Information Design in Smooth Games

## A. 3 Formalism Omitted in Section 5

Game Normalization The game normalization is achieved by defining $\omega^{\prime}=\omega-\mathbb{E}[\omega], a_{i}^{\prime}=$ $a_{i}+\frac{\mathbb{E}[\omega]+b_{1}}{q^{o}+q^{c}}, b=h \mathbb{E}[\omega]+b_{2}-2\left(p^{o}+p^{c}\right) \frac{\mathbb{E}[\omega]+b_{1}}{q^{o}+q^{c}}$, and ignoring the strategically irrelevant additive terms, i.e., those in $u_{i}$ that do not depend on $a_{i}$ and those in $v$ that do not depend on $a$.

Proof of Theorem 2, continued. If $k^{*} \notin[0, N]$, then, since $q^{o} p^{c}>p^{o} q^{c}$, either (i) $p^{o}<h q^{o}$, or (ii) $\left(h q^{o}-p^{o}\right)\left(q^{o}+q^{c}\right)>2\left(p^{c} q^{o}-p^{o} q^{c}\right)$.

Indeed, if neither (i) or (ii) holds then it must be the case that $p^{o} \geq h q^{o}$, and $\left(h q^{o}-p^{o}\right)\left(q^{o}+\right.$ $\left.q^{c}\right) \leq 2\left(p^{c} q^{o}-p^{o} q^{c}\right)$. But in this case, because of our assumptions $q^{o}<0, q^{o}\left(h q^{o}-p^{o}\right) \geq 0$ and therefore,

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

Consequently, if $k^{*} \notin[0, N]$, then either (i) or (ii) holds.
Case (i). If (i) holds, then no disclosure is optimal, which can be certified by an affine contract with $\left(x, x_{0}\right)=\left(-h,-\frac{b}{q^{o}+q^{c}}\right)$. Under this contract, the dual payoff function is

$$
w(a, \omega)=x_{0} \omega+\left(p^{o}-h q^{o}\right) \check{a}+\left(p^{c}-h q^{c}\right) \bar{a}^{2}=x_{0} \omega+\frac{1}{N^{2}} a^{T} M\left(\left(p^{o}-h q^{o}\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\right) a,
$$

where $M(x, y)$ denotes the $N \times N$ matrix whose diagonal entries equal $x$ and off-diagonal entries equal $y$. By Theorem 1, no disclosure is optimal if $a=(0, \ldots, 0)$ maximizes $w(a, \omega)$ for all $\omega$. This happens if $M\left(\left(p^{o}-h q^{o}\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\right)$ is negative semidefinite (NSD). Note that matrix $M(x, y)$ has two distinct eigenvalues: $x+(N-1) y$ and $x-y$. Therefore, $M(x, y)$ is NSD if $x+(N-1) y \leq 0$ and $x-y \leq 0$ which for $M\left(\left(p^{o}-h q^{o}\right) N+p^{c}-h q^{c}, p^{c}-h q^{c}\right)$ corresponds to:

$$
\begin{aligned}
p^{o} & \leq h q^{o} \\
p^{o}+p^{c} & \leq h\left(q^{o}+q^{c}\right)
\end{aligned}
$$

Subcase 1: $h q^{c} \geq p^{c}$. In this case, NSD is equivalent to $p^{o}-h q^{o} \leq 0$. In case $q^{o} p^{c}>p^{o} q^{c}$ and $p^{o}<h q^{o}$, this is satisfied.

Subcase 2: $h q^{c}<p^{c}$. In this case, NSD is equivalent to $p^{o}-h q^{o}+p^{c}-h q^{c} \leq 0$. In case $q^{o} p^{c}>p^{o} q^{c}$ and $p^{o}<h q^{o}$, note that $p^{c}=-p^{o}-\eta$ and $q^{c}=-q^{o}-\varepsilon$ for some $\eta$ and $\varepsilon>0$. Because $q^{o} p^{c}>p^{o} q^{c}$, we have $-q^{o} \eta>-p^{o} \varepsilon$, which implies $\eta / \varepsilon>p^{o} / q^{o}>h$ because $p^{o}<h q^{o}<0$. Thus, $\eta>h \varepsilon$; equivalently, $p^{o}+p^{c}<h\left(q^{o}+q^{c}\right)$.

Case (ii). If (ii) holds, then full disclosure is optimal, which can be certified by an affine
contract with $\left(x, x_{0}\right)=\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}},-\frac{b}{q^{o}+q^{c}}\right)$. Under this contract, the dual payoff function is

$$
\begin{aligned}
w(a, \omega) & =x_{0} \omega+\left(2 h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) \omega \bar{a}+\left(p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)\right) \check{a}+\left(p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)\right) \bar{a}^{2} \\
& =x_{0} \omega+\frac{1}{N^{2}}\left(\left(2 h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) N \omega 1_{N}^{T} a+a^{T} M(x, y) a\right),
\end{aligned}
$$

where $1_{N}=(1, \ldots, 1)^{T}$ is an $N$-dimensional vector of ones, $x=N\left(p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)\right)+p^{c}+$ $q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)$ and $y=p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)$.

