$R(s_{0})$ is convex in $[s_{0}[-\hat{n}],s_{0}[-\underline{n}]]$ and strictly
concave in $[s_{0}[-\bar{n}],s_{0}[-\hat{n}]]$. In scenario (2), the strictly
concave interval is empty.

Furthermore, since nodes with distances larger than $\bar{d}$ have similar
market sizes relative to distances, following the same argument as for nodes
with distances less than $\underline{d}$, we know that $R(s_{0})$ is convex in
$[\inf\mathcal{S}_{0},~{}s_{0}[-\bar{n}-1]]$. Therefore, we can conclude that
$R(s_{0})$ is strictly concave in an interval
$[s_{0}[-\bar{n}],s_{0}[-\hat{n}]]$ (which can be empty) and convex below
$s_{0}[-\bar{n}]$ and above $s_{0}[-\hat{n}]$. $\square$

######  Lemma 7 (Dworczak and Martini [2019]).

If there exists a cumulative distribution function $G$ and a convex function
$\nu:S_{0}\to\mathbb{R}$, with $\nu(s_{0})\geq R(s_{0})$ for all $s_{0}\in
S_{0}$, that satisfy

$\displaystyle supp(G)\subseteq\\{\mathcal{S}_{0}:R(s_{0})=\nu(s_{0})\\},$
(44a)
$\displaystyle\int_{\mathcal{S}_{0}}\nu(z)dG(z)=\int_{\mathcal{S}_{0}}\nu(z)dF(z),$
(44b) $F$ is a mean-preserving spread of $G$. (44c)

then $G$ is an optimal posterior distribution that maximizes the expected
total revenue.

_Proof of Theorem 1._

In case (i), we set $\nu(s_{0})=R(s_{0})$, which is convex, and $G=F$. We can
check that $(\nu,G)$ satisfies the conditions in Proposition 7, and thus $G=F$
is the optimal posterior distribution indicating that full information
revelation is optimal.

In case (ii), if the strictly concave interval is empty, then full information
provision is optimal following case (i). On the other hand, when the strictly
concave interval is nonempty, we construct a pooling region
$s_{0}\in[\underline{z},\bar{z}]$, where $\underline{z}\leq s[\hat{n}]$ and
$\bar{z}\geq s[\bar{n}]$ such that $\hat{n}$ (resp. $\tilde{n}$) is the regime
where $R(s_{0})$ changes from convex to strictly concave (resp. strictly
concave to convex), and $z^{*}=\mathbb{E}_{F}[S_{0}|\underline{z}\leq
S_{0}\leq\bar{z}]>\mathbb{E}_{F}[S_{0}]$.

We show that such $\underline{z}$ and $\bar{z}$ exist. For every regime
$\hat{n}-1\leq k\leq\bar{n}$, we define the linear function

$\displaystyle g_{k}(z)=\frac{dR(s_{0})}{ds_{0}}[k](z-s_{0}[k])+R(s_{0}[k]),$
(45)

where $\frac{dR(s_{0})}{ds_{0}}[k]$ is the derivative of $R(s_{0})$ in regime
$k$. We define

$\displaystyle z_{a}[k]$
$\displaystyle=\min\\{s_{0}\in\mathcal{S}_{0}|g_{k}(z)\geq R(z)\quad s_{0}\leq
z\leq s_{0}[k]\\},$ (46a) $\displaystyle z_{b}[k]$
$\displaystyle=\max\\{s_{0}\in\mathcal{S}_{0}|g_{k}(z)\geq R(z),\quad
s_{0}[k]\leq z\leq s_{0}\\}.$ (46b)

Since $g_{k}(s_{0})$ is tangent to the strictly concave piece of $R(s_{0})$,
we know that $g_{k}(z)>R(z)$ for all $z\in[z_{a}[k],z_{b}[k]]\setminus
s_{0}[k]$, and $g_{k}(s_{0}[k])=R(s_{0}[k])$. Moreover,
$z_{a}[k]<s_{0}[\hat{n}]$ and $z_{b}[k]>s_{0}[\bar{n}]$ for all $\hat{n}\leq
k\leq\bar{n}$. We also have

$\displaystyle z_{a}[\hat{n}-1]=s_{0}[\hat{n}-1],\quad
z_{b}[\bar{n}]=s_{0}[\bar{n}+1].$ (47)

We note that for any $z\leq s_{0}[\hat{n}]$ and any $\hat{n}-1\leq
k\leq\bar{n}$,

$\begin{split}&g_{k}(z)-g_{k-1}(z)\\\
=&\frac{dR(s_{0})}{ds_{0}}[k](z-s_{0}[k])+R(s_{0}[k])-\left(\frac{dR(s_{0})}{ds_{0}}[k-1](z-s_{0}[k-1])+R(s_{0}[k-1])\right)\\\
=&\left(\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]\right)z+\frac{dR(s_{0})}{ds_{0}}[k-1]s_{0}[k-1]-\frac{dR(s_{0})}{ds_{0}}[k]s_{0}[k]\\\
&+R(s_{0}[k])-R(s_{0}[k-1])\\\
=&\left(\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]\right)z+\frac{dR(s_{0})}{ds_{0}}[k-1]s_{0}[k-1]-\frac{dR(s_{0})}{ds_{0}}[k]s_{0}[k]\\\
&+R(s_{0}[k-1])+\frac{dR(s_{0})}{ds_{0}}[k-1](s_{0}[k]-s_{0}[k-1])-R(s_{0}[k-1])\\\
=&\left(\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]\right)z-\frac{dR(s_{0})}{ds_{0}}[k]s_{0}[k]+\frac{dR(s_{0})}{ds_{0}}[k-1]s_{0}[k]\\\
=&\left(\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]\right)(z-s_{0}[k])>0.\end{split}$
(48)

where the last inequality is due to the fact that
$\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]<0$ and
$z-s_{0}[k]<0$. Consequently, we know that $z_{a}[k]\leq z_{a}[k-1]$ for any
$\hat{n}-1\leq k\leq\bar{n}$, and the inequality is strict if
$z_{a}[k]>\inf\mathcal{S}_{0}$.

Similarly, for any $z\geq s_{0}[\bar{n}]$ and any $\hat{n}-1\leq
k\leq\bar{n}$, we have

$g_{k}(z)-g_{k-1}(z)=\left(\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]\right)(z-s_{0}[k])<0$

since $\frac{dR(s_{0})}{ds_{0}}[k]-\frac{dR(s_{0})}{ds_{0}}[k-1]<0$ and
$z-s_{0}[k]>0$. Thus, $z_{b}[k]\leq z_{b}[k-1]$ for any $\hat{n}-1\leq
k\leq\bar{n}$, and the inequality is strict if $z_{b}[k]<\sup\mathcal{S}_{0}$.

Therefore,

$\displaystyle\mathbb{E}_{F}[S_{0}|z_{a}[\hat{n}-1]\leq S_{0}\leq
z_{b}[\hat{n}-1]]-s_{0}[\hat{n}-1]>\mathbb{E}_{F}[S_{0}|z_{a}[\hat{n}]\leq
S_{0}\leq z_{b}[\hat{n}]]-s_{0}[\hat{n}]$ $\displaystyle>$
$\displaystyle\cdots>\mathbb{E}_{F}[S_{0}|z_{a}[k]\leq S_{0}\leq
z_{b}[k]]-s_{0}[k]>\mathbb{E}_{F}[S_{0}|z_{a}[k+1]\leq S_{0}\leq
z_{b}[k+1]]-s_{0}[k+1]$ $\displaystyle>$
$\displaystyle\cdots>\mathbb{E}_{F}[S_{0}|z_{a}[\bar{n}]\leq S_{0}\leq
z_{b}[\bar{n}]]-s_{0}[\bar{n}]$

Since $z_{a}[\hat{n}-1]=s_{0}[\hat{n}-1]$, we know that
$\mathbb{E}_{F}[S_{0}|z_{a}[\hat{n}-1]\leq S_{0}\leq
z_{b}[\hat{n}-1]]-s_{0}[\hat{n}-1]>0$.

