that the optimal information structure may exhibit a discontinuous change with respect to the consumer weight $\delta$ at $\hat{\delta}(\eta, c)$, accompanied by a change in equilibrium price volatility and correlations. ${ }^{17}$ Intuitively, the pricing induced under the optimal information structure seeks to approximate its first-best counterpart under direct price control. However, lacking the ability to enforce prices directly, the designer chooses an information regime that better approximates it.

This observation yields two insights. First, patterns of price volatility and correlations can serve as effective diagnostics for discerning which side of the market, consumers or producers, the information structure is designed to favor. Second, even a slight shift in the design objective can trigger a dramatic change in the information provision and in resulting market behavior. This, in turn, can serve as a cautionary tale, underscoring the potential for market instability introduced by algorithms, which may pursue shifting objectives and adapt their policies more rapidly than human decision makers.

[^14]


Information Design in Smooth Games

## 9 Conclusion

In this paper, we introduced a certification approach to solving information-design problems in games and demonstrated its effectiveness and tractability in symmetric linearquadratic settings. In doing so, we provided theoretical justification for the use of targeted disclosure, Gaussian coupling, and linear disclosure. Our findings shed light on disclosure practices that guide investment, on the socially efficient control of information in markets, and on the limits of Bayesian polarization.

Our analysis lays the groundwork and offers tools for studying information design in general smooth games. We see at least three promising avenues for further research. First, our theoretical framework could be applied to other important economic settings, such as contests, public-good provision, and labor or financial markets. Second, one could develop more general sufficient conditions for strong duality in non-compact settings. Third, the framework could be expanded to incorporate elements like information elicitation, information spillovers across players, and dynamic interaction. We expect all of these extensions to be feasible within the approach we have outlined.


Information Design in Smooth Games

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Information Design in Smooth Games

## A Appendix


Information Design in Smooth Games

## A. 1 Formalism Omitted in Section 2

Derivation of First-Order Condition (4) For $\varepsilon>0$, define

$$
\begin{aligned}
& \phi_{i \varepsilon}^{+}\left(a_{i}, a_{-i}, \omega\right) \triangleq \frac{u_{i}\left(a_{i}+\varepsilon, a_{-i}, \omega\right)-u_{i}\left(a_{i}, a_{-i}, \omega\right)}{\varepsilon}, \\
& \phi_{i \varepsilon}^{-}\left(a_{i}, a_{-i}, \omega\right) \triangleq \frac{u_{i}\left(a_{i}, a_{-i}, \omega\right)-u_{i}\left(a_{i}-\varepsilon, a_{-i}, \omega\right)}{\varepsilon} .
\end{aligned}
$$

Definition 6. (Admissible Equilibria) An equilibrium is admissible, if for any player i and equilibrium belief $\mu_{i} \in \Delta\left(A_{-i} \times \Omega\right)$, there exists a best response $a_{i}^{*} \in A_{i}, \Delta>0$, and $\psi:$ $A_{-i} \times \Omega \rightarrow \mathbb{R}$ such that $\int_{A_{-i} \times \Omega} \psi\left(a_{-i}, \omega\right) d \mu_{i}<+\infty$ and for all $\varepsilon \in(0, \Delta),\left(a_{-i}, \omega\right) \in A_{-i} \times \Omega$, $\left|\phi_{i \varepsilon}^{+}\left(a_{i}^{*}, a_{-i}, \omega\right)\right| \leq \psi\left(a_{-i}, \omega\right)$ and $\left|\phi_{i \varepsilon}^{-}\left(a_{i}^{*}, a_{-i}, \omega\right)\right| \leq \psi\left(a_{-i}, \omega\right)$.

This admissibility effectively requires players' payoff differences to be locally well behaved at the best responses. It ensures the interchangeability of operators in (4) and is satisfied in all equilibria that we characterize. (If $A$ and $\Omega$ were compact, then admissibility would be trivially satisfied whenever each $u_{i}(a, \omega)$ was continuously differentiable in $a_{i}$. For general $A$ and $\Omega$, any equilibrium is admissible if, for instance, each $u_{i}(a, \omega)$ is polynomial.)