By Theorem 1, full disclosure is optimal if $a=\left(-\frac{\omega}{q^{o}+q^{c}}, \ldots,-\frac{\omega}{q^{o}+q^{c}}\right)$ maximizes $w(a, \omega)$ for all $\omega$. If $M(x, y)$ is NSD, then the maximizing $a$ can be found via F.O.C.:

$$
\left(2 h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) N \omega 1_{N}+2 M(x, y) a=0,
$$

with a solution $a=\left(-\frac{\omega}{q^{o}+q^{c}}, \ldots,-\frac{\omega}{q^{o}+q^{c}}\right)$. Therefore, it remains to establish that $M(x, y)$ is NSD, $x+(N-1) y \leq 0$ and $x-y \leq 0$ which corresponds to

$$
\begin{aligned}
p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) & \leq 0, \\
p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)+p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) & \leq 0 .
\end{aligned}
$$

Subcase 1: $p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right) \leq 0$ and $p^{o}+p^{c} \neq h\left(q^{o}+q^{c}\right)$. We show that the inequalities hold strictly. It suffices to show $p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)<0$; equivalently, $\left(h q^{o}-p^{o}\right)\left(q^{o}+q^{c}\right)>$ $2\left(p^{c} q^{o}-p^{o} q^{c}\right)$. In case $p^{c} q^{o}>p^{o} q^{c}$ and $\left(h q^{o}-p^{o}\right)\left(q^{o}+q^{c}\right)>2\left(p^{c} q^{o}-p^{o} q^{c}\right)$, this is clearly satisfied.

Subcase 2: $p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)>0$ and $p^{o}+p^{c} \neq h\left(q^{o}+q^{c}\right)$. We show that the inequalities hold strictly. It suffices to show $p^{o}+q^{o}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)+p^{c}+q^{c}\left(h-\frac{2\left(p^{o}+p^{c}\right)}{q^{o}+q^{c}}\right)<0$; equivalently, $h\left(q^{o}+q^{c}\right)-p^{o}-p^{c}<0$. In case $q^{o} p^{c}-p^{o} q^{c}>0$ and $\left(h q^{o}-p^{o}\right)\left(q^{o}+q^{c}\right)>2\left(p^{c} q^{o}-p^{o} q^{c}\right)$, noting that $p^{c}=-p^{o}-\eta$ and $q^{c}=-q^{o}-\varepsilon$ for some $\eta \in \mathbb{R}$ and $\varepsilon>0$, we have $-q^{o} \eta>-p^{o} \varepsilon$ and $q^{o}(2 \eta-h \varepsilon)>p^{o} \varepsilon$, implying (by summing them side-by-side) $q^{o}(\eta-h \varepsilon)>0$; equivalently,
$\eta-h \varepsilon<0$. Thus,

$$
h\left(q^{o}+q^{c}\right)-p^{o}-p^{c}=\eta-h \varepsilon<0 .
$$


Information Design in Smooth Games

Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

Proof of Theorem 3, continued. By concavity conditions $-q^{o}>0$, and thus $\sigma^{2}>0$ if and only if

$$
\begin{aligned}
\beta\left(1+q^{c} \beta+q^{o} \beta\right) & >0, \\
\frac{\left(p^{o}-h q^{o}\right)}{2\left(p^{c} q^{o}-p^{o} q^{c}\right)}\left(1+\frac{\left(q^{c}+q^{o}\right)\left(p^{o}-h q^{o}\right)}{2\left(p^{c} q^{o}-p^{o} q^{c}\right)}\right) & >0, \\
\frac{\left(p^{o}-h q^{o}\right)\left(2\left(p^{c} q^{o}-p^{o} q^{c}\right)+\left(q^{c}+q^{o}\right)\left(p^{o}-h q^{o}\right)\right)}{4\left(p^{c} q^{o}-p^{o} q^{c}\right)^{2}} & >0 .
\end{aligned}
$$

Because $q^{o}<0$, setting $c_{1} \triangleq q^{o}\left(h q^{o}-p^{o}\right)$ and $c_{2} \triangleq q^{o}\left(2 p^{c}-h q^{c}\right)-p^{o} q^{c}$, the inequality holds if and only if

$$
\begin{aligned}
q^{o}\left(h q^{o}-p^{o}\right)\left(2\left(p^{c} q^{o}-p^{o} q^{c}\right)+\left(q^{c}+q^{o}\right)\left(p^{o}-h q^{o}\right)\right) & >0 \\
c_{1}\left(c_{2}-c_{1}\right) & >0
\end{aligned}
$$

Moreover,

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

The result follows, because for any $c_{1}, c_{2} \in \mathbb{R}, c_{1} / c_{2} \in(0,1)$ if and only if $c_{1}\left(c_{2}-c_{1}\right)>0$.


Information Design in Smooth Games

## A. 4 Formalism Omitted in Section 8.1

No Disclosure Under no disclosure, each player's action cannot depend on the state and is thus uniquely determined by condition

$$
\mathbb{E}\left[a_{i}\right]=\frac{1}{N+1} \mathbb{E}[\omega] .
$$

Full Disclosure For any $\omega$, the ensuing game admits a strictly concave potential $\Psi(a, \omega)=$ $\left(\omega-\frac{\breve{a}}{2}\right) \breve{a}-\frac{1}{2} \sum_{i=1}^{N} a_{i}^{2}$ and thus has a unique equilibrium. Parameterize a symmetric linear strategy profile:

$$
a_{i}(\omega)=k_{0}+k_{1} \omega .
$$

The best-response condition (43) can be rewritten as

$$
a_{i}(\omega)=-\frac{N-1}{2} k_{0}+\frac{1-k_{1}(N-1)}{2} \omega,
$$

which determines the equilibrium parameters at $k_{0}=0$ and $k_{1}=\frac{1}{N+1}$.