Consider the case that

$\mathbb{E}_{F}[S_{0}|z_{a}[\bar{n}]\leq
s_{0}<z_{b}[\bar{n}]]-s_{0}[\bar{n}]\leq 0,$

then there must exists $\hat{n}\leq k^{*}\leq\bar{n}$ such that

$\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq
z_{b}[k^{*}]]-s_{0}[k^{*}]>0,\quad\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}+1]\leq
S_{0}\leq z_{b}[k^{*}+1]]-s_{0}[k^{*}+1]\leq 0.$

There can be two cases:

  * (i)

$\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq
z_{b}[k^{*}]]-s_{0}[k^{*}+1]\leq 0$. In this case,
$\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq
z_{b}[k^{*}]]\in[s_{0}[k^{*}],s_{0}[k^{*}+1]]$, and thus
$\underline{z}=z_{a}[k^{*}]$, $\bar{z}=z_{b}[k^{*}]$, and
$z^{*}=\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq z_{b}[k^{*}]]$.

  * (ii)

$\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq
z_{b}[k^{*}]]-s_{0}[k^{*}+1]>0$. In this case, we consider a family of affine
functions

$g_{\gamma}(z)=\gamma(z-s_{0}[k^{*}+1])+R(s_{0}[k^{*}+1]),\quad\gamma\in\partial
R(s_{0}[k^{*}+1])=\left[\frac{dR(s_{0})}{ds_{0}}[k^{*}],\frac{dR(s_{0})}{ds_{0}}[k^{*}+1]\right].$

Similar to (46), we define
$z_{a}[\gamma]=\min\\{s_{0}\in\mathcal{S}_{0}|g_{k}(z)\geq R(z),s_{0}\leq
z\leq s_{0}[k]\\}$ and
$z_{b}[\gamma]=\max\\{s_{0}\in\mathcal{S}_{0}|g_{k}(z)\geq R(z),s_{0}[k]\leq
z\leq s_{0}\\}$. Following similar procedure as in (48), we know that
$z_{a}[\gamma]$ and $z_{b}[\gamma]$ increase in $\gamma$. Since

$\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}+1]\leq S_{0}\leq z_{b}[k^{*}+1]]\leq
s_{0}[k^{*}+1]\quad\mathbb{E}_{F}[S_{0}|z_{a}[k^{*}]\leq S_{0}\leq
z_{b}[k^{*}]]>s_{0}[k^{*}+1]$

there must exists $\gamma^{*}\in\partial R(s_{0}[k^{*}+1])$ such that

$\mathbb{E}_{F}[S_{0}|z_{a}[\gamma^{*}]\leq S_{0}\leq
z_{b}[\gamma^{*}]]=s_{0}[k^{*}+1].$

Thus, we have also identified $\underline{z}=z_{a}[\gamma^{*}]$,
$\bar{z}=z_{a}[\gamma^{*}]$, $z^{*}=s_{0}[k^{*}+1]$.

On the other hand, consider the case that

$\mathbb{E}_{F}[S_{0}|z_{a}[\bar{n}]\leq S_{0}\leq
z_{b}[\bar{n}]]-s_{0}[\bar{n}]>0.$

As we know from (47) that

$\mathbb{E}_{F}[S_{0}|z_{a}[\bar{n}]\leq S_{0}\leq
z_{b}[\bar{n}]]-s_{0}[\bar{n}+1]\leq 0,$

we must have $\underline{z}=z_{a}[\bar{n}]$,
$\bar{z}=z_{b}[\bar{n}]=s_{0}[\bar{n}+1]$, and
$z^{*}\in[s_{0}[\bar{n}],s_{0}[\bar{n}+1]]$. Therefore, we have found the
pooling interval $[\underline{z},\bar{z}]$ and $z^{*}$ in all cases.

Finally, we construct the following $\nu$ function:

$\displaystyle\nu(s_{0})=\left\\{\begin{array}[]{ll}R(s_{0}),&\quad
s_{0}\leq\underline{z},\\\ &\\\
\frac{R(\bar{z})-R(\underline{z})}{\bar{z}-\underline{z}}(s_{0}-z^{*})+R(z^{*}),&\quad
s_{0}\in[\underline{z},\bar{z}],\\\ &\\\ R(s_{0}),&\quad
s_{0}\geq\bar{z},\end{array}\right.$ (54)

and $G$ as follows:

$\displaystyle G(s_{0})=\left\\{\begin{array}[]{ll}F(s_{0}),&\quad
s_{0}\leq\underline{z},\\\
F(\underline{z}),&\quad\underline{z}<s_{0}<z^{*},\\\ F(\bar{z}),&\quad
z^{*}\leq s_{0}<\bar{z},\\\ F(s_{0}),&\quad
s_{0}\geq\bar{z}.\end{array}\right.$ (59)

We first note that $\nu(s_{0})\geq R(s_{0})$ since

$\frac{R(\bar{z})-R(\underline{z})}{\bar{z}-\underline{z}}(s_{0}-z^{*})+R(z^{*})\geq
R(s_{0}),\quad\forall s_{0}\in[\underline{z},\bar{z}].$

Additionally, we note that for the intervals
$[\inf\mathcal{S}_{0},s_{0}[\hat{n}]]\cup[s_{0}[\bar{n}],\sup\mathcal{S}_{0}]$,
the function $R(s_{0})$ is convex and
$\nu(s_{0})=\max\\{R(s_{0}),\frac{R(\bar{z})-R(\underline{z})}{\bar{z}-\underline{z}}(s_{0}-z^{*})+R(z^{*})\\}$.
Thus, $\nu(s_{0})$ is convex for
$s_{0}\in[\inf\mathcal{S}_{0},s_{0}[\hat{n}]]\cup[s_{0}[\bar{n}],\sup\mathcal{S}_{0}]$.
Since $\nu(s_{0})$ is also linear for
$s\in[\underline{z},\bar{z}]\supseteq[s_{0}[\hat{n}],s_{0}[\bar{n}]]$. Thus,
$R(s_{0})$ is convex in $s_{0}$.

Additionally, $G$ satisfies (44a) since $G$ pools states between
$[\underline{z},\bar{z}]$ to the mean $z^{*}$, and $\nu(z^{*})=R(z^{*})$. We
next show that (44b) is satisfied:

$\displaystyle\int_{S_{0}}\nu(z)dG(z)=\int_{s_{0}\leq\underline{z}}\nu(z)dG(z)+\int_{\underline{z}}^{\bar{z}}\nu(z)dG(z)+\int_{s_{0}\geq\bar{z}}\nu(z)dG(z)$
$\displaystyle=$
$\displaystyle\int_{s_{0}\leq\underline{z}}R(z)dG(z)+\int_{\underline{z}}^{\bar{z}}\left(\frac{R(\bar{z})-R(\underline{z})}{\bar{z}-\underline{z}}(s_{0}-z^{*})+R(z^{*})\right)dG(z)+\int_{s_{0}\geq\bar{z}}R(z)dG(z)$
$\displaystyle=$
$\displaystyle\int_{s_{0}\leq\underline{z}}R(z)dG(z)+R(z^{*})(G(\bar{z})-G(\underline{z}))+\int_{s_{0}\geq\bar{z}}R(z)dG(z)=\int_{S_{0}}R(z)dG(z).$

Finally, since $G$ is induced by a feasible information structure that pools
stats between $\underline{z}$ and $\bar{z}$ and reveal the rest of states, we
know that $G$ is a mean preserving spread of $F$. That is, $(\nu,G)$ satisfies
all the conditions in Lemma 7, and thus pooling $[\underline{z},\bar{z}]$ and
revealing the remaining states is an optimal information mechanism.

