Persuasion and Welfare

## 6 Conclusion

[^15]We provide a framework to study the potentially disparate impact of information policies in a population of heterogeneous individuals. Because information policies increasingly shape society's choices in high-stakes domains, the Bayes welfare set describes the limits of what society can achieve under such policies and the welfare trade-offs implied by the choice between different information policies. In the spirit of mechanism design and information design, our characterization of the Bayes welfare set provides a unifying tool to evaluate the welfare implications of different information policies across a wide array of objective functions.

We see several avenues worth exploring and left for future work. First, our model assumes any information can be provided about an individual's payoff-relevant type. However, this assumption does not necessarily hold in applications of interest in which an individual's type may include protected characteristics. In the online appendix, we extend our framework to accommodate limits on how much information can be disclosed about the individuals in the population. The analysis there, however, does not consider that these limits may be designed when a potentially malicious third party selects the information structure. Liang et al. (2022) consider this case in the context of decision-making algorithms, and we expect their insights to extend to the case of recommendations algorithms like the ones we consider.

Second, because the welfare function depends only on the first-order beliefs about an individual's type, our model only accounts for strategic interactions that follow the realization of a public signal (see, e.g., Laclau and Renou, 2017). Extending the analysis to account for general strategic interactions is worth exploring. Galperti et al. (2023), which studies the Bayes welfare profile that gives rise to the sender's maximum average payoff across all Bayes correlated equilibria, is a step in this direction.

Third, motivated by recent policies, Tirole (2021) studies the use of information in the form of a social score to incentivize good behavior in the population. Whereas Tirole (2021) studies this question in the context of a parametric family of information structures, our initial explorations show the Bayes welfare set allows us to extend his results by allowing any information structure. More generally, information has been suggested as a substitute for monetary incentives, and the Bayes welfare set describes what can be achieved with information alone.

Finally, whereas we characterize individuals' welfare as a function of their type, thinking of applications in which we care instead about the welfare of groups is natural. For instance, an individual's type could encompass their gender and their ability, and the social planner is concerned with the welfare different genders may obtain. This extension can inform the study of statistical discrimination, where recent work shows Bayesian persuasion tools can shed new light to this problem (Chambers and Echenique, 2021; Escudé et al., 2022; Deb and Renou, 2022). Whereas much of the existing litera-
ture focuses on statistical properties of discrimination, our work highlights the important aspect of economic welfare, offering a complementary perspective that enriches the ongoing dialogue between statistics and economics.


Persuasion and Welfare

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

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

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Persuasion and Welfare

## A Omitted results and proofs from the main text


Persuasion and Welfare

## A. 1 Omitted proofs from Section 3

For completeness, we include a proof of Claim 1, which follows from the analysis in Alonso and Câmara (2016, pp. 683-684), in present notation:

Proof of Claim 1. For an information structure $\Pi$, let $\operatorname{supp}\langle\Pi\rangle$ denote the support of the distribution over posterior beliefs induced by $\Pi$ and let $\operatorname{Pr}_{\Pi}(s)$ denote the unconditional probability of signal $s$ under the prior distribution $\mu_{0}$, i.e., $\operatorname{Pr}_{\Pi}(s)=$ $\sum_{\theta \in \Theta} \mu_{0}(\theta) \pi(s \mid \theta)$. Then, for a given type $\theta$, their welfare under information structure $\Pi$ can be written as follows:

$$
\begin{aligned}
w_{\Pi}(\theta)=\mathbb{E}_{\langle\Pi \mid \theta\rangle}[w(\tilde{\mu}, \theta)] & =\sum_{\mu \in \operatorname{supp}\langle\Pi\rangle} \sum_{s \in S: \mu_{s}=\mu} \pi(s \mid \theta) w(\mu, \theta) \\
& =\sum_{\mu \in \operatorname{supp}\langle\Pi\rangle} \sum_{s \in S: \mu_{s}=\mu} \operatorname{Pr}_{\Pi}(s) \frac{1}{\mu_{0}(\theta)} \frac{\mu_{0}(\theta) \pi(s \mid \theta)}{\operatorname{Pr}_{\Pi}(s)} w(\mu, \theta) \\
& =\sum_{\mu \in \operatorname{supp}\langle\Pi\rangle} \sum_{s \in S: \mu_{s}=\mu} \operatorname{Pr}_{\Pi}(s) \frac{\mu(\theta)}{\mu_{0}(\theta)} w(\mu, \theta) \\
& =\sum_{\mu \in \operatorname{supp}\langle\Pi\rangle} \sum_{s \in S: \mu_{s}=\mu} \operatorname{Pr}_{\Pi}(s) \hat{w}(\mu, \theta)=\mathbb{E}_{\langle\Pi\rangle}[\hat{w}(\tilde{\mu}, \theta)] .
\end{aligned}
$$ $\square$