Lemma 3. (First-Order Approach) Under Assumption 1, for any player $i$ and admissible equilibrium belief $\mu_{i} \in \Delta\left(A_{-i} \times \Omega\right), a_{i}$ is player $i$ 's best response if and only if condition (4) is satisfied.

Proof. For any $i$ and admissible equilibrium belief $\mu_{i}, a_{i}^{*}$ satisfies the following condition:

$$
\int_{A_{-i} \times \Omega} u_{i}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} \geq \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}, \quad \forall a_{i} \in A_{i} .
$$

Under Assumption 1, condition (53) is satisfied if and only if for any $\varepsilon>0$, the following holds:

$$
\begin{aligned}
& \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{+}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} \leq 0 \\
& \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{-}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} \geq 0
\end{aligned}
$$

Moreover, $\phi_{i \varepsilon}^{+}$is decreasing in $\varepsilon$ and $\phi_{i \varepsilon}^{-}$is increasing in $\varepsilon$; thus, conditions (54), (55) are equivalent to

$$
\begin{aligned}
0 \geq \lim _{\varepsilon \rightarrow 0} \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{+}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} & =\lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}^{*}+\varepsilon, a_{-i}, \omega\right)-u_{i}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} \\
& =\left.\frac{\partial^{+}}{\partial a_{i}} \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}\right|_{a_{i}=a_{i}^{*}}
\end{aligned}
$$

$$
\begin{aligned}
0 \leq \lim _{\varepsilon \rightarrow 0} \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{-}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i} & =\lim _{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}^{*}, a_{-i}, \omega\right)-u_{i}\left(a_{i}^{*}-\varepsilon, a_{-i}, \omega\right) \mathrm{d} \mu_{i} \\
& =\left.\frac{\partial^{-}}{\partial a_{i}} \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}\right|_{a_{i}=a_{i}^{*}}
\end{aligned}
$$

By Lebesgue's dominated convergence theorem (Theorem 11.32, Rudin (1976)), because belief $\mu_{i}$ arises in an admissible equilibrium, the following holds:

$$
\begin{aligned}
& \lim _{\varepsilon \rightarrow 0} \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{+}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}=\int_{A_{-i} \times \Omega} \lim _{\varepsilon \rightarrow 0} \phi_{i \varepsilon}^{+}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}=\int_{A_{-i} \times \Omega} \dot{u}_{i}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}, \\
& \lim _{\varepsilon \rightarrow 0} \int_{A_{-i} \times \Omega} \phi_{i \varepsilon}^{-}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}=\int_{A_{-i} \times \Omega} \lim _{\varepsilon \rightarrow 0} \phi_{i \varepsilon}^{-}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}=\int_{A_{-i} \times \Omega} \dot{u}_{i}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i},
\end{aligned}
$$

and thus

$$
\left.\frac{\partial}{\partial a_{i}} \int_{A_{-i} \times \Omega} u_{i}\left(a_{i}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}\right|_{a_{i}=a_{i}^{*}}=\int_{A_{-i} \times \Omega} \dot{u}_{i}\left(a_{i}^{*}, a_{-i}, \omega\right) \mathrm{d} \mu_{i}=0 .
$$ $\square$


Information Design in Smooth Games

## A. 2 Formalism Omitted in Section 3

Proof of Theorem 1 The first step is
Lemma 4. (Weak Duality) $V^{P} \leq V^{D}$.