Since $\underline{z}\leq s_{0}[\hat{n}]$ and $\bar{z}\geq s_{0}[\bar{n}]$, we
know that the posterior mean $z^{*}>\mathbb{E}_{F}[S_{0}]$. Additionally, when
$\underline{d}=0$, we know from Lemma 6 that $\hat{n}=1$. Thus, the strictly
concave interval is non-empty, indicating that the pooling region is non-
empty. Furthermore, when $\bar{d}=D$, we have $\bar{n}=K$. As a result, given
any affine function $g_{k}(s_{0})$ that is tangent to $R(s_{0})$ in the
strictly concave interval, we must have $z_{b}[k]=\sup\mathcal{S}_{0}$. As a
result, we know that $\bar{z}=\sup\mathcal{S}_{0}$.

In case (iii), analogous to case (ii), if the strictly concave interval is
empty, then full information revelation is optimal. Otherwise, we construct a
pooling region $[\underline{z},\bar{z}]$ and the posterior mean
$z^{*}=\mathbb{E}_{F}[S_{0}|\underline{z}\leq S_{0}\leq\bar{z}]$. In this
case, $\underline{z}\leq s_{0}[-\bar{n}]$ and $\bar{z}\geq s_{0}[-\hat{n}]$,
where $-\bar{n}$ (resp. $-\hat{n}$) is the regime boundary where $R(s_{0})$
changes from convex to strictly concave (resp. strictly concave to convex)
following Lemma 6. Following the same procedure as in case (ii), we can show
that such $\bar{z},\underline{z},z^{*}$ exist, and $(\nu(s),G)$ as in (54) –
(59) satisfy conditions in Lemma 7. Thus, the optimal information mechanism is
to pool states in $[\underline{z},\bar{z}]$ and reveal the remaining states.
Moreover, since $\underline{z}\leq s_{0}[-\bar{n}]$ and $\bar{z}\geq
s_{0}[-\hat{n}]$, $z^{*}<\mathbb{E}_{F}[S_{0}]$. When $\underline{d}=0$, we
know from Lemma 6 that $\hat{n}=-1$. Thus, the strictly concave interval is
non-empty, indicating that the pooling region is non-empty. Furthermore, when
$\bar{d}=D$, we have $\bar{n}=-\tilde{K}$, and
$\underline{z}=\inf\mathcal{S}_{0}$. $\square$

_Proof of Corollary 1._ Analogous to the proof of Lemma 6, if all nodes have
decreasing market sizes relative to distances, we can show that the function
$R(s_{0})$ is strictly concave for all $s_{0}=s_{0}[k]$ with $k=1,\dots,K$,
and strictly convex for all $s_{0}=s_{0}[-k]$ with $k=1,\dots,\tilde{K}$. In
this case, we know from the proof of Theorem 1 that the optimal information
mechanism has a non-empty pooling interval. Moreover, following the
construction of $g_{k}(z)$ as in (45) for each $k=1,\dots,K$, we can check
that $z_{b}[k]=\sup\mathcal{S}_{0}$ as in (46) since the derivative of
$g_{k}(z)$ is larger than the derivative of all linear pieces associated with
regimes $k^{\prime}>k$ so that $g_{k}(z)>R(z)$ for all $z>s_{0}[k]$. Thus, we
know from Theorem 1 that the optimal information mechanism fully reveal states
below a threshold, and pool states above the threshold.

Similarly, if all nodes have increasing market sizes relative to distances, we
can show that the function $R(s_{0})$ is strictly convex for all
$s_{0}=s_{0}[k]$ with $k=1,\dots,K$, and strictly concave for all
$s_{0}=s_{0}[-k]$ with $k=1,\dots,\tilde{K}$. The construction of $g_{k}(z)$
as in (45) for each $k=-1,\dots,-\tilde{K}$ satisfies that
$z_{a}[k]=\inf\mathcal{S}_{0}$. The optimal information mechanism fully reveal
states above a threshold, and pool states below the threshold. $\square$

_Proof of Corollary 2._ We can check that when $r\geq\bar{r}$, for any $i,j$
with $d_{i}\neq d_{j}$, we have

$\left|\frac{s_{i}-s_{j}}{d_{i}-d_{j}}\right|\leq\frac{1}{1-r}.$

Therefore, the function $R(s_{0})$ is convex in $s_{0}$, and full information
revelation is optimal. $\square$

The proof of Proposition 4 follows directly from the proof of Theorem 1, and
thus is omitted.

##  Appendix E Proof of statements in Section 4

_Proof of Lemma 2._ Following from (48), we know that any $k$ and function
$g_{k}(z)=\frac{dR(s_{0})}{ds_{0}}[k](z-s_{0}[k])+R(s_{0}[k])$ satisfy

$\displaystyle g_{k}(z)-g_{k-1}(z)\left\\{\begin{array}[]{ll}<0,&\quad\forall
z>s_{0}[k]\\\ >0,&\quad\forall z<s_{0}[k].\end{array}\right.$

Therefore, we know from (13a) and (13c) that there must exists $k_{\ell,a}\leq
k_{\ell}^{*}\leq k_{\ell,b}$ such that

$\displaystyle g_{k_{\ell}^{*}-1}(s_{0})-R(s_{0})>0,\quad\forall
s_{0}\in\left[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]\right],$
(60a) $\displaystyle\exists
s_{0}\in\left[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]\right],\quad
s.t.\quad g_{k_{\ell}^{*}}(s_{0})-R(s_{0})\leq 0.$ (60b)

Similarly, following from (13b) and (13d), there must exists
$k_{\ell^{\prime},a}\leq k_{\ell^{\prime}}^{*}\leq k_{\ell^{\prime},b}$ such
that

$\displaystyle g_{k_{\ell^{\prime}}^{*}}(s_{0})-R(s_{0})>0,\quad\forall
s_{0}\in\left[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]\right],$ (61a)
$\displaystyle\exists
s_{0}\in\left[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]\right],\quad s.t.\quad
g_{k_{\ell^{\prime}}^{*}-1}(s_{0})-R(s_{0})\leq 0.$ (61b)

We construct the function $g(z)$ as follows

$g(z)=\frac{R(s_{0}[k_{\ell^{\prime}}^{*}])-R(s_{0}[k_{\ell}^{*}])}{s_{0}[k_{\ell^{\prime}}^{*}]-s_{0}[k_{\ell}^{*}]}(z-s_{0}[k_{\ell}^{*}])+R(s_{0}[k_{\ell}^{*}]).$

Then, $g(s_{0}[k_{\ell}^{*}])=R(s_{0}[k_{\ell}^{*}])$ and
$g(s_{0}[k_{\ell^{\prime}}^{*}])=R(s_{0}[k_{\ell^{\prime}}^{*}])$. It remains
to prove that the following holds:

$\gamma=\frac{R(s_{0}[k_{\ell^{\prime}}^{*}])-R(s_{0}[k_{\ell}^{*}])}{s_{0}[k_{\ell^{\prime}}^{*}]-s_{0}[k_{\ell}^{*}]}\in\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}-1]\right]\cap\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]\right].$

From (60a), we know that
$g_{k_{\ell}^{*}-1}(s_{0}[k_{\ell^{\prime}}^{*}])>R(s_{0}[k_{\ell^{\prime}}^{*}])$,
thus, $\gamma<\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}-1]$. Proving that
$\gamma\geq\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]$, is equivalent to showing
that $g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])\leq
R(s_{0}[k_{\ell^{\prime}}^{*}])$. For the sake of contradiction, we assume
that
$g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])>R(s_{0}[k_{\ell^{\prime}}^{*}])$.
Then, we must have:

$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]\stackrel{{\scriptstyle(a)}}{{>}}\gamma,\quad\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]\stackrel{{\scriptstyle(b)}}{{<}}\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}],$

where (a) is due to
$g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell}^{*}])>R(s_{0}[k_{\ell}^{*}])$, and
(b) is due to the fact that $R(s_{0})$ is strictly concave at
$s_{0}[k_{\ell^{\prime}}^{*}]$. There are two cases (We note that
$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]=\gamma$ violates Assumption
4):

(Case 1):
$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]\in\left(\gamma,\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]\right)$.
In this case, we argue that
$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]<\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]$.
This is because