Proof of Theorem 1. By definition, the point $\left(\mu_{0}, \mathrm{w}\right) \in \operatorname{co}(\operatorname{graph} \hat{\mathrm{w}})$ if and only if a Bayes plausible distribution over posteriors $\tau$ exists such that $\mathbb{E}_{\tau}[\hat{\mathrm{w}}(\mu)]=\mathrm{w}$. At the same time, the distribution over posteriors induced by an information structure, $\langle\Pi\rangle$, is Bayes plausible, i.e., $\mathbb{E}_{\langle\Pi\rangle}[\tilde{\mu}]=\mu_{0}$. The result follows from Claim 1. $\square$

On closedness Proposition A. 1 collects the results discussed in Section 3. To introduce Proposition A.1, we first introduce the analogue of the Bayes welfare set when the population's welfare depends on the outside observer's actions and together with the information structure, we can flexibly choose a selection from the outside observer's best-response correspondence. Formally, upon observing the realization from
the information structure, the outside observer takes an action in a finite set $A$ to maximize their expected payoff. We denote the outside observer's utility by $u: A \times \Theta \mapsto \mathbb{R}$ and the welfare function by $v: A \times \Theta \mapsto \mathbb{R}$. Given an information structure, $\Pi$, let $\operatorname{supp}\langle\Pi\rangle$ denote the support of the distribution over posteriors $\langle\Pi\rangle$. For each $\mu_{s} \in \operatorname{supp}\langle\Pi\rangle$, let

$$
\alpha\left(\mu_{s}\right) \in \Delta\left(\arg \max _{a \in A} \sum_{\theta \in \Theta} \mu_{s}(\theta) u(a, \theta)\right) \equiv \Delta\left(a^{*}(\mu)\right),
$$

denote the outside observer's (possibly mixed) best response. Recall that the Theorem of the Maximum implies that the correspondence $a^{*}(\mu)$ is upper-hemicontinuous.

Denote by $F$ set of tuples $(\Pi, \alpha)$, such that selection $\alpha$ satisfies Equation A.2. Each $(\Pi, \alpha) \in F$ defines a welfare profile, $w_{\Pi, \alpha}: \Theta \mapsto \mathbb{R}^{N}$, such that

$$
w_{\Pi, \alpha}(\theta)=\sum_{s \in S} \pi(s \mid \theta) \sum_{a \in A} \alpha\left(\mu_{s}\right)(a) v(a, \theta)=\mathbb{E}_{\langle\Pi \mid \theta\rangle}\left[\sum_{a \in A} \alpha(\tilde{\mu})(a) v(a, \theta)\right] .
$$

The analogue of the Bayes welfare set, which we denote by $\mathrm{W}_{\mathrm{BP}}$, is then

$$
\mathrm{W}_{\mathrm{BP}}=\left\{\mathrm{w} \in \mathbb{R}^{N}:(\exists(\Pi, \alpha) \in F) \text { s.t. } \mathrm{w}_{i}=w_{\Pi, \alpha}\left(\theta_{i}\right) \forall i \in\{1, \ldots, N\}\right\} .
$$

We have the following result:
Proposition A.1. The following hold:

(a) If for each $\theta \in \Theta$, the welfare function $w(\cdot, \theta)$ is continuous on $\Delta(\Theta)$, the Bayes welfare set W is closed.
(b) The set $\mathrm{W}_{\mathrm{BP}}$ is closed.