Information Design in Smooth Games

## A. 5 Formalism Omitted in Section 6

Game Normalization Under any information structure, the expected marginal payoffs must equal zero. Thus,

$$
\mathbb{E}\left[\omega_{i} p^{o}+\breve{\omega}_{-i} p^{c}+b_{1}+q^{o} a_{i}+q^{c} \breve{a}_{-i}\right]=\mu\left(p^{o}+(N-1) p^{c}\right)+b_{1}+\mathbb{E}\left[q^{o} a_{i}+q^{c} \breve{a}_{-i}\right]=0 .
$$

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

$$
\mathbb{E}\left[a_{i}\right]=a_{0} \triangleq-\frac{\mu\left(p^{o}+(N-1) p^{c}\right)+b_{1}}{q^{o}+(N-1) q^{c}} .
$$

In the normalized game, the expected actions equal zero. The normalization is achieved by defining $\omega^{\prime}=(\omega-\mu) / \sigma, a_{i}^{\prime}=\left(a_{i}-a_{0}\right) / \sigma, b=\left(b_{2}+\left(\tilde{p}^{o}+(N-1) \tilde{p}^{c}\right) \mu+2\left(\tilde{q}^{o}+(N-1) \tilde{q}^{c}\right) a_{0}\right) / \sigma$, rescaling the payoffs by $\sigma^{2}$, and ignoring the strategically irrelevant additive terms, i.e., those in $u_{i}$ that do not depend on $a_{i}$ and those in $v$ that do not depend on $a$.

Proof of Lemma 1 Matrix $M(x, y)$ has two distinct eigenvalues: $x+(N-1) y$ and $x-y$. Therefore, it is negative semidefinite if and only if $x+(N-1) y \leq 0$ and $x-y \leq 0$; it is negative definite if and only if $x+(N-1) y<0$ and $x-y<0$. Therefore, $M\left(\bar{q}^{o}, \bar{q}^{c}\right)=M\left(\tilde{q}^{o}+x q^{o}, \tilde{q}^{c}+x q^{c}\right)$
is negative semidefinite if and only if

$$
\begin{aligned}
x\left(q^{o}+(N-1) q^{c}\right) & \leq-\tilde{q}^{o}-(N-1) \tilde{q}^{c} \\
x\left(q^{o}-q^{c}\right) & \leq-\left(\tilde{q}^{o}-\tilde{q}^{c}\right)
\end{aligned}
$$

By concavity assumptions, $q^{o}+(N-1) q^{c}<0$ and $q^{o}-q^{c}<0$. The result follows.

Proof of Lemma 2 Note that $M(x, y) M(z, w)=M(x z+(N-1) y w, x w+y z+(N-2) y w)$ and $M(x, y)^{-1}=M\left(\frac{x+(N-2) y}{(x-y)(x+(N-1) y)},-\frac{y}{(x-y)(x+(N-1) y)}\right)$, if invertible. Thus,

$$
\begin{aligned}
a^{*}(\omega, x) & =-\frac{1}{2} M^{-1}\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right) M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) \omega \\
& =-\frac{1}{2} \frac{M\left(\bar{q}^{o}(x) \bar{p}^{o}(x)+(N-2) \bar{q}^{c}(x) \bar{p}^{o}(x)-(N-1) \bar{q}^{c}(x) \bar{p}^{c}(x), \bar{q}^{o}(x) \bar{p}^{c}(x)-\bar{q}^{c}(x) \bar{p}^{o}(x)\right)}{\left(\bar{q}^{o}(x)-\bar{q}^{c}(x)\right)\left(\bar{q}^{o}(x)+(N-1) \bar{q}^{c}(x)\right)} \omega,
\end{aligned}
$$

and

$$
\begin{aligned}
W(x) & =-\frac{1}{4} \mathbb{E}\left[\omega^{T} M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) M^{-1}\left(\bar{q}^{o}(x), \bar{q}^{c}(x)\right) M\left(\bar{p}^{o}(x), \bar{p}^{c}(x)\right) \omega\right] \\
& =-\frac{N}{4} \frac{Z(x)}{\left(\bar{q}^{o}(x)-\bar{q}^{c}(x)\right)\left(\bar{q}^{o}(x)+(N-1) \bar{q}^{c}(x)\right)},
\end{aligned}
$$

where

$$
\begin{aligned}
Z(x) \triangleq & \bar{q}^{o}(x)\left(\bar{p}^{o}(x)^{2}+\bar{p}^{c}(x)^{2}(N-1)(1+(N-2) \rho)+2(N-1) \rho \bar{p}^{o}(x) \bar{p}^{c}(x)\right)+ \\
& +\bar{q}^{c}(x)\left(\bar{p}^{o}(x)^{2}(N-2-\rho(N-1))-\rho(N-1)^{2} \bar{p}^{c}(x)^{2}-2(N-1) \bar{p}^{o}(x) \bar{p}^{c}(x)\right) .
\end{aligned}
$$