$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]=\frac{g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])-g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell}^{*}])}{s_{0}[k_{\ell^{\prime}}^{*}]-s_{0}[k_{\ell}^{*}]},\quad\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]=\frac{g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])-g_{k_{\ell}^{*}}(s_{0}[k_{\ell}^{*}])}{s_{0}[k_{\ell^{\prime}}^{*}]-s_{0}[k_{\ell}^{*}]},$

and
$R(s_{0}[k_{\ell^{\prime}}^{*}])=g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])<g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])$
and
$g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell}^{*}])>R(s_{0}[k_{\ell}^{*}])=g_{k_{\ell}^{*}}(s_{0}[k_{\ell}^{*}])$.
As a result, we know that

$g_{k_{\ell}^{*}}(z)\stackrel{{\scriptstyle(a)}}{{>}}g_{k_{\ell^{\prime}}^{*}}(z)\stackrel{{\scriptstyle(b)}}{{\geq}}R(z),\quad\forall
z\in[s_{0}[k_{\ell^{\prime}}^{*}],s_{0}[k_{\ell^{\prime},b}]],$

where (a) follows from
$g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])>R(s_{0}[k_{\ell^{\prime}}^{*}])=g_{k_{\ell^{\prime}}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])$
and
$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]<\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]$,
and (b) is due to concavity of $R(s_{0})$.

Additionally, it follows from (61b) that there exists at least one
$\hat{s}_{0}\in[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]]$ such that
$g_{k_{\ell^{\prime}}^{*}-1}(\hat{s}_{0})\leq R(\hat{s}_{0})$. Since
$R(s_{0})$ is concave in $[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]]$, we have
$g_{k_{\ell}^{*}}(\hat{s}_{0})\geq R(\hat{s}_{0})$. Therefore,
$g_{k_{\ell}^{*}}(\hat{s}_{0})\geq g_{k_{\ell^{\prime}}^{*}-1}(\hat{s}_{0})$.
Since we have assumed that
$R(s_{0}[k_{\ell^{\prime}}^{*}])=g_{k_{\ell^{\prime}}^{*}-1}(s_{0}[k_{\ell^{\prime}}^{*}])<g_{k_{\ell}^{*}}(s_{0}[k_{\ell^{\prime}}^{*}])$,
we must have $g_{k_{\ell^{\prime}}^{*}-1}(z)<g_{k_{\ell}^{*}}(z)$ for all
$z\in[s_{0}[k_{\ell^{\prime},a},s_{0}[k_{\ell^{\prime}}^{*}]]$. Since
$R(s_{0})$ is concave in
$[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$, we have $R(z)\leq
g_{k_{\ell^{\prime}}^{*}-1}(z)$ for all
$z\in[s_{0}[k_{\ell^{\prime},a},s_{0}[k_{\ell^{\prime}}^{*}]]$. Therefore,
$R(z)<g_{k_{\ell}^{*}}(z)$ for all
$z\in[s_{0}[k_{\ell^{\prime},a},s_{0}[k_{\ell^{\prime}}^{*}]]$, and hence
$g_{k_{\ell}^{*}}(z)>R(z)$ for all
$z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$, which
contradicts (60b).

(Case 2): $\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]<\gamma$. In this
case, we note that
$\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]<\gamma<\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}]$,
and thus

$\gamma(z-s_{0}[k_{\ell^{\prime}}^{*}])+R(s_{0}[k_{\ell^{\prime}}^{*}])\geq
R(z),\quad\forall
z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]].$

Since we assumed that $\gamma<\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]$, we know
that

$g_{k_{\ell}^{*}}(z)>\gamma(z-s_{0}[k_{\ell^{\prime}}^{*}])+R(s_{0}[k_{\ell^{\prime}}^{*}])\geq
R(z),\quad\forall
z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]],$

which contradicts (60b).

We have derived contradiction in both cases. Therefore, we know that
$\gamma\geq\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}]$. We have thus proved that
$\gamma\in\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}-1]\right]$.
The proof of
$\gamma\in\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]\right]$
is analogous, and thus is omitted.

We next argue that such affine function $g(z)$ is unique. From the
construction of $k_{\ell}^{*}$ and $k_{\ell^{\prime}}^{*}$, we know that
$k_{\ell}^{*}$ and $k_{\ell^{\prime}}^{*}$ are unique. For any
$\hat{s}_{0}\in[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]]$, we define affine
function
$\tilde{g}(z|\hat{s}_{0})=\tilde{\gamma}(z-\hat{s}_{0})+R(\hat{s}_{0})$, where
$\tilde{\gamma}=\frac{dR(s_{0})}{ds_{0}}[k]$ if
$\hat{s}_{0}\in(s_{0}[k],s_{0}[k+1])$ and
$\tilde{\gamma}\in[\frac{dR(s_{0})}{ds_{0}}[k-1],\frac{dR(s_{0})}{ds_{0}}[k]]$
if $\hat{s}_{0}=s_{0}[k]$. We can check that for any
$s_{0}[k_{\ell,a}]\leq\hat{s}_{0}<s_{0}[k_{\ell}^{*}]$,
$\tilde{g}(z|\hat{s}_{0})>R(z)$ for all
$z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$. Additionally,
for any $s_{0}[k_{\ell}^{*}]<\hat{s}_{0}\leq s_{0}[k_{\ell,b}]$,
$\tilde{g}(z|\hat{s}_{0})<R(z)$ for all at least one
$z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$. Thus, there
does not exist another affine function that is tangent to $R(s_{0})$ in both
$\ell$ and $\ell^{\prime}$ concave intervals.

Finally, we prove that that if any one of the constraints (13) is violated,
then such affine function does not exist. Assume that constraint (13a) is
violated, i.e. there exists
$s_{0}\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$ such that
$g_{k_{\ell,a}}(s_{0})<R(s_{0})$. Then, for any
$\hat{s}_{0}\in[s_{0}[k_{\ell,a}],s_{0}[k_{\ell,b}]]$, there must exist at
least one $z\in[s_{0}[k_{\ell^{\prime},a}],s_{0}[k_{\ell^{\prime},b}]]$ such
that $g(z|\hat{s}_{0})<R(z)$. Thus, the tangent affine function does not
exist. We can analogously argue that the tangent affine function does not
exist when any of the other three constraints is violated. $\square$

######  Definition 3.

$G$ is a mean-preserving contraction of $F$, i.e., $G\preceq F$, if

$\displaystyle\int_{z\leq s_{0}}F(z)dz$ $\displaystyle\geq\int_{z\leq
s_{0}}G(z)dz,\quad\forall s_{0}\in\mathcal{S}_{0},$ (62a)
$\displaystyle\int_{z\in\mathcal{S}_{0}}F(z)dz$
$\displaystyle=\int_{z\in\mathcal{S}_{0}}G(z)dz.$ (62b)

######  Lemma 8 (Blackwell [1953], Gentzkow and Kamenica [2016]).

A posterior mean distribution $G$ is feasible given prior $F$ if and only if
$G$ is a mean preserving contraction of $F$.

######  Lemma 9 (Dworczak and Martini [2019]).

Suppose that $R(s_{0})$ is Lipchitz continuous. Then, for every optimal
posterior mean distribution $G^{*}$, there exists a convex and continuous
function $\nu:\mathcal{S}_{0}\to\mathbb{R}$ such that $\nu(s_{0})\geq
R(s_{0})$, and $(G^{*},\nu)$ satisfies (44a) – (44c).

######  Lemma 10 (Candogan and Strack [2021]).

For any optimal posterior mean distribution $G^{*}$, there exists a finite
number of intervals $\\{I_{j}\\}_{j\in J}$, where $I_{j}=[z_{j,a},z_{j,b}]$,
such that

  * -

States outside of $\cup_{j\in J}I_{j}$ are fully revealed, i.e.
$G^{*}(s_{0})=F(s_{0})$ for all $s\in\mathcal{S}_{0}\setminus\\{\cup_{j\in
J}I_{j}\\}$.