Proof of Proposition A.1. The proof of part (a) is immediate and hence omitted. Consider then part (b). Define the analogue of the truth-adjusted welfare function $\hat{v}(a, \mu, \theta)$ to be

$$
\hat{v}(a, \mu, \theta)=\frac{\mu(\theta)}{\mu_{0}(\theta)} v(a, \theta) .
$$

The same arguments as in Claim 1 imply that for any pair $(\Pi, \alpha) \in F$ and all $\theta \in \Theta$,

$$
w_{\Pi, \alpha}(\theta)=\mathbb{E}_{\langle\Pi\rangle}\left[\sum_{a \in A} \alpha(\tilde{\mu})(\theta) \hat{v}(a, \tilde{\mu}, \theta)\right] .
$$

Define the correspondence $\hat{V}: \Delta(\Theta) \rightrightarrows \mathbb{R}^{N}$ as follows

$$
\hat{V}(\mu)=\operatorname{co}\left\{\hat{\mathrm{v}}(a, \mu): a \in a^{*}(\mu)\right\} .
$$

Observe that $\hat{V}$ is a non-empty valued, convex-valued, and compact-valued correspondence. Furthermore, $\hat{V}$ is upper-hemicontinuous. The first part follows immediately from noting that $\hat{V}$ is the convex hull of finitely many vectors in $\mathbb{R}^{N}-a^{*}(\mu)$ is nonempty-and $\hat{v}$ is bounded. Upper-hemicontinuity of $\hat{V}$ follows from continuity of $\hat{\mathrm{v}}$ in $\mu$ and upper-hemicontinuity of $a^{*}(\mu)$. Consequently, the graph of $\hat{V}$,

$$
\operatorname{graph} \hat{V}=\left\{(\mu, \mathrm{v}) \in \Delta(\Theta) \times \mathbb{R}^{N}: \mathrm{v} \in \hat{V}(\mu)\right\},
$$

is closed.
We now show that similar to Theorem 1, the set $\mathrm{W}_{\mathrm{BP}}$ is the section at the prior of the convex hull of the graph of the correspondence $\hat{V}$, that is

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


Persuasion and Welfare

Clearly, if $\mathrm{w} \in \mathrm{W}_{\mathrm{BP}}$, then $\left(\mu_{0}, \mathrm{w}\right) \in \operatorname{co}(\operatorname{graph} \hat{V})$. To see that the opposite holds, let $\left(\mu_{0}, \mathrm{w}\right) \in \operatorname{co}($ graph $\hat{V})$. Then, a finite collection $\left(\tau_{k}, \mu_{k}, \tilde{\mathrm{v}}_{k}\right)_{k=1}^{M}$ of non-negative weights, beliefs, and correspondence values exists such that $\sum_{k=1}^{M} \tau_{k}=1, \sum_{k=1}^{M} \tau_{k} \mu_{k}=$ $\mu_{0}, \tilde{\mathrm{v}}_{k} \in \hat{V}\left(\mu_{k}\right)$ for all $k \in\{1, \ldots, M\}$, and

$$
\mathrm{w}=\sum_{k=1}^{M} \tau_{k} \tilde{\mathrm{v}}_{k} .
$$

Because for each $k \in\{1, \ldots, M\}, \tilde{\mathrm{v}}_{k} \in \hat{V}\left(\mu_{k}\right)$, the definition of $\hat{V}$ implies a finite collection of non-negative weights and actions, $\left\{\alpha_{k, l}, a_{l}\right\}_{l=1}^{L_{k}}$, exists such that $\sum_{l=1}^{L_{k}} \alpha_{l, k}=$ 1 , for all $l \in\left\{1, \ldots, L_{k}\right\}, a_{l} \in a^{*}\left(\mu_{k}\right)$ and for all $i \in\{1, \ldots, N\}$,

$$
\tilde{\mathrm{v}}_{k, i}=\sum_{l=1}^{L_{k}} \alpha_{k, l} \hat{v}\left(a_{l}, \mu_{k}, \theta_{i}\right) .
$$

Define $\Pi$ to be the information structure with signals $S=\left\{\mu_{1}, \ldots, \mu_{M}\right\}$, and signal distribution $\pi\left(\mu_{k} \mid \theta\right)=\left(\mu_{k}(\theta) / \mu_{0}(\theta)\right) \tau_{k}$. Furthermore, define $\alpha$ so that for belief $\mu_{k}$, $\alpha\left(\mu_{k}\right) \in \Delta\left(a^{*}\left(\mu_{k}\right)\right)$ coincides with $\left\{\alpha_{k, l}\right\}_{l=1}^{L_{k}}$. By construction, $(\Pi, \alpha) \in F$. Equations A. 5 and A. 6 together imply that

$$
\mathrm{w}=\sum_{k=1}^{M} \tau_{k} \sum_{l=1}^{L_{k}} \alpha_{k, l \hat{\mathrm{v}}}\left(a_{l}, \mu_{k}\right)=\mathbb{E}_{\langle\Pi\rangle}\left[\sum_{a \in A} \alpha(\tilde{\mu})(a) \hat{\mathrm{v}}(a, \mu)\right] .
$$