First, consider the case of $x \downarrow \underline{x}$. In this case, the term dividing $Z(x)$ is going to zero (from above). Thus, if $Z(\underline{x}) \neq 0$, then as $x \downarrow \underline{x}, W(x)$ diverges to $+\infty$.
(i) $\underline{x}=-\frac{\tilde{q}^{o}-\tilde{q}^{c}}{q^{o}-q^{c}}$. As $-\frac{\tilde{q}^{o}-\tilde{q}^{c}}{q^{o}-q^{c}} \geq-\frac{\tilde{q}^{o}+(N-1) \tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$, we have $\tilde{q}^{o} q^{c} \leq q^{o} \tilde{q}^{c}$, and by the genericity assumption, $\tilde{q}^{o} q^{c}<q^{o} \tilde{q}^{c}$. This implies that $\bar{q}^{o}=\bar{q}^{c}<0$. Thus,

$$
Z(\underline{x})=\bar{q}^{c}(N-1)(1-\rho)\left(\bar{p}^{o}-\bar{p}^{c}\right)^{2}<0 .
$$

(ii) $\underline{x}=-\frac{\tilde{q}^{o}+(N-1) \tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$. As $-\frac{\tilde{q}^{o}-\tilde{q}^{c}}{q^{o}-q^{c}} \leq-\frac{\tilde{q}^{o}+(N-1) \tilde{q}^{c}}{q^{o}+(N-1) q^{c}}$, we have $\tilde{q}^{o} q^{c} \geq q^{o} \tilde{q}^{c}$, and by the genericity assumption, $\tilde{q}^{o} q^{c}>q^{o} \tilde{q}^{c}$. This implies that $\bar{q}^{o}+(N-1) \bar{q}^{c}=0$, and $\bar{q}^{c}>0$. Thus,

$$
Z(\underline{x})=-\bar{q}^{c}(1+\rho(N-1))\left(\bar{p}^{o}+(N-1) \bar{p}^{c}\right)^{2}<0 .
$$

Now, consider the case of $x \uparrow+\infty$. In this case, for the denominator of the dual payoff, we have

$$
\frac{\left(\bar{q}^{o}-\bar{q}^{c}\right)\left(\bar{q}^{o}+(N-1) \bar{q}^{c}\right)}{x^{2}} \rightarrow\left(q^{o}-q^{c}\right)\left(q^{o}+(N-1) q^{c}\right)>0,
$$

while for the numerator we have

$$
\begin{aligned}
\frac{Z(x)}{x^{3}} \rightarrow & q^{o}\left(\left(p^{o}\right)^{2}+\left(p^{c}\right)^{2}(N-1)(1+(N-2) \rho)+2(N-1) \rho p^{o} p^{c}\right) \\
& +q^{c}\left(\left(p^{o}\right)^{2}(N-2-\rho(N-1))-\rho(N-1)^{2}\left(p^{c}\right)^{2}-2(N-1) p^{o} p^{c}\right) .
\end{aligned}
$$

It suffices to show that the right-hand side of (58) is strictly negative, because then the dual payoff is $x$ times a strictly positive constant, which diverges to infinity as $x$ grows.

First, note that

$$
\begin{aligned}
& \left(p^{o}\right)^{2}+\left(p^{c}\right)^{2}(N-1)(1+(N-2) \rho)+2(N-1) \rho p^{o} p^{c} \\
& =\left(p^{o}+\rho(N-1) p^{c}\right)^{2}+\left(p^{c}\right)^{2}\left((N-1)(1+(N-2) \rho)-\rho^{2}(N-1)^{2}\right) \\
& >\left(p^{c}\right)^{2}(N-1)\left(1+(N-2) \rho-\rho^{2}(N-1)\right) \\
& =\left(p^{c}\right)^{2}(N-1)(1+(N-1) \rho)(1-\rho) \geq 0 .
\end{aligned}
$$

(i) $q^{c}<0$. Because $q^{o}<q^{c}$, the right-hand side of (58) is lower than

$$
\begin{aligned}
& q^{c}\left(\left(p^{o}\right)^{2}+\left(p^{c}\right)^{2}(N-1)(1+(N-2) \rho)+2(N-1) \rho p^{o} p^{c}\right) \\
& +q^{c}\left(\left(p^{o}\right)^{2}(N-2-\rho(N-1))-\rho(N-1)^{2}\left(p^{c}\right)^{2}-2(N-1) p^{o} p^{c}\right) \\
& =q^{c}\left(p^{o}-p^{c}\right)^{2}(N-1)(1-\rho)<0 .
\end{aligned}
$$

(ii) $q^{c}>0$. Because $q^{o}+(N-1) q^{c}<0$, the right-hand side of (58) is lower than
$$
\begin{aligned}
& -(N-1) q^{c}\left(\left(p^{o}\right)^{2}+\left(p^{c}\right)^{2}(N-1)(1+(N-2) \rho)+2(N-1) \rho p^{o} p^{c}\right) \\
& +q^{c}\left(\left(p^{o}\right)^{2}(N-2-\rho(N-1))-\rho(N-1)^{2}\left(p^{c}\right)^{2}-2(N-1) p^{o} p^{c}\right) \\
& =-q^{c}\left(p^{o}+(N-1) p^{c}\right)^{2}(1+\rho(N-1))<0
\end{aligned}
$$

The result follows.