  * -

Each interval is either pooled to a single atom
$x=\mathbb{E}_{F}[S_{0}|S_{0}\in I_{j}]$ or pooled to two atoms $x,y$
according to a double-interval structure, where
$x=\mathbb{E}_{F}[S_{0}|S_{0}\in[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]]$,
$y=\mathbb{E}_{F}[S_{0}|S_{0}\in[z_{j}^{{}^{\prime}},z_{j}^{"}]]$ and
$z_{j,a}<z_{j}^{{}^{\prime}}<z_{j}^{"}<z_{j,b}$.

######  Lemma 11.

Consider the sequence of intervals $\\{I_{j}\\}_{j\in J}$, where
$I_{j}=[z_{j,a},z_{j,b}]$, associated with the optimal posterior mean
distribution $G^{*}$, we must have:

$\displaystyle\int_{z\leq z_{j,a}}F(z)dz$ $\displaystyle=\int_{z\leq
z_{j,a}}G^{*}(z)dz,\quad\int_{z\leq z_{j,b}}F(z)dz=\int_{z\leq
z_{j,b}}G^{*}(z)dz\quad\forall j\in J.$ (63)

_Proof of Lemma 11._ The proof builds on Lemmas 8 and 10. We note that the
posterior mean distribution $G^{*}$ is a mean-preserving contraction of $F$ in
each interval $I_{j}$. Therefore, following from (62b)

$\int_{z_{j,a}}^{z_{j,b}}G^{*}(z)dz=\int_{z_{j,a}}^{z_{j,b}}F(z)dz,\quad\forall
j\in J.$

Since $F(s_{0})=G^{*}(s_{0})$ for all
$s_{0}\in\mathcal{S}_{0}\setminus\\{\cup_{j\in J}I_{j}\\}$, we can conclude
that (63) holds. $\square$

######  Lemma 12.

Given the prior distribution $F$, if interval $[z_{j,a},z_{j,b}]$ with two
atoms $z_{j,a}<x<y<z_{j,b}$ is generated by a double interval structure given
the optimal information mechanism, then there must exist $p_{x},p_{y}$ that
satisfies:

$\displaystyle p_{x}+p_{y}=F(z_{j,b})-F(z_{j,a}),$ (64a) $\displaystyle
xp_{x}+yp_{y}=\int_{z_{j,a}}^{z_{j,b}}zdF(z),$ (64b)
$\displaystyle(F^{-1}(F(z_{j,a})+p_{x})-x)p_{x}<\int_{z_{j,a}}^{F^{-1}(F(z_{j,a})+p_{x})}(F(z)-F(z_{j,a}))dz,$
(64c) $\displaystyle p_{x}>0,\quad p_{y}>0.$ (64d)

Moreover, $(p_{x},p_{y})$ that satisfies (64) is unique.

_Proof of Lemma 12._ If $[z_{j,a},z_{j,b}]$ has a double-interval information
structure associated with atoms $x,y$, then $p_{x},p_{y}$ must satisfy (64a) –
(64b), and (64d), where (64a) and (64d) ensure that $(p_{x},p_{y})$ is a valid
probability vector given $F$ in the interval $[z_{j,a},z_{j,b}]$ and (64b)
ensures that $(p_{x},p_{y})$ is mean-preserving. Additionally, we note that
the optimal posterior mean distribution $G^{*}$ with $(p_{x},p_{y})$ is given
by:

$\displaystyle G^{*}(s)=\left\\{\begin{array}[]{ll}F(z_{j,a}),&\quad\forall
s_{0}\in[z_{j,a},x),\\\ F(z_{j,a})+p_{x},&\quad\forall s_{0}\in[x,y),\\\
F(z_{j,b})&\quad\forall s_{0}\in[y,z_{j,b}].\end{array}\right.$

We need to ensure that $\int_{z\leq s_{0}}F(z)dz-\int_{z\leq
s_{0}}G^{*}(z)dz\geq 0$ for all $s_{0}\in[z_{j,a},z_{j,b}]$. We define
$\Delta(s_{0})=\int_{z\leq s_{0}}F(z)dz-\int_{z\leq s_{0}}G^{*}(z)dz$, and

$\displaystyle\frac{d\Delta(s_{0})}{ds_{0}}=F(s_{0})-G^{*}(s_{0})\left\\{\begin{array}[]{ll}=0,&\quad
s_{0}=z_{j,a},\\\ >0,&\quad\forall s_{0}\in(z_{j,a},x)\\\ <0,&\quad\forall
s_{0}\in[x,F^{-1}(F(z_{j,a})+p_{x})),\\\ =0,&\quad
s_{0}=F^{-1}(F(z_{j,a})+p_{x}),\\\ >0,&\quad\forall
s_{0}\in(F^{-1}(F(z_{j,a})+p_{x}),y),\\\ <0,&\quad\forall
s_{0}\in[y,z_{j,b})\\\ =0,&\quad s_{0}=z_{j,b}.\end{array}\right.$ (72)

Therefore, we only need to verify that $\Delta(s_{0})\geq 0$ for
$s\in\\{z_{j,a},F^{-1}(F(z_{j,a})+p_{x}),z_{j,b}\\}$. We know from Lemma 11
that $\Delta(z_{j,a})=\Delta(z_{j,b})=0$. Thus, we need to satisfy:

$\displaystyle\Delta(F^{-1}(F(z_{j,a})+p_{x}))=\int_{z_{j,a}}^{F^{-1}(F(z_{j,a})+p_{x})}F(z)dz-\int_{z_{j,a}}^{F^{-1}(F(z_{j,a})+p_{x})}G^{*}(z)dz$
$\displaystyle=$
$\displaystyle\int_{z_{j,a}}^{F^{-1}(F(z_{j,a})+p_{x})}F(z)dz-\left(F(z_{j,a})(x-z_{j,a})+(F(z_{j,a})+p_{x})(F^{-1}(F(z_{j,a})+p_{x})-x)\right)$
$\displaystyle=$
$\displaystyle\int_{z_{j,a}}^{F^{-1}(F(z_{j,a})+p_{x})}(F(z)-F(z_{j,a}))dz-
p_{x}(F^{-1}(F(z_{j,a})+p_{x})-x)\geq 0.$

We note that when the above inequality is tight, the double-interval structure
degenerates to two disjoint pooling intervals each with a single atom $x$ and
$y$ separately. Therefore, $p_{x}$ must satisfy (64c). $\square$

######  Lemma 13.

Suppose $[z_{j,a},z_{j,b}]$ admits a double-interval partition with given
atoms $x,y$. Define

$\displaystyle h(\epsilon)$
$\displaystyle=x(F(z_{j,a})-F(z_{j,a}-\epsilon))-\int_{z_{j,a}-\epsilon}^{z_{j,a}}zdF(z)-(y-x)p_{y},$
(73a) $\displaystyle\tilde{h}(\epsilon)$
$\displaystyle=\int_{z_{j,b}}^{z_{j,b}+\epsilon}zdF(z)-y(F(z_{j,b}+\epsilon)-F(z_{j,b}))-(y-x)p_{x}.$
(73b)

Let $\epsilon_{a}$ (resp. $\epsilon_{b}$) be defined as the smallest solution
of $h(\epsilon)=0$ (resp. $\tilde{h}(\epsilon)=0$) when
$h(z_{j,a}-\inf\mathcal{S}_{0})\geq 0$ (resp.
$\tilde{h}(\sup\mathcal{S}_{0}-z_{j,b})\geq 0$), and
$z_{j,a}-\inf\mathcal{S}_{0}$ (resp. $\sup\mathcal{S}_{0}-z_{j,b}$) otherwise.
For any $\epsilon\in(0,\epsilon_{a})$ (resp. $\epsilon\in(0,\epsilon_{b})$),
$[z_{j,a}-\epsilon,z_{j,b}]$ (resp. $[z_{j,a},z_{j,b}+\epsilon]$) is also
associated with a double-interval structure with atoms $x,y$. Furthermore,
$\mathbb{E}_{F}[S_{0}|z_{j,a}-\epsilon_{a}\leq S_{0}\leq z_{j,b}]=x$ when
$\epsilon_{a}<z_{j,a}-\inf\mathcal{S}_{0}$, and
$\mathbb{E}_{F}[S_{0}|z_{j,a}\leq S_{0}\leq z_{j,b}+\epsilon_{b}]=y$ when
$\epsilon_{b}<\sup\mathcal{S}_{0}-z_{j,b}$.