Proof. Take any dual variables $(\lambda, \gamma)$ that satisfy the constraints of the dual problem (8). Take any measure $\pi$ that satisfies the constraints of the primal problem (5). Integrating both sides
of the dual constraints over $a \in A$ and $\omega \in \Omega$ against measure $\pi$ yields:

$$
\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi \leq \int_{A \times \Omega} \sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) \mathrm{d} \pi+\int_{A \times \Omega} \gamma(\omega) \mathrm{d} \pi=\int_{\Omega} \gamma(\omega) \mathrm{d} \mu_{0},
$$

where the equality follows because $\pi$ satisfies the primal constraints. The left-hand side of (56) is the value of the primal problem given measure $\pi$. The right-hand side of (56) is the value of the dual problem given dual variables $(\lambda, \gamma)$. As the inequality (56) holds for any allowed values of primal measure and dual variables, it also holds at the respective maximization and minimization limits. $\square$

Continuing with the proof of Theorem 1, take any primal measure $\pi$ implementable by information, i.e., that satisfies the constraints of the primal problem (5). If it is implementable by incentives, then there exist dual variables $\lambda$ that implement this measure in the dual problem (10), and

$$
\begin{aligned}
V^{D} & =\inf _{\lambda^{\prime} \in \times_{i} \mathcal{M}\left(A_{i}\right)} \mathbb{E}_{\mu_{0}}\left[\sup _{a \in A} w\left(a, \omega, \lambda^{\prime}\right)\right] \\
& \leq \mathbb{E}_{\pi}[w(a, \omega, \lambda)] \\
& =\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi-\int_{A \times \Omega} \sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) \mathrm{d} \pi \\
& =\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi \leq V^{P},
\end{aligned}
$$

where the first inequality follows from the implementability of $\pi$ in the dual problem and the last three steps follow from the feasibility of $\pi$ in the primal problem.

Furthermore, by Lemma $4, V^{D} \geq V^{P}$. Combining the two inequalities, we obtain

$$
V^{D}=\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi=V^{P},
$$

which proves the optimality of measure $\pi$.

Proof of Proposition 1 For any optimal measure $\pi$, the following holds:

$$
\begin{gathered}
\mathbb{E}_{\pi}[v(a, \omega)]=V^{P}, \\
\int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) \mathrm{d} \pi=0 \quad \forall i=1, \ldots, N, \text { measurable } A_{i}^{\prime} \subseteq A_{i} .
\end{gathered}
$$

As such, if the dual agent is offered contract $\lambda$ and plays according to an optimal measure then his expected payoff is

$$
\begin{aligned}
\mathbb{E}_{\pi}[w(a, \omega, \lambda)] & =\mathbb{E}_{\pi}\left[v(a, \omega)-\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega)\right] \\
& =\mathbb{E}_{\pi}[v(a, \omega)]-\mathbb{E}_{\pi}\left[\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega)\right]=V^{P} .
\end{aligned}
$$

That is, the dual agent obtains the same expected payoff of $V^{P}$ by playing according to an optimal measure, whether $\pi$ or $\pi^{\prime}$.

Moreover, if $\lambda$ certifies the optimality of measure $\pi$, then playing according to $\pi$ is a best response of the dual agent to contract $\lambda$. However, since, as shown above, $\pi^{\prime}$ delivers the same expected dual payoff as $\pi$, it follows that $\pi^{\prime}$ is also a best response of the dual agent to contract $\lambda$. Indeed, for every $\omega \in \Omega$ and $a \in A, w(a, \omega, \lambda) \leq \sup _{\tilde{a} \in A} w(\tilde{a}, \omega, \lambda)$. Since $\lambda$ certifies $\pi$, we have $V^{D}=V^{P}$, and since $\pi^{\prime}$ is optimal, $\mathbb{E}_{\pi^{\prime}}[w(a, \omega, \lambda)]=V^{P}$. Hence

$$
0=V^{D}-\mathbb{E}_{\pi^{\prime}}[w(a, \omega, \lambda)]=\int_{\Omega}\left(\sup _{a \in A} w(a, \omega, \lambda)-\int_{A} w(a, \omega, \lambda) \pi^{\prime}(d a \mid \omega)\right) \mu_{0}(d \omega)
$$

The integrand is nonnegative, so it must be equal to zero for $\mu_{0}$-almost every $\omega$. Therefore, for $\mu_{0}$-almost every $\omega$, the measure $\pi^{\prime}(\cdot \mid \omega)$ is supported on $\arg \max _{a \in A} w(a, \omega, \lambda)$, that is, $\pi^{\prime}$ is also implementable by incentives under contract $\lambda$.