Information Design in Smooth Games

## A. 6 Formalism Omitted in Section 8.3

We can microfound the linear demand as being generated by a continuum of consumers that differ in their tastes. Each consumer has a type $\omega=\left(\omega_{1}, \omega_{2}\right) \in \mathbb{R}^{2}$ and decides how much of the firms' products to consume, $q=\left(q_{1}, q_{2}\right)$. The payoff of a type- $\omega$ consumer who consumes quantities $q$ at prices $a$ is

$$
-a^{T} q+\frac{1}{2}(\omega-q)^{T} H^{-1}(\omega-q),
$$

where $H$ is an $N \times N$ negative semidefinite matrix:

$$
H \triangleq\left(\begin{array}{cc}
-1 & \eta \\
\eta & -1
\end{array}\right) .
$$

For any price vector $a \in A$, the quantities demanded by a type- $\omega$ consumer are $q(a, \omega)=\omega+H a$. Consumer and producer surpluses can be written as

$$
\begin{aligned}
& C S(a, \omega)=a^{T} B_{C S} \omega-\frac{1}{2} a^{T} C_{C S} a \\
& P S(a, \omega)=a^{T}(\omega+H a)-c(\omega+H a)^{T}(\omega+H a)=-c \omega^{T} \omega+a^{T} B_{P S} \omega-\frac{1}{2} a^{T} C_{P S} a,
\end{aligned}
$$

where the payoff coefficient matrices are

$$
\begin{aligned}
& B_{C S} \triangleq-I, C_{C S} \triangleq H \\
& B_{P S} \triangleq I-2 c H, C_{P S} \triangleq-2 H+2 c H^{2} \\
& B \triangleq \delta B_{C S}+(1-\delta) B_{P S}, C \triangleq \delta C_{C S}+(1-\delta) C_{P S}
\end{aligned}
$$

Price Control The designer's first-order condition is

$$
B \omega-C a=0,
$$

which results in the first-best responsiveness matrix $R^{F B}=C^{-1} B$, and thus,

$$
\begin{aligned}
r_{o}^{F B} & =\frac{2+\delta(6 \delta-7)+4 c^{2}(1-\delta)^{2}\left(1-\eta^{2}\right)+2 c(1-\delta)\left(3-5 \delta-(1-\delta) \eta^{2}\right)}{\left(1-\eta^{2}\right)(2-3 \delta+2 c(1-\delta)(1-\eta))(2-3 \delta+2 c(1-\delta)(1+\eta))}, \\
r_{c}^{F B} & =\frac{\eta\left(2+\delta(6 \delta-7)+4 c^{2}(1-\delta)^{2}\left(1-\eta^{2}\right)+4 c(1-\delta)(1-2 \delta)\right)}{\left(1-\eta^{2}\right)(2-3 \delta+2 c(1-\delta)(1-\eta))(2-3 \delta+2 c(1-\delta)(1+\eta))} .
\end{aligned}
$$

The threshold value $\bar{\delta}$ is the one that equalizes the determinant of $C$ to zero:

$$
\bar{\delta}=\frac{2+2 c(1-|\eta|)}{3+2 c(1-|\eta|)} .
$$

No Disclosure and Full Disclosure Equilibrium pricing behavior is derived from the system of first-order conditions:

$$
\mathbb{E}_{\mu}\left[q_{i}\left(a_{i}, a_{-i}, \omega\right)+\frac{\partial q_{i}\left(a_{i}, a_{-i}, \omega\right)}{\partial a_{i}}\left(a_{i}-2 c q_{i}\left(a_{i}, a_{-i}, \omega\right)\right)\right]=0, \quad i=1,2 .
$$

Proof of Proposition 5 The setting is an instance of the general framework (23-24) with $N=2, \rho=0, b_{1}=b_{2}=0$ and $p_{o}=1+2 c, p_{c}=0, q_{o}=-2(1+c), q_{c}=\eta(1+2 c)$, $\tilde{p}_{o}=\delta(-1)+(1-\delta)(1+2 c), \tilde{p}_{c}=(1-\delta)(-2 c \eta), \tilde{q}_{o}=\delta / 2+(1-\delta)\left(-1-c\left(1+\eta^{2}\right)\right)$, $\tilde{q}_{c}=-\delta \eta / 2+(1-\delta) \eta(1+2 c)$.

The non-genericity conditions (26-28) reduce to

$$
\delta \neq \hat{\delta}(\eta, c) \triangleq \frac{2+6 c+4 c^{2}-8 c|\eta|-8 c^{2}|\eta|+2 c \eta^{2}+4 c^{2} \eta^{2}}{5+8 c+4 c^{2}-|\eta|-10 c|\eta|-8 c^{2}|\eta|+2 c \eta^{2}+4 c^{2} \eta^{2}} .
$$

If $\eta<0$, at $\delta=\hat{\delta}$ condition (27) is violated. If $\eta>0$, at $\delta=\hat{\delta}$ condition (28) is violated. Otherwise, the conditions hold.

The resulting lower bound on $x$ is

$$
\underline{x}=\max \left\{-\frac{\tilde{q}_{o}-\tilde{q}_{c}}{q_{o}-q_{c}},-\frac{\tilde{q}_{o}+\tilde{q}_{c}}{q_{o}+q_{c}}\right\} .
$$

The expected dual payoff reduces to

$$
W(x)=-\frac{\sigma^{2}\left(\bar{q}_{o}(x) \bar{p}_{o}(x)^{2}+\bar{q}_{o}(x) \bar{p}_{c}(x)^{2}-2 \bar{q}_{c}(x) \bar{p}_{o}(x) \bar{p}_{c}(x)\right)}{2\left(\bar{q}_{o}(x)^{2}-\bar{q}_{c}(x)^{2}\right)} .
$$