_Proof of Lemma 13._ We first prove for the interval
$[z_{j,a}-\epsilon,z_{j,b}]$. We denote the interval associated with $x$
(resp. $y$) as $[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]$ (resp.
$[z_{j}^{{}^{\prime}},z_{j}^{"}]$), and the probability as $p_{x}$ (resp.
$p_{y}$). We define

$u(\epsilon)=F(z_{j,a})-F(z_{j,a}-\epsilon),\quad
v(\epsilon)=\int_{z_{j,a}-\epsilon}^{z_{j,a}}zdF(z).$

We construct $p_{x}^{\prime}(\epsilon)=p_{x}+u(\epsilon)+\delta(\epsilon)$ and
$p_{y}^{\prime}(\epsilon)=p_{y}(\epsilon)-\delta(\epsilon)$, where

$\delta(\epsilon)=\frac{xu(\epsilon)-v(\epsilon)}{y-x}.$

We note that for any $\delta(\epsilon)\in(0,p_{y})$, there exists a sub-
interval
$[\tilde{z}^{{}^{\prime}}_{j},\tilde{z}^{"}_{j}]\subseteq[z_{j}^{{}^{\prime}},z_{j}^{"}]$
such that $\mathbb{E}_{F}[S_{0}|\tilde{z}^{{}^{\prime}}_{j}\leq
S_{0}\leq\tilde{z}^{"}_{j}]=y$ and
$F(\tilde{z}^{"}_{j})-F(\tilde{z}^{{}^{\prime}}_{j})=p_{y}-\delta$. Thus, the
double-interval structure
$[z_{j,a}-\epsilon,\tilde{z}^{{}^{\prime}}_{j}]\cup[\tilde{z}^{"}_{j},z_{j,b}]$
(resp. $[\tilde{z}^{{}^{\prime}}_{j},\tilde{z}^{"}_{j}]$) generates $x$ (resp.
$y$) with probability $p_{x}^{\prime}(\epsilon)$ (resp.
$p_{y}^{\prime}(\epsilon)$) for the extended interval
$[z_{j,a}-\epsilon,z_{j,b}]$.

For any $\epsilon<\inf\mathcal{S}_{0}-z_{j,a}$, we can check that the value of
$\delta(\epsilon)$ is non-decreasing in $\epsilon$, and strictly increasing if
$f(z_{j,a}-\epsilon)>0$, where $f(\cdot)$ is the probability density function
corresponding to $F(\cdot)$:

$\frac{d\delta(\epsilon)}{d\epsilon}=\frac{1}{y-x}\left(xf(z_{j,a}-\epsilon)-(z_{j,a}-\epsilon)f(z_{j,a}-\epsilon)\right)=\frac{f(z_{j,a}-\epsilon)(x-z_{j,a}+\epsilon)}{y-x}\geq
0.$

We note that $\delta(0)=0$. Since $\delta(\epsilon)$ is non-decreasing in
$\epsilon$, if $\delta(z_{j,a}-\inf\mathcal{S}_{0})<p_{y}$, then
$\delta(\epsilon)<p_{y}$ for any $\epsilon\in[0,z_{j,a}-\inf\mathcal{S}_{0}]$.
This indicates that for any $\epsilon\in[0,z_{j,a}-\inf\mathcal{S}_{0}]$, the
extended interval is also associated with a double-interval structure with
atoms $x$ and $y$.

On the other hand, if $\delta(z_{j,a}-\inf\mathcal{S}_{0})\geq p_{y}$, then
(73a), which is equivalent to $\delta(\epsilon)=p_{y}$, has a solution
$\hat{\epsilon}$. This solution is unique when $f(\epsilon)>0$ in a local
neighborhood of $\hat{\epsilon}$. In this case $\epsilon_{a}=\hat{\epsilon}$.
Additionally, if there exists a local neighborhood of $\hat{\epsilon}$ such
that $f(\epsilon)=0$, then the solution set of (73a) is a closed interval, and
$\epsilon_{a}$ equals to the the smallest solution. In both cases, for any
$\epsilon\in[0,\epsilon_{a})$, the extended interval is also associated with a
double-interval structure with atoms $x$ and $y$. Moreover, when
$\epsilon=\epsilon_{a}$, the double interval structure becomes degenerate and
$p_{y}^{\prime}(\epsilon)=0$. Thus,
$\mathbb{E}_{F}[S_{0}|z_{j,a}-\epsilon_{a},z_{j,b}]=x$.

Similarly, we can show that the interval $[z_{j,a},z_{j,b}+\epsilon]$ is
associated with $p_{x}^{\prime}(\epsilon)=p_{x}-\tilde{\delta}(\epsilon)$ and
$p_{y}^{\prime}=p_{y}+\tilde{u}(\epsilon)+\tilde{\delta}(\epsilon)$, where

$\tilde{u}(\epsilon)=F(z_{j,b}+\epsilon)-F(z_{j,b}),\quad\tilde{v}(\epsilon)=\int_{z_{j,b}}^{z_{j,b}+\epsilon}zdF(z),\quad\tilde{\delta}(\epsilon)=\frac{\tilde{v}-\tilde{u}y}{y-x}.$

Additionally, there exists a double-interval structure
$[z_{j,a},\tilde{z}^{{}^{\prime}}_{j}]\cup[\tilde{z}^{"}_{j},z_{j,b}]$ and
$[\tilde{z}^{{}^{\prime}}_{j},\tilde{z}^{"}_{j}]$ associated atoms $x,y$,
where
$[\tilde{z}^{{}^{\prime}}_{j},\tilde{z}^{"}_{j}]\supseteq[z_{j}^{{}^{\prime}},z_{j}^{"}]$.
Similarly, the value of $\tilde{\delta}(\epsilon)$ is non-decreasing in
$\epsilon$. If $\tilde{\delta}(\sup\mathcal{S}_{0}-z_{j,b})<p_{x}$, then
$\tilde{\delta}(\epsilon)<p_{x}$ for any
$\epsilon\in[0,\sup\mathcal{S}_{0}-z_{j,b}]$, indicating that for any
$\epsilon\in[0,\sup\mathcal{S}_{0}-z_{j,b}]$, the extended interval is also
associated with a double-interval structure with atoms $x$ and $y$. On the
other hand, if $\tilde{\delta}(\sup\mathcal{S}_{0}-z_{j,b})\geq p_{x}$, then
by taking $\epsilon_{b}$ equals to the smallest solution of (73b), the
extended interval is also associated with a double-interval structure with
atoms $x$ and $y$ for any $\epsilon\in[0,\epsilon_{b})$. Moreover, when
$\epsilon=\epsilon_{b}$, the double interval structure becomes degenerate and
$\mathbb{E}_{F}[S_{0}|z_{j,a},z_{j,b}+\epsilon_{b}]=y$.

