Persuasion and Welfare

## A. 3 Omitted proofs in Section 5

Proof of Theorem 3. As explained in the main text, necessity follows from noting

$$
\mathrm{w} \in \mathrm{~W} \Rightarrow \mathrm{w}=\mathrm{D}_{0} \sum_{m=1}^{M} \alpha_{m} \mu_{m} \mu_{m}^{T} \rho \equiv \mathrm{D}_{0} \mathrm{C} \rho,
$$

where $M \leq 2 N$ follows from Corollary 1. C is completely positive because it is the convex combination of rank-one non-negative matrices, $\mu_{m} \mu_{m}^{T}$. That $\mathrm{Ce}=\mu_{0}$ follows from the martingale property of beliefs.

For sufficiency, consider $\mathrm{w}=\mathrm{D}_{0} \mathrm{C} \rho$, for some completely positive matrix C, such that $\mathrm{Ce}=\mu_{0}$. Then, $\left\{\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{M}}\right\} \subseteq \mathbb{R}_{+}^{N}$ exist such that

$$
\mathrm{C}=\sum_{m=1}^{M} \mathrm{x}_{\mathrm{m}} \mathrm{x}_{\mathrm{m}}^{T} .
$$

Let $\sqrt{\alpha_{m}}=\sum_{j=1}^{N} \mathrm{x}_{\mathrm{m} j}$ and note $\mathrm{x}_{\mathrm{m}} /\left(\sqrt{\alpha_{m}}\right) \equiv \mu_{m} \in \Delta(\Theta)$.

$$
\mathrm{C}=\sum_{m=1}^{M} \alpha_{m}\left(\frac{\mathrm{x}_{\mathrm{m}}}{\sqrt{\alpha_{m}}}\right)\left(\frac{\mathrm{x}_{\mathrm{m}}}{\sqrt{\alpha_{m}}}\right)^{T}=\sum_{m=1}^{M} \alpha_{m} \mu_{m} \mu_{m}^{T} .
$$

It remains to show $\sum_{m=1}^{M} \alpha_{m}=1$ and $\sum_{m=1}^{M} \alpha_{m} \mu_{m}=\mu_{0}$. Note that for all $i \in$ $\{1, \ldots, N\}$,

$$
(\mathrm{Ce})_{i}=\sum_{m=1}^{M} \alpha_{m} \mu_{m i} \sum_{j=1}^{N} \mu_{m j}=\sum_{m=1}^{M} \alpha_{m} \mu_{m i}=\mu_{0}\left(\theta_{i}\right) .
$$

Furthermore,

$$
\sum_{i=1}^{N} \mu_{0}\left(\theta_{i}\right)=1=\sum_{i=1}^{N} \sum_{m=1}^{M} \alpha_{m} \mu_{m i}=\sum_{m=1}^{M} \alpha_{m} .
$$

Thus, an information structure exists that generates the distribution over posteriors $\left\{\alpha_{m}, \mu_{m}\right\}_{m=1}^{M}$. Therefore, $\mathrm{w} \in \mathrm{W}$. $\square$

Proof of Claim 2. If $\operatorname{Pr}(X)=0$, the statement is trivial. If $\operatorname{Pr}(X)>0$, denote by $\mathrm{P}_{\mathrm{i}}$. the i-th row of the matrix $\mathrm{P} \equiv \mathrm{D}_{0} \mathrm{C}$, presented as a row-vector. By Bayes' rule, $\operatorname{Pr}(X)=\mu_{0}^{T} \beta$ and $\operatorname{Pr}\left(\theta_{i} \mid X\right)=\left(\mu_{0 i} \beta_{i}\right) /\left(\mu_{0}^{T} \beta\right)$, so

$$
\begin{gathered}
\mathbb{E}_{\Pi}\left[\operatorname{Pr}(X \mid s) \mid \theta_{i}\right]=\sum_{j=1}^{N} \mathbb{E}_{\Pi}\left[\operatorname{Pr}\left[\theta_{j} \mid s\right] \mid \theta_{i}\right] \operatorname{Pr}\left(X \mid \theta_{j}\right)=\mathrm{P}_{\mathrm{i} \cdot} \beta \\
\mathbb{E}_{\Pi}[\operatorname{Pr}(X \mid s) \mid X]=\sum_{i=1}^{N} \operatorname{Pr}\left(\theta_{i} \mid X\right) \mathbb{E}_{\Pi}\left[\operatorname{Pr}(X \mid s) \mid \theta_{i}\right]=\sum_{i=1}^{N} \frac{\mu_{0 i} \beta_{i}}{\mu_{0}^{T} \beta} \operatorname{Pi}_{\mathrm{i} \cdot \beta} .
\end{gathered}
$$