By Theorem 4 and normalization (A.5), whenever $\delta \neq \hat{\delta}$, the optimal direct information structure is unique and is given by (52) with

$$
\begin{aligned}
& r_{o}^{*}=-\frac{\bar{q}_{o}\left(x^{*}\right) \bar{p}_{o}\left(x^{*}\right)-\bar{q}_{c}\left(x^{*}\right) \bar{p}_{c}\left(x^{*}\right)}{2\left(\bar{q}_{o}\left(x^{*}\right)^{2}-\bar{q}_{c}\left(x^{*}\right)^{2}\right)}, \\
& r_{c}^{*}=-\frac{\bar{q}_{o}\left(x^{*}\right) \bar{p}_{c}\left(x^{*}\right)-\bar{q}_{c}\left(x^{*}\right) \bar{p}_{o}\left(x^{*}\right)}{2\left(\bar{q}_{o}\left(x^{*}\right)^{2}-\bar{q}_{c}\left(x^{*}\right)^{2}\right)},
\end{aligned}
$$

and $x^{*}$ being any minimizer of (61) over $x^{*}>\underline{x}$. The constant term $a_{0}^{*}$ can be found via mean action invariance: $a_{0}^{*}=a_{i}^{N D}-\left(r_{o}^{*}+r_{c}^{*}\right) \omega_{0}$.


Information Design in Smooth Games

## B Bounded Action Spaces

Consider the concave information-design problem as in the main text, but let the action space of each player be $A_{i}=\left[\underline{a}_{i}, \bar{a}_{i}\right],-\infty<\underline{a}_{i}<\bar{a}_{i}<+\infty .{ }^{18}$ For any given $\mu \in \Delta\left(A_{-i} \times \Omega\right)$, the player's best-response action $a_{i}^{*}(\mu)$, if interior, must be unimprovable by local deviations to

[^15]lower and higher actions and hence satisfies the first-order condition
$$
\mathbb{E}_{\mu}\left[\dot{u}_{i}\left(a_{i}^{*}, a_{-i}, \omega\right)\right]=0 .
$$
In contrast, the optimal boundary actions must only be unimprovable by one-sided local deviations. As such, the player's best response if located on the boundary must satisfy the following:
$$
\begin{aligned}
& \mathbb{E}_{\mu}\left[\dot{u}_{i}\left(\underline{a}_{i}^{*}, a_{-i}, \omega\right)\right] \leq 0, \\
& \mathbb{E}_{\mu}\left[\dot{u}_{i}\left(\bar{a}_{i}^{*}, a_{-i}, \omega\right)\right] \geq 0 .
\end{aligned}
$$
We can write the resulting primal information-design problem as follows:
$$
\begin{aligned}
V_{B}^{P} \triangleq & \sup _{\pi \in \Delta(A \times \Omega)} \int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi \\
& \text { s.t. } \int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) \mathrm{d} \pi=0 \quad \forall i=1, \ldots, N, \text { measurable } A_{i}^{\prime} \subseteq\left(\underline{a}_{i}, \bar{a}_{i}\right), \\
& \int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) \mathrm{d} \pi \leq 0 \quad \forall i=1, \ldots, N, \text { measurable } A_{i}^{\prime} \subseteq\left[\underline{a}_{i}, \bar{a}_{i}\right), \\
& \int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \dot{u}_{i}(a, \omega) \mathrm{d} \pi \geq 0 \quad \forall i=1, \ldots, N, \text { measurable } A_{i}^{\prime} \subseteq\left(\underline{a}_{i}, \bar{a}_{i}\right], \\
& \int_{A \times \Omega^{\prime}} \mathrm{d} \pi=\int_{\Omega^{\prime}} \mathrm{d} \mu_{0} \quad \forall \text { measurable } \Omega^{\prime} \subseteq \Omega .
\end{aligned}
$$
This primal problem entails the dual problem
$$
\begin{aligned}
V_{B}^{D} \triangleq & \inf _{\lambda \in x_{i} \mathcal{M}\left(A_{i}\right), \gamma \in \mathcal{M}(\Omega)} \int_{\Omega} \gamma(\omega) \mathrm{d} \mu_{0} \\
& \text { s.t. } \sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega)+\gamma(\omega) \geq v(a, \omega) \forall a \in A, \omega \in \Omega, \\
& \quad \lambda_{i}\left(\underline{a}_{i}\right) \geq 0, \lambda_{i}\left(\bar{a}_{i}\right) \leq 0 \quad \forall i=1, \ldots, N .
\end{aligned}
$$
The presence of additional obedience constraints (66), (67) in the primal problem translates into the sign constraints on the Lagrange multipliers (71) in the dual problem. As in the case
of unbounded actions, the dual problem (69) can be simplified and rewritten as
$$
\begin{aligned}
V_{B}^{D} & =\inf _{\lambda \in \times_{i} \mathcal{M}\left(A_{i}\right)} \mathbb{E}_{\mu_{0}}\left[\sup _{a \in A} w(a, \omega, \lambda)\right] \\
\text { s.t. } & \lambda_{i}\left(\underline{a}_{i}\right) \geq 0, \lambda_{i}\left(\bar{a}_{i}\right) \leq 0 \quad \forall i=1, \ldots, N
\end{aligned}
$$
The adversarial-contracting interpretation remains intact, but the space of allowed contracts is limited at the boundary actions by the presence of sign constraints.