_Proof of Proposition 5._ Assume for the sake of contradiction that there does
not exist an optimal information mechanism that is monotone partitional. From
Lemma 10, we know that the intervals $\\{I_{j}\\}_{j\in J}$ associated with
the optimal posterior mean distribution $G^{*}$ must have at least one $j$
such that $I_{j}=[z_{j,a},z_{j,b}]$ has a double-interval structure: pooling
states in the intervals $[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]$
to generate atom $x$ and pooling states in the interval
$[z_{j}^{{}^{\prime}},z_{j}^{"}]$ to generate atom $y$, where
$z_{j,a}<z_{j}^{{}^{\prime}}<z_{j}^{"}<z_{j,b}$ and $z_{j,a}<x<y<z_{j,b}$.
Moreover, we denote the probability of atom $x$ as $p_{x}$ and that of atom
$y$ as $p_{y}$. Then, we must have:

$\displaystyle p_{x}$
$\displaystyle=\mathrm{Pr}(S_{0}\in[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}])=\int_{z_{j,a}}^{z_{j}^{{}^{\prime}}}dF(z)+\int_{z_{j}^{"}}^{z_{j,b}}dF(z),$
(74a) $\displaystyle p_{y}$
$\displaystyle=\mathrm{Pr}(S_{0}\in[z_{j}^{{}^{\prime}},z_{j}^{"}])=\int_{z_{j}^{{}^{\prime}}}^{z_{j}^{"}}dF(z),$
(74b) $\displaystyle x$
$\displaystyle=\frac{1}{p_{x}}\left(\int_{z_{j,a}}^{z_{j}^{{}^{\prime}}}zdF(z)+\int_{z_{j}^{"}}^{z_{j,b}}zdF(z)\right),$
(74c) $\displaystyle y$
$\displaystyle=\frac{1}{p_{y}}\left(\int_{z_{j}^{{}^{\prime}}}^{z_{j}^{"}}zdF(z)\right).$
(74d)

In our setting, $R(s_{0})$ is a Lipchitz continuous function. Thus, we know
from Lemma 9 that for any optimal posterior mean distribution $G^{*}$, there
must exist a convex upper closure function $\nu(s_{0})$ such that conditions
$(G^{*},\nu)$ satisfies (44a) – (44c). We note that

$\displaystyle\int_{z\in[z_{j,a},z_{j,b}]}\nu(z)dF(z)=\int_{z\in[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]}\nu(z)dF(z)+\int_{z\in[z_{j}^{{}^{\prime}},z_{j}^{"}]}\nu(z)dF(z)$
$\displaystyle=$ $\displaystyle
p_{x}\int_{z\in[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]}\frac{\nu(z)dF(z)}{p_{x}}+p_{y}\int_{z\in[z_{j}^{{}^{\prime}},z_{j}^{"}]}\frac{\nu(z)dF(z)}{p_{y}}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\geq}}$ $\displaystyle
p_{x}\nu\left(\frac{\int_{z\in[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]}zdF(z)}{p_{x}}\right)+p_{y}\nu\left(\frac{\int_{z\in[z_{j}^{{}^{\prime}},z_{j}^{"}]}zdF(z)}{p_{y}}\right)\stackrel{{\scriptstyle(b)}}{{=}}p_{x}\nu\left(x\right)+p_{y}\nu(y)$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}$
$\displaystyle\int_{z\in[z_{j,a},z_{j,b}]}\nu(z)dG(z),$ (75)

where (a) is due to the fact that $\nu(s_{0})$ is convex, (b) follows from
(74c) – (74d) and (c) is due to the fact that $G^{*}$ assigns probability mass
$p_{x}$ to atom $x$ and $p_{y}$ to $y$. From (44b), we know that the
inequality in (75) must be equality, indicating that the function $\nu(s_{0})$
is an affine function in $[z_{j,a},z_{j,b}]$. We denote this function as
$\nu(s_{0})=\gamma s_{0}+\lambda$.

Furthermore, we know from Lemma 9 that $\nu(s_{0})\geq R(s_{0})$ for
$s_{0}\in[z_{j,a},z_{j,b}]$, and $R(x)=\nu(x)$, $R(y)=\nu(y)$. We argue that
the following statements are true:

  * (i)

$x,y$ are not in the same equilibrium regime, i.e. not belong to the same
linear piece of $R(s)$, since otherwise the double interval can be replaced by
either fully revealing the state in $[z_{j,a},z_{j,b}]$ or pooling all states
in $[z_{j,a},z_{j,b}]$ without changing the expected revenue.

  * (ii)

$x,y$ are not interior points of two different regimes. This is because if $x$
(or $y$) is in the interior of a regime $k$ such that the derivative
$\frac{dR(s_{0})}{ds_{0}}[k]\neq\gamma$, then there must exist
$0<\epsilon<\min\\{x-z_{j,a},z_{j,b}-y\\}$ such that $R(z)>\nu(z)$ for some
$z\in(x-\epsilon,x+\epsilon)$ (or $z\in(y-\epsilon,y+\epsilon)$),
contradicting to the fact that $\nu(s_{0})\geq R(s_{0})$ for all
$s_{0}\in[z_{j,a},z_{j,b}]$. On the other hand, if
$\frac{dR(s_{0})}{ds_{0}}[k]=\gamma$, then Assumption 4 is violated.

Thus, $x,y$ must be two regime boundaries. We further show that $x,y$ are the
boundaries of $s_{0}[k_{\ell}^{*}]$ and $s_{0}[k_{\ell^{\prime}}^{*}]$
respectively, and they belong to two concave intervals $\ell<\ell^{\prime}$.
From Assumption 4, we know that $R(s_{0})$ must be either strictly convex or
strictly concave at each regime boundary. If $x$ (or $y$) is at a regime
boundary that is strictly convex, then there must exist
$\epsilon<\min\\{x-z_{j,a},z_{j,b}-y\\}$ such that $R(z)>\nu(z)$ for some
$z\in(x-\epsilon,x+\epsilon)$ (or $(y-\epsilon,y+\epsilon)$). Thus, $x$ and
$y$ must be two regime boundaries such that $R(s_{0})$ is strictly concave.
Moreover, the two regime boundaries cannot belong to the same concave interval
since otherwise $x,y$ must be associated with the same regime, which has been
ruled out in (i). Therefore, we must have $x=s_{0}[k_{\ell}^{*}]$,
$y=s_{0}[k_{\ell^{\prime}}^{*}]$, where $\ell<\ell^{\prime}$ are two concave
intervals. Furthermore, we must have:

$\displaystyle\gamma\in\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell}^{*}-1]\right]\cap\left[\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}],\frac{dR(s_{0})}{ds_{0}}[k_{\ell^{\prime}}^{*}-1]\right].$

Under condition (C1), such two concave intervals $\ell<\ell^{\prime}$ do not
exist, and thus the optimal information mechanism cannot have a double-
interval structure, indicating that the optimal information mechanism is
monotone partitional.

Under condition (C2), we know from Lemma 2 that such affine function
$\nu(s_{0})$ does not exist for any two concave intervals
$\ell<\ell^{\prime}$. Thus, an optimal information mechanism must be monotone
partitional.

Under condition (C3), assume for the sake of contradiction that there exists a
pair of concave intervals $\ell<\ell^{\prime}$ such that the optimal
information mechanism has a double-interval structure associated with the
affine function $g(s_{0})$ tangent to $R(s_{0})$ in the two concave intervals,
and the two atoms $x=s_{0}[k_{\ell}^{*}]$, $y=s_{0}[k_{\ell^{\prime}}^{*}]$.
Then, we know from Lemma 9 that there must exist a convex function
$\nu(s_{0})$ such that $(G^{*},\nu)$ satisfies (44), where $G^{*}$ is the
optimal posterior mean distribution.

We know from the argument above that when the optimal information mechanism
contains the double interval structure associated with $x,y$, there must exist
an interval
$[\underline{z},\bar{z}]\subseteq[\underline{z}_{\ell},\bar{z}_{\ell^{\prime}}]$
such that $\underline{z}<x<y<\bar{z}$ and $\nu(z)=g(z)$ for
$z\in[\underline{z},\bar{z}]$. Here, we recall that
$\underline{z}_{\ell}=\max\\{z<x|R(z)\geq g(z)\\}$ and
$\bar{z}_{\ell^{\prime}}=\min\\{z>y|R(z)\leq g(z)\\}$.