Hence, the truth-drifting condition can be restated as:

$$
\sum_{i=1}^{N} \frac{\mu_{0 i} \beta_{i}}{\mu_{0}^{T} \beta} P_{i \cdot} \beta \geq \mu_{0}^{T} \beta .
$$

Define $\hat{\mathrm{C}} \equiv \mathrm{PD}_{0}=\mathrm{D}_{0} \mathrm{CD}_{0}$. By Theorem 3, $\hat{\mathrm{C}}$ is a completely positive matrix such that $\hat{\mathrm{C}} \mu_{0}=\mathrm{e}$ and $\mu_{0}^{T} \hat{\mathrm{C}} \mu_{0}=1$. Hence, the truth-drifting condition can be restated in a matrix form as:

$$
\left(\frac{\mu_{0} * \beta}{\mu_{0}^{T} \beta}\right)^{T} \hat{\mathrm{C}}\left(\frac{\mu_{0} * \beta}{\mu_{0}^{T} \beta}\right) \geq \mu_{0}^{T} \hat{\mathrm{C}} \mu_{0} .
$$

The term $\zeta \equiv\left(\mu_{0} * \beta\right) /\left(\mu_{0}^{T} \beta\right)$ is an element of the simplex $\Delta(\Theta)$, equal to $\mu_{0}$ when $\beta=$ e. Hence, showing that $\mu_{0}$ is a minimizer of a quadratic form $\zeta^{T} \hat{\mathrm{C}} \zeta$ among all $\zeta \in \Delta(\Theta)$ is enough to prove the result. Noting that we can rely on the Lagrangian approach, at $\zeta=\mu_{0}$, the derivative of the quadratic form is collinear to e and hence, collinear to the space $\Delta(\Theta)$. Thus, first-order conditions are satisfied. At the same time, $\hat{\mathrm{C}}$ is completely positive and thus positive semi-definite. Thus, second-order conditions are satisfied. The result follows. $\square$

Example A. $4\left(\mathrm{w}^{N D}\right.$ and $\mathrm{w}^{F D}$ not on the boundary of W). Consider the case of binary types, $\Theta=\left\{\theta_{1}, \theta_{2}\right\}$. Denote by $\mu \in[0,1]$ the probability of type $\theta_{2}$ and let $\mu_{0}=1 / 2$. Consider the following welfare function:

$$
w\left(\mu, \theta_{1}\right)=\frac{\sin (2 \pi \mu)}{2(1-\mu)}, w\left(\mu, \theta_{2}\right)=\frac{\sin (4 \pi \mu)}{2 \mu},
$$

with $w\left(1, \theta_{1}\right)$ and $w\left(0, \theta_{2}\right)$ defined by continuity as equal to $-\pi$ and $2 \pi$, respectively. Given this welfare function, the truth-adjusted welfare function is

$$
\hat{w}\left(\mu, \theta_{1}\right)=\sin (2 \pi \mu), \hat{w}\left(\mu, \theta_{2}\right)=\sin (4 \pi \mu) .
$$

The corresponding indirect utility in the supporting Bayesian persuasion problem in the direction $\lambda=\left(\lambda_{1}, \lambda_{2}\right)$ is equal to

$$
\hat{v}_{\lambda}(\mu)=\lambda_{1} \sin (2 \pi \mu)+\lambda_{2} \sin (4 \pi \mu) .
$$

For any $\lambda \in \mathbb{R}^{2} \backslash\{0\}, \hat{v}_{\lambda}\left(\mu_{0}\right)=\hat{v}_{\lambda}(1 / 2)=\hat{v}_{\lambda}(0)=\hat{v}_{\lambda}(1)=0$. Hence, both full disclosure and no disclosure results in zero payoff. At the same time, for any such $\lambda$, $\hat{v}_{\lambda}(\mu)$ is a non-constant continuous function anti-symmetric around $\mu=1 / 2$. Hence, it achieves strictly positive values on [0, 1] and $\operatorname{cav} \hat{v}_{\lambda}\left(\mu_{0}\right)>0$ so that optimal disclosure outperforms both full disclosure and no disclosure. Theorem 2-extended to all boundary points-implies that $\mathrm{w}^{N D}$ and $\mathrm{w}^{F D}$ are not on the boundary of W.

The proofs of Proposition 3 and Corollary 4 rely on the graph-theoretic approach in Rayo and Segal (2010), the main properties of which we summarize in Remark A.1:


Persuasion and Welfare

Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.

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Remark A. 1 (Rayo and Segal, 2010). Rayo and Segal (2010) propose the following graphical depiction of an information structure, II. Given a direction $\lambda$, let the prospect values $\left(\frac{\lambda\left(\theta_{i}\right)}{\mu_{0}\left(\theta_{i}\right)}, \rho\left(\theta_{i}\right)\right)=\left(\gamma_{i}, \rho_{i}\right)$ for $i=1, \ldots, N$ be vertices of a graph in $\mathbb{R}^{2}$. Connect the points $\left(\gamma_{i}, \rho_{i}\right)$ and $\left(\gamma_{j}, \rho_{j}\right)$ by an edge if and only if a signal $s$ exists such that $\pi\left(s \mid \theta_{j}\right) \pi\left(s \mid \theta_{i}\right)>0$. The set of types that have positive probability under $s$ is called the pooling set of signal $s$.