Finally, Equation A. 4 allows us to conclude that the set $\mathrm{W}_{\text {BP }}$ is closed: it is the section at the prior of the convex hull of the graph of the correspondence $\hat{V}$, which is closed. $\square$


Persuasion and Welfare

## A. 2 Omitted proofs from Section 4

Proof of Theorem 2. To complete the proof of the first direction of Theorem 2, we provide the steps to show that if $\mathrm{w} \in \mathrm{W}_{\mathrm{P}}$, a direction $\lambda \in \mathbb{R}_{+}^{N} \backslash\{0\}$ exists such that

$$
\lambda^{T} \mathrm{w}=\max \left\{\lambda^{T} \mathrm{w}^{\prime}: \mathrm{w}^{\prime} \in \mathrm{W}\right\},
$$

The opposite direction in Theorem 2 immediately follows.
Fix a Pareto efficient w and let $\Gamma=\left\{\mathrm{w}^{\prime} \in \mathbb{R}^{N}: \mathrm{w}^{\prime} \geq \mathrm{w}\right\}$. Clearly, $\Gamma$ is convex and int $\Gamma$ is non-empty. Because w is Pareto efficient, int $\Gamma \cap \mathrm{W}=\emptyset$. By Minkowski's separating hyperplane theorem, a direction $\lambda \in \mathbb{R}^{N} \backslash\{0\}$ exists such that for all $\mathrm{w}^{\prime \prime} \in \mathrm{W}$ and $\mathrm{w}^{\prime} \in \Gamma$,

$$
\lambda^{T} \mathrm{w}^{\prime \prime} \leq \lambda^{T} \mathrm{w}^{\prime} .
$$

Because $\mathrm{w} \in \Gamma$, we have that $\lambda^{T} \mathrm{w}^{\prime \prime} \leq \lambda^{T} \mathrm{w}$ for all $\mathrm{w}^{\prime \prime}$ in Bayes welfare set. Thus, $\lambda^{T} \mathrm{w}=\max \left\{\lambda^{T} \mathrm{w}^{\prime \prime}: \mathrm{w}^{\prime \prime} \in \mathrm{W}\right\}$ (cf. Equation 6). Similarly, because $\mathrm{w} \in \mathrm{W}$, then we have that $\lambda^{T} \mathrm{w} \leq \lambda^{T} \mathrm{w}^{\prime}$ for all $\mathrm{w}^{\prime} \in \Gamma$.

We now show that $\lambda \geq 0$. Let $\mathrm{t}_{i}$ denote the canonical vector that has a 1 in coordinate $\theta_{i}$ and 0 otherwise. Then, $\mathrm{w}+\mathrm{t}_{i} \in \Gamma$, so that

$$
\lambda^{T} \mathrm{w} \leq \lambda^{T}\left(\mathrm{w}+\mathrm{t}_{i}\right) \Rightarrow 0 \leq \lambda\left(\theta_{i}\right) .
$$

By definition, $\lambda \neq 0$ so that without loss of generality $\lambda \in \Delta(\Theta)$. $\square$

Proposition A.2. Suppose $w(\cdot, \theta)$ is upper-semicontinuous for all $\theta \in \Theta .{ }^{21}$ Suppose $\mathrm{w}^{*}$ is a limit point of $\mathrm{W}_{\mathrm{P}}$. Then, a point $\mathrm{w}^{* *} \in \mathrm{~W}_{\mathrm{P}}$ exists such that $\mathrm{w}^{* *} \geq \mathrm{w}^{*}$. Consequently, any weakly monotone social welfare function attains a solution in W .