Finally, as an optimal measure, $\pi^{\prime}$ satisfies the constraints of the primal problem. Hence, $\pi^{\prime}$ is implementable by information and by incentives; by definition, $\lambda$ certifies the optimality of $\pi^{\prime}$.

Proof of Claim 1 Assume that $\Omega$ and all $A_{i}$ are compact (metric) spaces, and that $v$ and each $\dot{u}_{i}$ are continuous on $X \triangleq \Omega \times A$. We first introduce an auxiliary dual problem in which
the multipliers are restricted to be continuous:

$$
\begin{aligned}
V^{D, c} \triangleq & \inf _{\gamma \in C(\Omega), \lambda_{i} \in C\left(A_{i}\right)} \int_{\Omega} \gamma(\omega) d \mu_{0} \\
& \text { s.t. } \quad \gamma(\omega)+\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) \geq v(a, \omega) \quad \forall(a, \omega) \in A \times \Omega
\end{aligned}
$$

Because every continuous function is measurable, the feasible set of $V^{D, c}$ is contained in the feasible set of $V^{D}$. Hence $V^{D} \leq V^{D, c}$. By weak duality, $V^{P} \leq V^{D}$. Therefore it is enough to prove that $V^{D, c}=V^{P}$.

Define $E \triangleq C(X), Y \triangleq C(\Omega) \times \prod_{i=1}^{N} C\left(A_{i}\right)$, each endowed with the sup norm. Since $X$ is compact metric, both are Banach spaces, and by the Riesz-Markov theorem (Rudin, 1987, Theorem 6.19) $E^{*}=M(X), Y^{*}=M(\Omega) \times \prod_{i=1}^{N} M\left(A_{i}\right)$, where $M(\cdot)$ denotes the finite signed Radon measures.

Define the continuous linear operator $T: Y \rightarrow E$ by

$$
T(\gamma, \lambda)(a, \omega) \triangleq \gamma(\omega)+\sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) .
$$

Also define the proper convex functions $\phi: E \rightarrow \mathbb{R} \cup\{+\infty\}$ and $\psi: Y \rightarrow \mathbb{R}$ by

$$
\phi(f) \triangleq\left\{\begin{array}{ll}
0, & \text { if } f(x) \geq v(x) \forall x \in X, \\
+\infty, & \text { otherwise }
\end{array} \quad \psi(\gamma, \lambda) \triangleq \int_{\Omega} \gamma(\omega) d \mu_{0}\right.
$$

Then,


Information Design in Smooth Games

$$
V^{D, c}=\inf _{y \in Y}\{\psi(y)+\phi(T y)\} .
$$

Now choose $M>\|v\|_{\infty}$ and let $y_{0}=(M, 0, \ldots, 0) \in Y$. Then $T y_{0} \equiv M>v$ on $X$, so $\phi$ is continuous at $T y_{0}$. Hence the Banach-space Fenchel-Rockafellar theorem applies and yields (Rockafellar, 1967, Theorems 1 and 3)

$$
V^{D, c}=\sup _{\pi \in E^{*}}\left\{-\psi^{*}\left(T^{*} \pi\right)-\phi^{*}(-\pi)\right\} .
$$

We now compute the two conjugates. First,

$$
\begin{aligned}
-\phi^{*}(-\pi) & =-\sup _{f \in E}\left\{\int_{X} f d(-\pi)-\phi(f)\right\} \\
& =\inf _{f \in C(X), f \geq v} \int_{X} f d \pi
\end{aligned}
$$