Lemmas 2-5 in Rayo and Segal (2010) establish that under any optimal information structure, the following hold:

(a) the posterior expectations of the prospect values induced by any two signals are ranked (in vector order), that is, for any two signals $s, s^{\prime}$, either $\left(\mathbb{E}_{\mu_{s}}[\gamma(\tilde{\theta})], \mathbb{E}_{\mu_{s}}[\rho(\tilde{\theta})]\right) \geq$ $\left(\mathbb{E}_{\mu_{s^{\prime}}}[\gamma(\tilde{\theta})], \mathbb{E}_{\mu_{s^{\prime}}}[\rho(\tilde{\theta})]\right)$ or the opposite inequality holds.
(b) prospects appear in the support of some signal only if they lie on a straight line with non-positive slope,
(c) if the pooling segments ${ }^{22}$ of two signals do not lie on the same line, they can intersect only if they share an endpoint, and
(d) if two prospect values are ranked and appear in the support of two signals, then the posterior expectations induced by these signals are ranked in the same way.

Proof of Proposition 3. By the arguments presented in the main text, any optimal information structure solves the instance of the problem of Rayo and Segal (2010) in which the prospect values are ( $0, \rho\left(\theta_{j}\right)$ ) for $j \neq i$ and $\left(1, \rho\left(\theta_{i}\right)\right)$ for the sender and for the receiver, respectively.

[^17]Given the structure of the prospect values in our problem, the property in part (b) implies that $\theta_{i}$ is never pooled with lower-index types. Furthermore, whenever it is pooled with some type, it is pairwise pooled. The property in part (c) implies that whenever $\theta_{i}$ is pooled with some type $\theta_{j}$, then no types $\theta_{k}, \theta_{l}$ with $k<j<l$ can be pooled. Together with the property in part (d), this observation implies that whenever $\theta_{i}$ is pooled with some type $\theta_{j}$, it is also pooled with all types $\theta_{k}$ with $k>j$. Moreover, as $\theta_{i}$ is pooled with increasingly higher-index types, the corresponding posterior expectations increase in vector order, which means that higher signals induce higher reputation yet have a relatively higher proportion of $\theta_{i}$ (if all types are equally likely, then the probability of pooling $\theta_{i}$ with $\theta_{j}$ increases in $j$ ).

It is left to show that the threshold type-the lowest type with which $\theta_{i}$ is pooled-is not pooled with any type of lower index. However, because the threshold type is of higher index than $\theta_{i}$, such pooling could clearly be improved by pooling the threshold type exclusively with $\theta_{i}$. $\square$

Calculations for Example 3. Define the following parameterized family of information structures (rows correspond to types and columns to signals):

$$
\begin{aligned}
& \Pi_{1}(\alpha, \beta)=\left(\begin{array}{ccc}
\alpha & 1-\alpha & 0 \\
1-\beta & \beta & 0 \\
0 & 0 & 1
\end{array}\right), \quad \Pi_{2}(\alpha)=\left(\begin{array}{cc}
\alpha & 1-\alpha \\
1 & 0 \\
0 & 1
\end{array}\right), \\
& \Pi_{3}(\beta)=\left(\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\beta & 1-\beta
\end{array}\right), \quad \Pi_{4}(\alpha, \beta)=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & \alpha & 1-\alpha \\
0 & 1-\beta & \beta
\end{array}\right) .
\end{aligned}
$$

Note information structures $\Pi_{1}(1,1)$ and $\Pi_{4}(1,1)$ coincide and correspond to full disclosure. Likewise, information structures $\Pi_{2}(0)$ and $\Pi_{3}(1)$ both correspond to full pooling of types $\theta_{1}$ and $\theta_{3}$. Information structures $\Pi_{2}$ and $\Pi_{3}$ are the $\theta_{1}$-noisy-priority policy and the $\theta_{3}$-noisy-degrading policies, respectively. Like the $\theta_{3}$-noisy-priority policy, $\Pi_{1}$ separates $\theta_{3}$ from $\theta_{1}$ and $\theta_{2}$, but unlike the $\theta_{3}$-noisy-priority policy, it allows for $\theta_{1}$ and $\theta_{2}$ to be pooled, which does not affect $\theta_{3}$ 's expected reputation. Similarly, $\Pi_{4}$ separates $\theta_{1}$ from $\theta_{2}$ and $\theta_{3}$ like the $\theta_{1}$-noisy-degrading policy, but unlike this policy, it allows for $\theta_{2}$ and $\theta_{3}$ to be pooled.