Proof of Proposition A.2. Let $\left(\mathrm{w}_{n}\right)_{n \in \mathbb{N}} \subset \mathrm{~W}_{\mathrm{P}}$ be such that $\mathrm{w}_{n} \rightarrow \mathrm{w}^{*}$ as $n \rightarrow \infty$. For each $n \in \mathbb{N}$ a direction $\lambda_{n} \in \Delta(\Theta)$ exists such that

$$
\left(\forall \mathrm{w}^{\prime} \in \mathrm{W}\right) \lambda_{n}^{T} \mathrm{w}_{n} \geq \lambda_{n}^{T} \mathrm{w}^{\prime} .
$$

Since $\Delta(\Theta)$ is compact, then up to a subsequence $\lambda_{n} \rightarrow \lambda_{*}$. Linearity of $\lambda_{n}^{T} \mathrm{w}^{\prime}$ implies that taking limits on both sides of Equation A. 8 we obtain

$$
\left(\forall \mathrm{w}^{\prime} \in \mathrm{W}\right) \lambda_{*}^{T} \mathrm{w}^{*} \geq \lambda_{*}^{T} \mathrm{w}^{\prime} .
$$

[^16]Thus, if $\mathrm{w}^{*} \in \mathrm{~W}$, Theorem 2 implies that $\mathrm{w}^{*} \in \mathrm{~W}_{\mathrm{P}}$ and $\lambda_{*} \in \Delta(\Theta)$ is the direction that witnesses this.

Now, for each $\mathrm{w}_{n}$, a distribution over posteriors $\tau_{n} \in \Delta_{\mu_{0}}(\Delta(\Theta))$ exists such that $\mathrm{w}_{n}=\mathbb{E}_{\tau_{n}}[\hat{\mathrm{w}}]$. Because $\Delta_{\mu_{0}}(\Delta(\Theta))$ is compact, we have that, up to a subsequence, $\tau_{n} \rightarrow \tau^{*} \in \Delta_{\mu_{0}}(\Delta(\Theta))$. Aliprantis and Border (2013, Theorem 15.5) implies $\mathbb{E}_{\tau}[\hat{\mathrm{w}}]$ is upper-semicontinuous as a function of $\tau$, thus for all $i \in\{1, \ldots, N\}$ we have that

$$
\mathbb{E}_{\tau^{*}}\left[\hat{\mathrm{w}}_{i}\right] \geq \lim _{n \rightarrow \infty} \mathbb{E}_{\tau_{n}}\left[\hat{\mathrm{w}}_{i}\right]=\lim _{n \rightarrow \infty} \mathrm{w}_{n, i}=\mathrm{w}_{i}^{*} .
$$

Let $\mathrm{w}^{* *} \equiv \mathbb{E}_{\tau^{*}}[\hat{\mathrm{w}}]$ and note that it is an element of W that dominates $\mathrm{w}^{*}$ coordinateby-coordinate. Moreover, because $\lambda_{*} \in \Delta(\Theta)$, Equation A. 10 implies that $\lambda_{*}^{T} \mathrm{w}^{* *} \geq$ $\lambda_{*}^{T} \mathrm{w}^{*}$. This, together with Equation A.9, implies $\mathrm{w}^{* *} \in \mathrm{~W}_{\mathrm{P}}$, which completes the proof. $\square$

The proof of the statements in Observation 1 follows from the following result:
Corollary A. 1 ((No) Benefit from disclosure). The following hold:

1. All types benefit from disclosure if for all $\theta \in \Theta, \hat{w}(\cdot, \theta)$ is strictly convex in a neighborhood of the prior. Furthermore, if $\hat{w}(\cdot, \theta)$ is everywhere strictly convex, then full disclosure is uniquely Pareto efficient.
2. No disclosure is Pareto efficient if either
    (a) a type $\theta \in \Theta$ exists such that $\hat{w}(\mu, \theta)$ is concave in $\mu$, or
    (b) a vector $a \in \mathbb{R}_{+}^{N}$ and a concave function $w: \Delta(\Theta) \mapsto \mathbb{R}$ exist such that for all $\theta \in \Theta$, the welfare function is given by $a(\theta) w(\mu)+b(\theta)$.

Proof of Corollary A.1. Consider first the conditions in part 1. Toward a contradiction, suppose that $\mathrm{w}^{N D}$ is in the Pareto frontier. Then, by Corollary 3 a direction $\tilde{\lambda} \in \Delta(\Theta)$ exists such that no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\tilde{\lambda}$. However, under the conditions in part 1, the indirect utility function is strictly convex in a neighborhood of the prior for all directions $\lambda \in \Delta(\Theta)$, contradicting that no disclosure is a solution to the supporting Bayesian persuasion problem in some direction $\tilde{\lambda}$ in $\Delta(\Theta)$. Furthermore, when $\hat{w}(\cdot, \theta)$ is strictly convex, so is $\hat{v}_{\lambda}$ for all $\lambda \in \Delta(\Theta)$ and the value of full disclosure strictly dominates that of any other information structure.