Therefore

$$
-\phi^{*}(-\pi)= \begin{cases}\int_{X} v d \pi, & \text { if } \pi \in M_{+}(X) \\ -\infty, & \text { otherwise }\end{cases}
$$

Indeed, if $\pi \in M_{+}(X)$, the infimum is attained at $f=v$. If $\pi \notin M_{+}(X)$, then by the Riesz-Markov theorem there exists $h \in C(X)$ with $h \geq 0$ and $\int_{X} h d \pi<0$; hence for $f_{n} \triangleq v+n h$ we have $f_{n} \geq v$ and

$$
\int_{X} f_{n} d \pi=\int_{X} v d \pi+n \int_{X} h d \pi \rightarrow-\infty .
$$

Second, identify $T^{*} \pi \in Y^{*}$ explicitly. For $\pi \in M(X)$, let $\pi_{0} \in M(\Omega)$ be the $\Omega$-marginal of $\pi$, and for each $i$ define a finite signed measure $\nu_{i} \in M\left(A_{i}\right)$ by

$$
\nu_{i}(B) \triangleq \int_{B \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) d \pi \quad \forall \text { Borel } B \subseteq A_{i}
$$

Since $\dot{u}_{i}$ is continuous on the compact set $X$, it is bounded, so $\nu_{i}$ is well defined. Moreover, for every $(\gamma, \lambda) \in Y$,

$$
\begin{aligned}
\left\langle T^{*} \pi,(\gamma, \lambda)\right\rangle & =\int_{X} \gamma(\omega) d \pi+\sum_{i=1}^{N} \int_{X} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) d \pi \\
& =\int_{\Omega} \gamma d \pi_{0}+\sum_{i=1}^{N} \int_{A_{i}} \lambda_{i} d \nu_{i}
\end{aligned}
$$

Hence, under the identification $Y^{*}=M(\Omega) \times \prod_{i=1}^{N} M\left(A_{i}\right)$,

$$
T^{*} \pi=\left(\pi_{0}, \nu_{1}, \ldots, \nu_{N}\right) .
$$

Now $\psi(\gamma, \lambda)=\int_{\Omega} \gamma d \mu_{0}$ depends only on $\gamma$, so its conjugate is

$$
\psi^{*}\left(\eta_{0}, \eta_{1}, \ldots, \eta_{N}\right)= \begin{cases}0, & \text { if } \eta_{0}=\mu_{0} \text { and } \eta_{i}=0 \forall i, \\ +\infty, & \text { otherwise }\end{cases}
$$

Therefore

$$
-\psi^{*}\left(T^{*} \pi\right)= \begin{cases}0, & \text { if } \pi_{0}=\mu_{0} \text { and } \nu_{i}=0 \forall i \\ -\infty, & \text { otherwise }\end{cases}
$$

Combining the two conjugate calculations gives

$$
V^{D, c}=\sup _{\pi \in M_{+}(X)}\left\{\int_{X} v d \pi \mid \pi_{0}=\mu_{0}, \nu_{i}=0 \forall i\right\} .
$$

Since $\pi_{0}=\mu_{0}$ and $\mu_{0}$ is a probability measure, every such $\pi$ has total mass one, so $\pi \in \Delta(X)$.
Finally, the condition $\pi_{0}=\mu_{0}$ is exactly the Bayes-plausibility constraint. Also, $\nu_{i}=0$ is equivalent, by the uniqueness part of the Riesz-Markov theorem, to

$$
\int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) d \pi=0 \quad \forall \text { Borel } A_{i}^{\prime} \subseteq A_{i}
$$

which is exactly the obedience constraint in the primal problem. Hence the feasible set above is precisely the primal feasible set, and therefore $V^{D, c}=V^{P}$. We conclude that

$$
V^{P} \leq V^{D} \leq V^{D, c}=V^{P},
$$

so $V^{D}=V^{P}$.