Lemma 5. (Weak Duality with Bounded Action Spaces) $V_{B}^{P} \leq V_{B}^{D}$.
Proof. Take any dual variables $(\lambda, \gamma)$ that satisfy the constraints of the dual problem (69). Take any measure $\pi$ that satisfies the constraints of primal problem (64). Integrating both sides of the dual constraints over $a \in A$ and $\omega \in \Omega$ against measure $\pi$ yields

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

where the second inequality follows because $\pi$ satisfies the primal constraints and the Lagrange multipliers satisfy the dual constraints. (This inequality holds as equality in the case of unbounded actions.) The left-hand side of (74) is the value of the primal problem given measure $\pi$, whereas the right-hand side of (74) is the value of the dual problem given dual variables $(\lambda, \gamma)$. As inequality (74) holds for any allowed values of primal measure and dual variables, it also holds at the respective maximization and minimization limits. $\square$

In the case of bounded actions, we call a measure $\pi \in \Delta(A \times \Omega)$ implementable by information if it satisfies the constraints of the primal problem (64) and implementable by incentives if there exists a feasible contract in the dual problem (72) that induces this measure as a best response and satisfies the complementarity slackness condition: for all $i$ and measurable $A_{i}^{\prime} \subseteq\left[\underline{a}_{i}, \bar{a}_{i}\right], \int_{A_{i}^{\prime} \times A_{-i} \times \Omega} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) \mathrm{d} \pi=0$.

Theorem 5. (Optimality Certification with Bounded Action Spaces) In the case of bounded actions, 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 adversarialcontracting problem, and (iii) $V_{B}^{P}=V_{B}^{D}$.

Proof. Take any primal measure $\pi$ implementable by information, i.e., one that satisfies the constraints of the primal problem (64). If it is implementable by incentives, then there exist dual variables $\lambda$ that implement this measure in the dual problem (72) and satisfy complementarity slackness. Therefore,

$$
\begin{aligned}
V_{B}^{D} & =\inf _{\lambda^{\prime} \in \times_{i} \mathcal{M}\left(A_{i}\right)+(73)} \mathbb{E}_{\mu_{0}}\left[\sup _{a \in A} w\left(a, \omega, \lambda^{\prime}\right)\right] \\
& \leq \mathbb{E}_{\pi}[w(a, \omega, \lambda)] \\
& =\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi-\int_{A \times \Omega} \sum_{i=1}^{N} \lambda_{i}\left(a_{i}\right) \dot{u}_{i}(a, \omega) \mathrm{d} \pi \\
& =\int_{A \times \Omega} v(a, \omega) \mathrm{d} \pi \leq V_{B}^{P}
\end{aligned}
$$


Information Design in Smooth Games

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

Furthermore, by Lemma 5, $V_{B}^{D} \geq V_{B}^{P}$. Hence,

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

which proves the optimality of measure $\pi$. $\square$

Claim 2. (Strong Duality with Bounded Action Spaces) If $\Omega$ is compact, and $v$ and each $\dot{u}_{i}$ are continuous, then $V_{B}^{D}=V_{B}^{P}$.

Proof. The proof is analogous to that of Claim 1, except that one carries the boundary sign restrictions through the auxiliary continuous dual. In particular, one defines $V_{B}^{D, c}$ exactly as in the proof of Claim 1, but with continuous multipliers $\lambda_{i} \in C\left(A_{i}\right)$ constrained to satisfy $\lambda_{i}\left(\underline{a}_{i}\right) \geq 0$ and $\lambda_{i}\left(\bar{a}_{i}\right) \leq 0$ for each $i$. Then $V_{B}^{D} \leq V_{B}^{D, c}$, while weak duality gives $V_{B}^{P} \leq V_{B}^{D}$, so it suffices to show $V_{B}^{D, c}=V_{B}^{P}$. Applying the Fenchel-Rockafellar duality with the same spaces and linear operator as in the proof of Claim 1, the only change is that the functional on the multiplier space now includes the indicators of the cones $K_{i}=\left\{\lambda_{i} \in C\left(A_{i}\right): \lambda_{i}\left(\underline{a}_{i}\right) \geq 0, \lambda_{i}\left(\bar{a}_{i}\right) \leq 0\right\}$. Accordingly, the conjugate condition in the dual becomes $\pi_{0}=\mu_{0}$ together with $\nu_{i} \in K_{i}^{\circ}$ for
every $i$, where $\nu_{i}$ is the weighted marginal measure induced by $\dot{u}_{i}$ and $K_{i}^{\circ}$ is the polar cone of $K_{i}$. A direct characterization of $K_{i}^{\circ}$ shows that this is equivalent to the bounded-action obedience constraints (65)-(67): zero weighted mass on interior subsets and the appropriate weak inequalities on sets touching the lower or upper boundary. Thus the feasible set in the Fenchel-Rockafellar dual coincides exactly with the primal feasible set, so $V_{B}^{D, c}=V_{B}^{P}$, and therefore $V_{B}^{P} \leq V_{B}^{D} \leq V_{B}^{D, c}=V_{B}^{P}$, implying $V_{B}^{D}=V_{B}^{P}$. $\square$

[^0]:    Smolin: Toulouse School of Economics, University of Toulouse Capitole and CEPR, alexey.v.smolin@gmail.com. Yamashita: Osaka University, tytakuroy@gmail.com. For valuable suggestions and comments, we would like to thank the Coeditor Rakesh Vohra, two anonymous referees, Refine.ink, Dirk Bergemann, Deniz Dizdar, Laura Doval, Philippe Jehiel, Jiangtao Li, Xiao Lin, Elliot Lipnowski, Stephen Morris, Alessandro Pavan, Antonio Penta, Jacopo Perego, Fedor Sandomirskiy, Ludvig Sinander, Takashi Ui, Xavier Vives, and Alexander Wolitzky, as well as seminar participants at Toulouse School of Economics, Western University, University of Toronto, University of Surrey, University of Oxford, Hitotsubashi University, Singapore Management University, CMU/Pittsburgh, Pennsylvania State University, MIT/Harvard, Bonn Winter Theory Workshop 2022, SWET 2022, CMid2022, EC 2022, EEA-ESEM 2022, EARIE 2022, and Venice Winter Theory Workshop 2023. An extended abstract of a previous version of this paper appeared as "Information Design in Concave Games" in the proceedings to EC'22. Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and through the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002), as well as the hospitality of Northwestern University and Columbia Business School, where parts of this paper were completed. Yamashita gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010), funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant 714693), and JSPS KAKENHI (grant 23K01311).