By Proposition 2, any solution to the supporting Bayesian persuasion problem solves the instance of the problem of Rayo and Segal (2010) with prospect values $\left\{\left(\gamma_{i}, \rho_{i}\right):\right.$ $i \in\{1,2,3\}\}$ for the sender and for the receiver, respectively. For simplicity, we assume that the types are strictly ranked under $\rho$, i.e., $\rho_{1}<\rho_{2}<\rho_{3}$.

The property in part (b) in Remark A. 1 implies that if $\left(\gamma_{i}, \rho_{i}\right)<\left(\gamma_{j}, \rho_{j}\right)$, then types $\theta_{i}$ and $\theta_{j}$ are never pooled (cf. Corollary 4). We can then immediately establish the properties of the boundary information structures in the following cases:

- If $\gamma_{1}<\gamma_{2}<\gamma_{3}$, then the uniquely optimal information structure is full disclosure.
- If $\gamma_{2}<\gamma_{1}<\gamma_{3}$, then any optimal information structure separates type $\theta_{3}$ and belongs to class $\Pi_{1}(\alpha, \beta)$.
- If $\gamma_{1}<\gamma_{3}<\gamma_{2}$, then any optimal information structure separates type $\theta_{1}$ and belongs to class $\Pi_{4}(\alpha, \beta)$.
- If $\gamma_{2}<\gamma_{3}<\gamma_{1}$, then an optimal information structure never pools types $\theta_{2}$ and $\theta_{3}$ and belongs to class $\Pi_{2}(\alpha)$.
- If $\gamma_{3}<\gamma_{1}<\gamma_{2}$, then an optimal information structure never pools types $\theta_{1}$ and $\theta_{2}$ and belongs to class $\Pi_{3}(\beta)$.


Persuasion and Welfare

In the remaining case $\gamma_{3}<\gamma_{2}<\gamma_{1}$, no two prospects are ranked. However, by the property in part (a) in Remark A.1, the induced posterior expectations are necessarily ranked. Hence, if all three prospects lie on a straight line, then no disclosure is optimal. In contrast, if the three prospects do not lie on a straight line, then an optimal information structure separates either types $\theta_{1}$ and $\theta_{2}$ or types $\theta_{2}$ and $\theta_{3}$, and thus belongs to either class $\Pi_{2}(\alpha)$ or to class $\Pi_{3}(\beta)$.

Finally, it is easy to see that by the same arguments, an optimal information structure for the cases in which $\gamma_{i}=\gamma_{j}$ for some $i$ and $j$ belongs to one of the same four classes of information structures.

Knowing the classes of boundary information structures, we can plot the Bayes welfare set in Example 3 by direct calculation. $\square$


Persuasion and Welfare

## Online Appendix


Persuasion and Welfare

## B Data Limits

The analysis in the paper assumes that the information structure can arbitrarily condition on an individual's payoff-relevant type. However, this assumption does not necessarily hold in many applications of interest. For instance, regulation may prevent the disclosure of protected characteristics, such as gender or race. Thus, when $\theta$ encompasses such characteristics, considering information structures that respect these restrictions is natural.

In this section, we extend our analysis by removing this assumption. Formally, we consider the following extension of the model in Section 2. Together with the individuals' types, we are given a data source that is potentially informative about these types. The data source has realizations in a finite set $D \equiv\left\{d_{1}, \ldots, d_{M}\right\}$. We describe the joint distribution over payoff-relevant types and data via the prior distribution on $\Theta, \mu_{0}$, and a system of conditional probabilities $\left\{\nu_{0}(\cdot \mid \theta): \theta \in \Theta\right\}$, describing the distribution of the data source $d$ conditional on the types $\theta$. We let $\eta_{0} \in \Delta(D)$ denote the induced marginal distribution on $D .{ }^{23}$ The model in Section 2 corresponds to the case in which $\Theta=D$ and $\nu_{0}(d \mid \theta)=\mathbb{1}[d=\theta]$.