Persuasion and Welfare

The proof of part 2 follows from Theorem 2 by looking at the solution of the supporting Bayesian persuasion problem in certain directions. Under the conditions in part 2a, no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\lambda \in \Delta(\Theta)$ such that $\lambda(\theta)=1$ and $\lambda\left(\theta^{\prime}\right)=0$ for $\theta^{\prime} \neq \theta$. Instead, under the conditions
in part 2b, no disclosure is a solution to the supporting Bayesian persuasion problem in direction $\lambda(\theta)=\mu_{0}(\theta) / a(\theta)$. $\square$

Proof of Observation 2. The result follows from Corollary 3. For the first part, the condition ensures $\mu(\theta) w(\mu, \theta)$ is strictly convex for all $\theta \in\left\{\theta_{1}, \theta_{2}\right\}$, so the indirect utility function $\hat{v}_{\lambda}$ is strictly convex for any $\lambda \in \Delta(\Theta)$ and full disclosure is the unique solution to the supporting Bayesian persuasion problem. For the second part, the condition ensures $\mu(\theta) w(\mu, \theta)$ is weakly concave for type $\theta$, and the result follows from looking at the direction $\lambda$ such that $\lambda(\theta)=1$ and $\lambda\left(\theta^{\prime}\right)=0$ for $\theta^{\prime} \neq \theta$. $\square$


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

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

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

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


Optimal Technology Design

> Machine-readable author manuscript.
> Authors: Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes.
> Canonical citation: Garrett, Daniel F., George Georgiadis, Alex Smolin, and Balázs Szentes. “Optimal Technology Design.” Journal of Economic Theory 209 (2023): 105621.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Optimal Technology Design

# Optimal Technology Design

**Authors:** Daniel F. Garrett; George Georgiadis; Alex Smolin; Balázs Szentes

**Manuscript date:** 2023-01

#### Abstract

This paper considers a moral hazard model with agent limited liability. Prior to interacting with the principal, the agent designs the production technology, which is a specification of his cost of generating each output distribution. After observing the production technology, the principal offers a payment scheme and then the agent chooses a distribution over outputs. We show that there is an optimal design involving only binary distributions (i.e., the cost of any other distribution is prohibitively high), and we characterize the equilibrium technology defined on the binary distributions. Notably, the equilibrium payoff of both players is $1 / e$.

JEL classification: D82, D86
Keywords: moral hazard, limited liability, contract theory

[^0]


Optimal Technology Design

## 1 Introduction

A central result in contract theory is that agency rents are a key source of economic welfare. When analyzing environments with asymmetric information, most microeconomic models take the determinants of these agency frictions as given. In hidden-information models, for example, the distribution of types, which determines information rents, is typically treated as exogenous. Similarly, in principal-agent problems with hidden actions, the production technology available to the agent, which governs the principal's cost of implementing various actions, is usually part of the model description. However, if an agent's payoff depends on agency frictions, then he is likely to pursue generating these frictions in a way that enhances his payoff. The goal of this paper is to reconsider the standard limited-liability moral hazard problem and understand how an agent might maximize rents by optimally designing the production technology.

For a potential application where such a problem may arise, consider an entrepreneur who is starting a business, and will eventually need venture capital backing to grow it. Prior to contracting with venture capitalists, he must make a host of choices pertaining to the product, the business model, the product market strategy, and so on. ${ }^{1}$ If the venture capitalist has strong bargaining power, the entrepreneur benefits from making choices that exacerbate the moral hazard problem to increase agency rents. Even if there were more profitable alternatives, they may not be considered in the contractual negotiations if the venture capitalist is unaware of them.