[^1]:    ${ }^{1}$ See, for example, Dworczak and Martini (2019), Dizdar and Kováč (2020), and Dworczak and Kolotilin (2024). The single-player case also covers scenarios in which there are many players but the information is required to be public.

[^2]:    ${ }^{2}$ The duality methodology is routinely used in many disciplines to solve optimization problems. In mechanism design, duality methods have been recently used to study optimal delegation (Amador, Werning, and Angeletos (2006); Amador and Bagwell (2013)), matching (Chiappori, Salanié, and Weiss (2017); Galichon and Salanié (2022)), robust selling mechanisms (Carroll (2017); Du (2018); Brooks and Du (2021)), mediation (Salamanca (2021), Ortner, Sugaya, and Wolitzky (2024)), and limited commitment (Lin and Liu (2024)) among others. For a unified treatment, see Vohra (2011).
    ${ }^{3}$ Alternatively, one can develop original arguments tailored to the studied problem; see, for example, Arieli and Babichenko (2019), Chan, Gupta, Li, and Wang (2019), Elliott, Galeotti, Koh, and Li (2022), Arieli, Babichenko, and Sandomirskiy (2023), and Candogan and Strack (2023).

[^3]:    ${ }^{4}$ In their Section 4.4, Bergemann et al. (2017) argue that Gaussian structures, possibly asymmetric and with extraneous noise, span all implementable covariance matrices of equilibrium actions.

[^4]:    ${ }^{5}$ We write $\mathbb{E}_{\mathcal{I}, \sigma_{i}, \sigma_{-i}}[\cdot]$ for the expectation under information structure $\mathcal{I}$ and strategy profile $\left(\sigma_{i}, \sigma_{-i}\right)$. Throughout the paper an integral is left undefined whenever the integrand is not integrable with respect to the relevant measure.

[^5]:    ${ }^{6}$ This notion of a concave game is related to but distinct from the notion of a concave game of Rosen (1965). In particular, it requires neither differentiability nor continuity of a player's payoff in the other players' actions or the state.

[^6]:    ${ }^{7}$ This observation resonates with the research on robust information design in regime change games, such as Inostroza and Pavan (2025) and Morris, Oyama, and Takahashi (2024), which is fundamentally concerned with multiple informational implementations of the same aggregate behavior.


Information Design in Smooth Games

[^7]:    ${ }^{8}$ This includes the case when the denominator in (20) is zero.
    ${ }^{9}$ Note that conditions (i) and (ii) are not equivalent to $k^{*}<0$ and $k^{*}>N$. Examples exist in which full disclosure is optimal even when $k^{*}<0$, and others in which no disclosure is optimal even when $k^{*}>N$.

[^8]:    ${ }^{10}$ By Theorem 2, if $k^{*} \notin(0, N)$, then either full disclosure or no disclosure is optimal.

[^9]:    ${ }^{11}$ For a more detailed analysis, see the working version of the paper.

[^10]:    ${ }^{12}$ This phenomenon parallels cream-skimming among competing contractors on online platforms, which can likewise be mitigated through information design (Romanyuk and Smolin (2019)).

[^11]:    ${ }^{13}$ Because the expected investment and the expected investment costs are invariant to information, this payoff also captures the objective of maximizing the total investor welfare.

[^12]:    ${ }^{14}$ The setting in this section can also be viewed as Cournot competition with linear production costs and uncertain linear demand. In that interpretation, the rent dissipation under both the no-disclosure and full-disclosure benchmarks parallels the zero-profit outcome of large competitive markets. Exclusive disclosure then amounts to informing only a single firm about the demand state, and this policy maximizes total producer surplus. The structure bears some resemblance to collusion that designates one firm as a monopolist, but information control is weaker than direct production control, because every firm still chooses a positive output even when uninformed. This perspective helps illuminate the role of trade associations, which cannot engage in illegal collusion yet can manage information flows (see, e.g., Kirby (1988); Vives (1990)).

[^13]:    ${ }^{15}$ This formalization complements that adopted by Elliott et al. (2022), who study information design in unit-demand competition. Naturally, the details of their optimal information structures differ from ours; however, their interpretation of information provision as market segmentation and their broader discussion of how information shapes competitive outcomes can be applied in our setting.
    ${ }^{16}$ As standard, this specification allows the prices and quantities to be negative.

[^14]:    ${ }^{17}$ Formally, while the optimal certificate changes continuously for all $\delta \in(0,1)$, at $\delta=\hat{\delta}$ the certificate allows a range of the dual agent's best responses connecting one branch of best responses to another, so that the implemented allocation rule displays a jump.

[^15]:    ${ }^{18}$ The extension to half-bounded spaces is straightforward.


Information Design in Smooth Games

## Citation and provenance

**Authors:** Alex Smolin; Takuro Yamashita

**Canonical citation:** Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.

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