We assume information can be provided to the outside observer only about datasource realizations, but not an individual's type. Formally, an information structure $\Pi=(\pi, S)$ consists of a countable set of labels $S$ and a mapping $\pi$, which associates to each data-source realization, $d$, a distribution over signals $\pi(\cdot \mid d) \in \Delta(S)$. Given an information structure $\Pi$ and a signal realization $s \in S$, updated beliefs about $\theta$ depend only on the updated belief about the realization of $d$. Indeed,

$$
\mu_{s}(\theta)=\sum_{d \in D} \frac{\mu_{0}(\theta) \nu_{0}(d \mid \theta)}{\eta_{0}(d)} \eta_{s}(d),
$$

where $\eta_{s}$ is the marginal on $D$ of the updated joint belief on $\Theta \times D$. It follows that we can define the welfare function as depending on beliefs about $d$ rather than about $\theta$. That is, we can define the function $w_{\dagger}: \Delta(D) \times \Theta \mapsto \mathbb{R}$ as follows:

$$
w_{\dagger}(\eta, \theta)=w(\mu(\eta), \theta),
$$

where the function $\mu(\eta)$ is determined by Equation B.1.
Given an information structure ( $\pi, S$ ), the welfare of an individual of type $\theta$ is

$$
w_{\Pi}(\theta) \equiv \mathbb{E}_{\langle\Pi \mid \theta\rangle}\left[w_{\dagger}(\tilde{\eta}, \theta)\right]=\sum_{s \in S} \sum_{d \in D} \nu_{0}(d \mid \theta) \pi(s \mid d) w_{\dagger}\left(\eta_{s}, \theta\right),
$$

[^18]and the Bayes welfare set continues to be defined as the set of Bayes welfare profiles.
We now show the analysis in the main text extends verbatim. Indeed, by the same arguments as in Section 3, the welfare of an individual of type $\theta$ under information structure $\Pi=(\pi, S)$ can be written as:
$$
w_{\Pi}(\theta)=\mathbb{E}_{\langle\Pi \mid \theta\rangle}\left[w_{\dagger}(\tilde{\eta}, \theta)\right]=\mathbb{E}_{\langle\Pi\rangle}\left[\hat{w}_{\dagger}(\tilde{\eta}, \theta)\right],
$$
where the truth-adjusted welfare function $\hat{w}_{\dagger}$ now takes the form:
$$
\hat{w}_{\dagger}(\eta, \theta)=\sum_{d \in D} \nu_{0}(d \mid \theta) \frac{\eta(d)}{\eta_{0}(d)} w_{\dagger}(\eta, \theta) .
$$
By separating the variable on which welfare is conditioned on-the payoff-relevant types, $\theta$-from the variable about which information is provided-the data source, $d-$Equation B. 4 allows us to provide further insight into the truth-adjusted welfare function in the model in Section 2. Indeed, note the likelihood correction is based on the variable $d$, highlighting that it corresponds to the variable about which information is provided. Similar to before, we can interpret the likelihood-ratio adjustment as describing that each data-source realization $d$ has a budget $\eta_{0}(d)$ to be distributed across different (data) posteriors $\eta$. Unlike the analysis before, individuals of type $\theta$ only own a fraction $\nu_{0}(d \mid \theta)$ of this ratio.

Equation B. 3 implies Theorem 1 immediately extends to this setting:
Theorem B.1. The Bayes welfare set W satisfies the following:

$$
\mathrm{W}=\left\{\mathrm{w} \in \mathbb{R}^{N}:\left(\eta_{0}, \mathrm{w}\right) \in \operatorname{co}\left(\operatorname{graph} \hat{\mathrm{w}}_{\dagger}\right)\right\} .
$$

In what follows, we explore how the Bayes welfare set changes as we change the informativeness of the data source. Intuitively, we would expect that the Bayes welfare set shrinks as data becomes less precise. Proposition B. 1 below shows that this intuition holds when the notion of less precise coincides with the notion of garbling in Blackwell (1953).

Formally, given the distribution of payoff-relevant types $\mu_{0} \in \Delta(\Theta)$, we wish to understand the effect of different data sources, as described by data-source realizations $D^{\prime}$ and conditional probability systems $\left\{\nu_{0}^{\prime}(\cdot \mid \theta) \in \Delta\left(D^{\prime}\right): \theta \in \Theta\right\}$. Following Blackwell (1953), we say $\left(D^{\prime}, \nu_{0}^{\prime}\right)$ is a garbling of ( $D, \nu_{0}$ ) if a stochastic matrix $G: D \mapsto \Delta\left(D^{\prime}\right)$ exists such that for every data-type pair $\left(d^{\prime}, \theta\right)$,

$$
\nu_{0}^{\prime}\left(d^{\prime} \mid \theta\right)=\sum_{d \in D} G\left(d^{\prime} \mid d\right) \nu_{0}(d \mid \theta)
$$

Let $\mathrm{W}\left(\mu_{0}, w, D, \nu_{0}\right)$ denote the Bayes welfare set for prior type distribution $\theta$ and welfare function $w$, as we vary the (informativeness of the) data source $\left(D, \nu_{0}\right)$. We then have the following:

Proposition B. 1 (Data Comparison). $\mathrm{W}\left(\mu_{0}, w, D^{\prime}, \nu_{0}^{\prime}\right) \subseteq \mathrm{W}\left(\mu_{0}, w, D, \nu_{0}\right)$ for all welfare functions $w$ and type distributions $\mu_{0}$ if and only if $\left(D^{\prime}, \nu_{0}^{\prime}\right)$ is a garbling of $\left(D, \nu_{0}\right)$.