Consider the scenario where $x,y<\mathbb{E}_{F}[S_{0}]$ and
$\bar{z}_{\ell^{\prime}}=\sup\mathcal{S}_{0}$. Since $\nu(s_{0})$ is a convex
function, we know that

$\displaystyle\nu(s_{0})\stackrel{{\scriptstyle(a)}}{{\geq}}g(s_{0})\stackrel{{\scriptstyle(b)}}{{>}}R(s_{0}),\quad\forall
s_{0}>y,$ (76)

where (a) is due to the convexity of $\nu(s_{0})$ and (b) follows from the
definition of $\bar{z}_{\ell^{\prime}}$ and the fact that
$\bar{z}_{\ell^{\prime}}=\sup S_{0}$. Consequently, we know that
$\\{\mathcal{S}_{0}|R(s_{0})=\nu(s_{0})\\}\cap\\{\mathcal{S}_{0}|s_{0}>y\\}=\emptyset$.
Following from (44a)
$supp(G^{*})\subseteq\\{\mathcal{S}_{0}|R(s_{0})=\nu(s_{0})\\}$ and the fact
that $y<\mathbb{E}_{F}[S_{0}]$, we know that
$supp(G^{*})\cap\\{\mathcal{S}_{0}|s_{0}\geq\mathbb{E}_{F}[S_{0}]\\}=\emptyset$,
which contradicts the fact that $G^{*}$ is a mean-preserving contraction of
the prior $F$.

Similarly, for the other scenario where $x,y>\mathbb{E}_{F}[S_{0}]$ and
$\underline{z}_{\ell}=\inf\mathcal{S}_{0}$, we can show that
$supp(G^{*})\cap\\{\mathcal{S}_{0}|s_{0}\leq\mathbb{E}_{F}[S_{0}]\\}=\emptyset$,
which again contradicts the fact that $G^{*}$ is a mean preserving contraction
of $F$.

Therefore, we can conclude that in both cases, the optimal information
mechanism does not contain a double-interval structure associated with concave
intervals $\ell<\ell^{\prime}$. Since this argument holds for all such pairs,
we know that the optimal information mechanism must be monotone partitional.

We next prove that when the prior distribution $F$ satisfies condition (C4),
then for any pair of $\ell<\ell^{\prime}$ that violates (C2), the two atoms
$x,y$ associated with the two concave intervals cannot be generated by a
double interval structure. For the sake of contradiction, we assume that there
exists a pair of $\ell<\ell^{\prime}$ such that the associated two atoms
$x=s_{0}[k_{\ell}^{*}]$, $y=s_{0}[k_{\ell^{\prime}}^{*}]$ are generated by a
feasible double-interval
$[z_{j,a},z_{j}^{{}^{\prime}}]\cup[z_{j}^{"},z_{j,b}]$ and
$[z_{j}^{{}^{\prime}},z_{j}^{"}]$, respectively. Since $\nu(s_{0})\geq
R(s_{0})$, we know that the interval
$[z_{j,a},z_{j,b}]\subseteq[\underline{z}_{\ell},\bar{z}_{\ell^{\prime}}]$,
where $\underline{z}_{\ell}$ and $\bar{z}_{\ell^{\prime}}$ are given by (15).
Moreover, from Lemma 12, we know that there must exist $p_{x}>0$ and $p_{y}>0$
that satisfies (64) given the interval
$[z_{j,a},z_{j,b}]\subseteq[\underline{z}_{\ell},\bar{z}_{\ell^{\prime}}]$.

From Lemma 13, we know that there exists $\epsilon\in(0,\epsilon_{a})$ (resp.
$\epsilon\in(0,\epsilon_{b})$) such that $[z_{j,a}-\epsilon,z_{j,b}]$ (resp.
$[z_{j,a},z_{j,b}+\epsilon]$) is also associated with a double-interval
structure with atoms $x,y$. Consequently, the interval
$[z_{j,a},z_{j,b}]\subseteq[\underline{z}_{\ell},\bar{z}_{\ell^{\prime}}]$
being associated with a double-interval structure implies that one of the
following three cases holds:

  1. (a)

$\epsilon_{a}\leq z_{j,a}-\underline{z}_{\ell}$, i.e. there exists
$z^{\dagger}\in(\underline{z}_{\ell},z_{j,a})$ such that

$\displaystyle\mathbb{E}_{F}[S_{0}|z^{\dagger}\leq S_{0}\leq z_{j,b}]=x.$ (77)

  2. (b)

$\epsilon_{b}\leq\bar{z}_{\ell^{\prime}}-z_{j,b}$, i.e. there exists
$z^{\ddagger}\in(z_{j,b},\bar{z}_{\ell^{\prime}})$ such that

$\mathbb{E}_{F}[S_{0}|z_{j,a}\leq S_{0}\leq z^{\ddagger}]=y.$

  3. (c)

$\epsilon_{a}\geq z_{j,a}-\underline{z}_{\ell}$,
$\epsilon_{b}\geq\bar{z}_{\ell^{\prime}}-z_{j,b}$, and thus the interval
$[\underline{z}_{\ell},\bar{z}_{\ell^{\prime}}]$ can be generated by a double
interval structure associated with atoms $x,y$.

In case (a), (77) implies that

$\mathbb{E}_{F}[S_{0}|\underline{z}_{\ell}\leq S_{0}\leq
y]<\mathbb{E}_{F}[S_{0}|z^{\dagger}\leq S_{0}\leq
z_{j,b}]=x=s_{0}[k_{\ell}^{*}],$

which contradicts (14) in condition (C4). Therefore, under condition (C4),
case (a) does not hold.

Similarly, in case (b), we have

$\mathbb{E}_{F}[S_{0}|x\leq
S_{0}\leq\bar{z}_{\ell^{\prime}}]>\mathbb{E}_{F}[S_{0}|z_{j,a}\leq S_{0}\leq
z^{\ddagger}]=y=s_{0}[k_{\ell^{\prime}}^{*}],$

which also contradicts (14) in condition (C4), and thus case (b) does not
hold.

Moreover, case (c) implies that there exists $p_{x}^{\prime},p_{y}^{\prime}$
such that

$\displaystyle
p_{x}^{\prime}+p_{y}^{\prime}=F(\bar{z}_{\ell^{\prime}})-F(\underline{z}_{\ell}),$
(78a) $\displaystyle
xp_{x}^{\prime}+yp_{y}^{\prime}=\int_{\underline{z}_{\ell}}^{\bar{z}_{\ell^{\prime}}}zdF(z),$
(78b)
$\displaystyle(F^{-1}(F(\underline{z}_{\ell})+p_{x}^{\prime})-x)p_{x}^{\prime}<\int_{\underline{z}_{\ell}}^{F^{-1}(F(\underline{z}_{\ell})+p_{x}^{\prime})}(F(z)-F(\underline{z}_{\ell}))dz,$
(78c) $\displaystyle p_{x}^{\prime}>0,\quad p_{y}^{\prime}>0.$ (78d)

We denote $z^{*}=\int_{\underline{z}_{\ell}}^{\bar{z}_{\ell^{\prime}}}zdF(z)$.
Then, from (78a) and (78b), we have:

$p_{x}^{\prime}=\frac{y\left(F(\bar{z}_{\ell^{\prime}})-F(\underline{z}_{\ell})\right)-z^{*}}{y-x}.$

In case (c), we have $p_{x}^{\prime}\in(0,1)$ since cases (a) and (b) do not
happen. Thus, (78c) indicates that

$\displaystyle(F^{-1}(F(\underline{z}_{\ell})+p_{x}^{\prime})-x)p_{x}^{\prime}=\left(F^{-1}\left(\frac{yF(\bar{z}_{\ell^{\prime}})-xF(\underline{z}_{\ell})-z^{*}}{y-x}\right)-x\right)\frac{y\left(F(\bar{z}_{\ell^{\prime}})-F(\underline{z}_{\ell})\right)-z^{*}}{y-x}$
$\displaystyle<$
$\displaystyle\int_{\underline{z}_{\ell}}^{F^{-1}\left(\frac{yF(\bar{z}_{\ell^{\prime}})-xF(\underline{z}_{\ell})-z^{*}}{y-x}\right)}(F(z)-F(\underline{z}_{\ell}))dz,$

which contradicts (14) of condition (C4). Thus, under (C4), all three cases
cannot hold, which implies that there exists an optimal monotone partitional
information mechanism. $\square$

The proof of Proposition 6 directly builds on Lemma 3 and Candogan [2019], and
thus is omitted.