Identifying the agent-optimal technology turns out to be useful even in those environments where the agent has no meaningful way of influencing the production technology and, from his viewpoint, it is given exogenously. Indeed, we are able to express predictions regarding surplus sharing in principal-agent models with limited liability that are robust to the set of available technologies. Specifically, we characterize the entire set of payoff combinations in such models which can arise for some production technology. ${ }^{2}$ The agent-optimal technology corresponds to an extreme point in this set at which the agent's payoff is maximal. This exercise is similar in spirit to that of Bergemann et al. (2015), who characterize the set of consumer and seller payoffs in a model of third-degree price discrimination for some information of the seller. ${ }^{3}$

In the baseline setup, we consider a risk-neutral agent who can choose a production technology

[^1](or "project") before interacting with a principal. ${ }^{4}$ A production technology specifies the agent's cost of each output distribution with support contained in [0, 1]. That is, the only restriction on the available projects is that output is uniformly bounded. Such a bound may represent a physical constraint and is normalized to one. After observing the agent's project, the principal offers a wage contract, which is a mapping from output realizations to monetary compensation. We assume that the agent has limited liability and hence the payment must be non-negative. Finally, the agent chooses an output distribution at a cost determined by his first-stage choice.

In order to focus on the incentives to generate agency rents we consider a particularly stylized model which abstracts from a number of forces that may have first-order importance in applications. Perhaps most importantly, the agent in our model does not incur any costs in choosing a production technology. In practice, developing a project is likely to require a substantial amount of irreversible investment. In fact, the necessity of such investments may prevent the technology from being renegotiated at the contracting stage. The reason is that, even if both parties are aware of production technologies that are more profitable than the one put forward by the agent, they would not be implemented if modifying the agent's project is too expensive. ${ }^{5}$ Since the agent chooses the production technology before the contracting stage, our model is a hold-up problem. The expense of modifications could also prevent the agent from secretly adjusting the technology after contracting to make the chosen output distribution less costly, helping to make the agent's commitment to the technology credible. ${ }^{6}$

We also recognize that, in practice, the agent's ability to shape the production technology may be severely limited. For example, some output distributions may be impossible to generate. The constraints on the set of available technologies are likely to be important determinants of the optimal design. However, even when the space of projects is restricted, our main result holds as long as this space includes the agent-optimal technology. Moreover, in the Discussion Section, we demonstrate that our analysis can accommodate certain moment constraints on the domain of production technologies.

Our first main result is that the optimal project involves only binary distributions on $\{0,1\} .^{7}$

[^2]In other words, the cost of all other distributions can be assumed to be so high that the principal never wants to implement them, and the agent would never choose them irrespective of the payment scheme. This means that the equilibrium project can be thought of as a task which yields a positive payoff only if completed. The production technology specifies the cost of each probability of completion. The principal's wage contract can be viewed as a bonus paid for project completion, with payments set to zero otherwise.


Optimal Technology Design

Let us explain the optimality of binary projects. Just like in standard moral hazard problems, output plays a dual role in our model. On the one hand, it is the principal's revenue, and on the other hand, it is an informative signal about the output distribution chosen by the agent, which is used by the principal to incentivize the agent. By the Informativeness Principle, if this signal is made less informative, incentivizing the agent becomes more expensive. The key observation is that each binary distribution with support \{0, 1\} can be viewed as a garbling of a distribution with the same mean. Consider now a transformation of each project so that, if the agent incurs a cost of a distribution, output is distributed according to the binary distribution with the same mean. This means that the agent's cost of inducing a given level of expected output remains the same in the transformed project but the principal's cost of implementing it goes up. In this sense, such a transformation exacerbates the moral hazard problem. We show how this observation can be used to replace any project with a binary one for which the agent's payoff is at least as high.

Our second main result is a full characterization of the optimal binary project. In this project, the cost of completing the task with probability $1 / e$ is zero. That is, even if the agent incurs no cost, project completion can be achieved with probability $1 / e$. In equilibrium, the principal offers a bonus which induces the agent to complete the project with probability one. Furthermore, the principal is indifferent between offering this bonus and anything less than that. This indifference condition pins down the cost to the agent of any probability of success between $1 / e$ and one. Since the marginal cost of the success probability is less than one and the maximal output is produced surely, the equilibrium is ex-post efficient. That is, given the equilibrium production technology, the allocation is efficient. It turns out that the optimal project yields an equal split of surplus: both the principal and the agent earn payoff $1 / e$.