Persuasion and Welfare

Proof of Proposition B.1. One direction is straightforward: if $\left(D^{\prime}, \nu_{0}^{\prime}\right)$ is a garbling of $\left(D, \nu_{0}\right)$, any distribution of signals conditional on payoff-relevant types induced by some information structure under data source ( $D^{\prime}, \nu_{0}^{\prime}$ ) is feasible under data source $\left(D, \nu_{0}\right)$. Consequently, any welfare profile that can be induced by some information structure under ( $D^{\prime}, \nu_{0}^{\prime}$ ) can be induced under ( $D, \nu_{0}$ ).

To obtain the other direction, toward a contradiction, assume $\left(D^{\prime}, \nu_{0}^{\prime}\right)$ is not a garbling of $\left(D, \nu_{0}\right)$. Then, by Blackwell (1953), a prior $\mu_{0} \in \Delta(\Theta)$ and a payoff function $u:$ $A \times \Theta \rightarrow \mathbb{R}$ exist such that a decision maker with utility $u$ derives strictly greater value from having access to $\left(D^{\prime}, \nu_{0}^{\prime}\right)$ than to $\left(D, \nu_{0}\right)$. That is, letting $U(\mu)$ denote the decision maker's indirect utility, $\max _{a \in A} \mathbb{E}_{\mu}[u(a, \theta)]$, we have that:

$$
\sum_{\theta \in \Theta} \mu_{0}(\theta) \mathbb{E}_{\left(D^{\prime}, \nu_{0}^{\prime}\right)}[U(\mu) \mid \theta]>\sum_{\theta \in \Theta} \mu_{0}(\theta) \mathbb{E}_{\left(D, \nu_{0}\right)}[U(\mu) \mid \theta] .
$$

Consider now the Bayes welfare sets given the welfare function $w(\mu, \theta)=U(\mu)$ and prior distribution $\mu_{0}$, under data sources $\left(D, \nu_{0}\right)$ and $\left(D^{\prime}, \nu_{0}^{\prime}\right)$. We have that

$$
\begin{aligned}
\sum_{\theta \in \Theta} \mu_{0}(\theta) \mathbb{E}_{\left(D^{\prime}, \nu_{0}^{\prime}\right)}[U(\mu) \mid \theta] & =\max _{\mathrm{w} \in \mathrm{~W}\left(\cdot, D^{\prime}, \nu_{0}^{\prime}\right)} \sum_{\theta \in \Theta} \mu_{0}(\theta) \mathrm{w}(\theta) \\
& \leq \max _{\mathrm{w} \in \mathrm{~W}\left(\cdot, D, \nu_{0}\right)} \sum_{\theta \in \Theta} \mu_{0}(\theta) \mathrm{w}(\theta)=\sum_{\theta \in \Theta} \mu_{0}(\theta) \mathbb{E}_{\left(D, \nu_{0}\right)}[U(\mu) \mid \theta],
\end{aligned}
$$

where the equalities follow because the maximal ex ante payoff is obtained by having full access to available data, and the inequality follows from the assumption that $\mathrm{W}\left(\mu_{0}, w, D^{\prime}, \nu_{0}^{\prime}\right) \subseteq \mathrm{W}\left(\mu_{0}, w, D, \nu_{0}\right)$ for all $w$ and $\mu_{0}$. Comparing Equations B. 6 and B. 7 leads to the desired contradiction and the result follows. $\square$

Remark B. 1 (When to blind an algorithm). Proposition B. 1 stands in contrast with the recommendation in the algorithmic fairness literature to "blind" algorithms to sensitive inputs such as race or gender. Indeed, having taken into account the impact of the outside observer's incentives in the population's welfare, allowing the information structure to condition on the individuals' payoff-relevant types leads to the largest Bayes welfare set, thereby (weakly) increasing the value of any social welfare function that is used to choose what information structure to implement.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure B.1: Noisy data in the online marketplace. Bayes welfare sets for different values of $\sigma \in\{0.55,0.7,0.85,1\}$.

There may be other reasons outside our model that could justify blinding the information structure to the individuals' types. For instance, Liang et al. (2022) show that when an agent different from the social planner selects a decision-making algorithm, the social planner may prefer to restrict the inputs into the agent's algorithm. Even though in our model algorithms are information structures that send non-binding recommendations, a similar result would hold in our setting.

We conclude this section with an example that illustrates the following two points. First, whereas Proposition B. 1 shows that less precise data sources limit the ability to generate and distribute welfare via information, the example shows that this effect is not uniform across individuals of different types. Second, the Bayes welfare set may collapse to the no-disclosure Bayes welfare profile for data sources that are strictly more informative than no information in the Blackwell order.

Example B. 2 (Example 2 continued; Noisy Data). Suppose the online marketplace only has access to a noisy estimate of the consumer's type, perhaps from past purchases or undeleted cookies. We model this as a data source that reveals a consumer's type with a fixed precision $\sigma \in[1 / 2,1]: D=\left\{d_{1}, d_{2}\right\}$ and $\nu_{0}\left(d_{i} \mid \theta_{i}\right)=\sigma$. When $\sigma=1$, the data source is perfectly informative about a consumer's type; when $\sigma=1 / 2$, the data source is pure noise. More generally, if $\sigma<\sigma^{\prime}$, the data source that corresponds to $\sigma$ is a garbling of the data source that corresponds to $\sigma^{\prime}$.

Figure B. 1 illustrates the Bayes welfare set W for different precision values. Three features are worth noting. First, in line with Proposition B.1, Bayes welfare sets resulting from data sources with lower precision are subsets of those with higher precision.

When $\sigma=1$, the Bayes welfare set naturally coincides with the one in Figure 3b in the main text. Second, at high values of $\sigma$, lower data precision has asymmetric effects across types: it decreases the maximal payoff of $\theta_{L}$-consumers without affecting their minimal payoff, yet it increases the minimal payoff of $\theta_{H}$-consumers without affecting their maximal payoff. Indeed, for sufficiently low values of $\sigma$, the unique Pareto efficient information structure is the one that maximizes the payoff of $\theta_{H}$-consumers. That is, in this example, lower data precision benefits $\theta_{H}$-consumers. Finally, whereas it is immediate that the Bayes welfare set coincides with the no-disclosure profile $\mathrm{w}^{N D}$ when $\sigma=1 / 2$, the Bayes welfare set actually collapses to this point at $\sigma=3 / 5$ : Once $\sigma<3 / 5$, generating Bayes plausible distributions over posteriors with support outside the interval $[1 / 2,3 / 4)$ is not possible, and on this interval, $w$ is constant. This feature highlights that an incrementally more informative data source may have a discontinuous impact on welfare redistribution possibilities. $\square$


Persuasion and Welfare

[^0]:    *We thank the Editor, Emir Kamenica, and three anonymous referees for feedback that has greatly improved this paper. For valuable suggestions and comments, we would like to thank Ricardo Alonso, Odilon Câmara, Navin Kartik, Elliot Lipnowski, Antonio Penta, Jean Tirole, and Kai Hao Yang, as well as seminar participants at Toulouse School of Economics, Columbia, Bonn Winter Theory Workshop 2021, Warwick Theory Workshop 2022, Stony Brook 2022, ESSET 2022, EEA-ESEM 2022, and Virtual Seminars in Economic Theory. We thank Shunsuke Matsuno for excellent research assistance. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010).
    ${ }^{\dagger}$ Columbia Business School and CEPR. E-mail: laura.doval@columbia.edu.
    ${ }^{\ddagger}$ Toulouse School of Economics and CEPR. E-mail: alexey.v.smolin@gmail.com.

[^1]:    ${ }^{1}$ A matrix $\mathrm{C} \in \mathbb{R}^{N \times N}$ is completely positive if non-negative vectors $\mathrm{c}_{1}, \ldots, \mathrm{c}_{K} \in \mathbb{R}_{+}^{N}$ exist such that $\mathrm{C}=\sum_{i=1}^{K} \mathrm{c}_{i} \mathrm{c}_{i}^{T}$ (Berman, 1988).

[^2]:    ${ }^{2}$ Because maximizing uncertainty can sometimes be accomplished by providing some information, this notion of data privacy differs from an unambiguous preference for no information disclosure.

[^3]:    ${ }^{3}$ Note that the welfare function $w$ differs from the sender's indirect utility function in Kamenica and Gentzkow (2011), usually denoted by $\hat{v}$. The indirect utility function is the expectation under $\mu$ of the welfare function, $w(\mu, \cdot)$. That is, $\hat{v}(\mu)=\sum_{\theta \in \Theta} \mu(\theta) v(a(\mu), \theta)=\sum_{\theta \in \Theta} \mu(\theta) w(\mu, \theta)$.

[^4]:    ${ }^{4}$ Claim 2 provides a more general version of this result based on Theorem 3.
    ${ }^{5}$ Formally, for any information structure, $\Pi$, and type $\theta \in \Theta, \mathbb{E}_{\langle\Pi\rangle}\left[\tilde{\mu}(\theta) / \mu_{0}(\theta)\right]=1$.

[^5]:    ${ }^{6}$ Rosar (2017) and Quigley and Walther (2019) similarly observe that the distribution over posteriors conditional on an individual's type can be written in terms of the modified unconditional distribution.

[^6]:    ${ }^{7}$ For a real-valued function $f, \operatorname{cav} f$ denotes the smallest concave function that dominates $f$ and vex $f$ denotes the highest convex function dominated by $f$ (Hiriart-Urruty and Lemaréchal, 2004).
    ${ }^{8}$ See also Aumann and Maschler (1995) and Rayo and Segal (2010).

[^7]:    ${ }^{9}$ The maximum is attained because by definition $\mathrm{w} \in \mathrm{W}$. As we show in Appendix A.2, that $\lambda \in$ $\mathbb{R}_{+}^{N} \backslash\{0\}$ follows from $\mathrm{w} \in \mathrm{W}_{\mathrm{P}}$.

[^8]:    ${ }^{10}$ Thus, one can always interpret the heterogeneous priors model in Alonso and Câmara (2016) as a model in which the sender and the receiver share the same prior, but the sender assigns weights different than those under the prior $\mu_{0}$ to each of his possible types.

[^9]:    ${ }^{11}$ For instance, the midpoint $\mathrm{w}=(3 / 4,1 / 2)$ can only be generated by an information structure that employs at least three signals. One such information structure is given by

    $$
    \begin{array}{c|ccc}
    \theta_{H} & 1 / 4 & 3 / 8 & 3 / 8 \\
    \theta_{L} & 0 & 1 / 4 & 3 / 4
    \end{array} .
    $$

[^10]:    ${ }^{12}$ Appendix A. 2 provides weaker conditions under which all types (do not) benefit from disclosure.

[^11]:    ${ }^{13}$ For an even starker example, consider $\Theta=\left\{\theta_{1}, \theta_{2}\right\}$ and $w\left(\mu, \theta_{1}\right)=-\mu^{2}-2 \mu+3, w\left(\mu, \theta_{2}\right)=$ $-\mu^{2}+4 \mu$, where $\mu \equiv \mu\left(\theta_{2}\right)$. For each type, the welfare function is concave; however, for any prior distribution, the truth-adjusted welfare functions are globally convex. As a result, the unique Pareto efficient information structure is full disclosure.

[^12]:    ${ }^{14}$ Completely positive matrices have been studied extensively as they play an important role in optimization theory, machine learning, and other applications (Berman and Shaked-Monderer, 2003). A completely positive matrix is symmetric and positive-semidefinite, with positive elements; for $N \leq 4$, the converse is also true.

[^13]:    ${ }^{15}$ An analogous characterization appears in concurrent work by Sayin and Başar (2021), who discuss the computational advantages of working with completely positive matrices to solve information design problems.
    ${ }^{16} \mathrm{P}^{2} \mathrm{e}=\mathrm{Pe}=\mathrm{e}$ and $\mathrm{P}^{2}=\mathrm{D}_{0} \mathrm{C}^{\prime}$, where $\mathrm{C}^{\prime} \equiv \mathrm{CD}_{0} \mathrm{C}$ is completely positive because C is symmetric.

[^14]:    ${ }^{17}$ For concreteness, $X$ can be seen as a subset of $\Theta \times[0,1]$ equipped with a probability measure that agrees with $\mu_{0}$ on $\Theta$ (Green and Stokey, 1978; Gentzkow and Kamenica, 2017).
    ${ }^{18}$ In a setting in which information respects the state space's ordinal structure, Kartik et al. (2021) formalize the sense in which the drift toward the truth is stronger for more Blackwell informative information structures.
    ${ }^{19}$ Recall that the relative interior of a set $X$ is the interior of $X$ within its affine hull, which is the set of all affine combinations of elements in $X$.

[^15]:    ${ }^{20}$ We follow the terminology in Galperti et al. (2023), who highlight that individuals with certain types are valuable precisely because of the possibility of pooling them with other types.

[^16]:    ${ }^{21}$ Taking $\Theta$ to be a compact Polish space, we endow the set of Borel probability measures on $\Theta, \Delta(\Theta)$, and on $\Delta(\Theta), \Delta(\Delta(\Theta))$, with the weak* topology, so they are also compact Polish (Aliprantis and Border, 2013, Theorems 15.11 and 15.12).

[^17]:    ${ }^{22}$ By part (b), the pooling set of a signal lies on a segment.

[^18]:    ${ }^{23}$ Although we should index $\eta_{0}$ by $\left(\mu_{0},\left(\nu_{0}(\cdot \mid \theta)\right)_{\theta \in \Theta}\right)$, we omit this dependence to simplify notation.


Persuasion and Welfare

## Citation and provenance

**Authors:** Laura Doval; Alex Smolin

**Canonical citation:** Doval, Laura, and Alex Smolin. “Persuasion and Welfare.” Journal of Political Economy 132, no. 7 (2024): 2451–2487.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/persuasion-and-welfare.md

**Source record:** https://arxiv.org/abs/2109.03061

**Published record:** https://doi.org/10.1086/729067

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