The first-best social surplus in our model is one since projects that can generate output one at no cost are feasible. Of course, the agent does not choose such a project because then the principal could achieve the maximal output without making any payment. To earn rents, the agent designs the technology so that generating high expected output is artificially costly. In fact, since the
equilibrium output is one with probability one, the only source of distortion induced by the optimal design relates to this cost. This might be considered a form of "cost padding", different from others identified in the literature. ${ }^{8}$

As mentioned above, an identifying feature of the optimal project is that the principal is indifferent between implementing a large range of completion probabilities. Let us explain the economic reasoning behind this feature. Note that, since the agent receives the bonus offered by the principal with the probability of completion, her marginal benefit at each completion probability is the bonus. Hence, at the agent's optimal completion probability, the marginal cost equals the bonus. This means that the principal's expected payment to implement a given completion probability is increasing in the marginal cost at that probability, so lowering this marginal cost makes it more attractive to the principal to implement the completion probability. However, if the principal strictly prefers to implement the equilibrium probability of completion to implementing some smaller probabilities, then the marginal costs at these smaller probabilities can be lowered without affecting the principal's equilibrium choice. Since the cost of a completion probability is the integral of the marginal costs of smaller probabilities, such a modification of the project decreases the agent's total cost and thus increases his overall payoff (while the principal is still willing to offer the same bonus that implements the equilibrium completion probability). The agent can improve any project in this way unless the principal is indifferent between implementing any completion probability which has a positive marginal cost.

We demonstrate that our main results remain valid even if the agent is risk averse. ${ }^{9}$ In particular, the search for an optimal project can still be restricted to the set of binary projects. Moreover, the optimal binary project is still ex-post efficient; that is, the principal implements completion probability one. The optimal binary project is still characterized by the requirement that the principal must be indifferent between offering the equilibrium bonus and anything less than that. Of course, the equilibrium payoffs of the principal and the agent are no longer $1 / e$ and they depend on the agent's concave utility function.

Despite the fact that our model is stylized, we believe that there are a number of lessons emerging from our analysis which may be useful in applications. First, a general message is that the agent in a technology design problem will often commit to a project that is not cost-efficient. By padding costs, he may be able to earn additional rents. Second, there is reason to expect that, where possible,

[^3]an agent's optimal technology will be binary. That is, the project essentially specifies a task and the agent either completes it or fails to solve it. More generally, the agent is likely to favor designs where the information content of output regarding his effort is limited. The reason is that this makes it more expensive for the principal to induce the target level of effort. A third observation that we expect to generalize is the importance of principal incentive constraints in the design of the agent's project. The agent's design must dissuade the principal from choosing ungenerous incentive schemes that give the agent little rent. Where such incentive constraints are slack, however, it may be possible for the agent to redesign the project to lower his equilibrium cost.


Optimal Technology Design

Finally, we characterize the payoff combinations that can arise in limited-liability moral hazard problems for some exogenously given technology where output is restricted to the interval [0, 1]. This follows by first identifying the largest payoff the agent can achieve as a function of a given profit of the principal. The domain of this function is the interval [0, 1] because the principal can always guarantee a nonnegative profit by offering zero wage and she cannot get more than the maximal output. This function is shown to be strictly concave and zero at the boundaries of its domain. We argue that a payoff profile can be generated by some production technology if and only if it lies weakly below this curve.

The rest of the paper is as follows. Next we discuss related literature. Section 2 introduces the model, Section 3 provides our characterization of the agent's optimal project, and Section 4 provides discussion and extensions, including the characterization of the set of possible player payoffs. Appendix A provides proofs not given in the main text and Appendix B discusses the uniqueness of the agent's optimal project. The Online Appendix solves the agent's project design problem when the agent is risk averse.

Literature. - The limited liability model of moral hazard for a risk-neutral agent is a staple of introductory courses on contract theory, where a restriction to binary output (which emerges endogenously in our setting) is often made for tractability. A classic reference for limited-liability moral hazard is Innes (1990), who demonstrates the optimality of simple debt contracts in a model with a continuum of outputs. More recent treatments of moral hazard with limited liability include Poblete and Spulber (2012) for a model with a continuum of outputs and Ollier and Thomas (2013) for a model with binary output, but complicated by the presence of adverse selection. ${ }^{10}$

While the models discussed above feature a risk-neutral agent with limited liability, the al-

[^4]ternative friction commonly explored is risk aversion. Seminal work for the moral hazard model with a risk-averse agent includes Mirrlees (1976), Holmstrom (1979), Grossman and Hart (1983), Rogerson (1985) and MacLeod (2003); see Bolton and Dewatripont (2005) and Holmstrom (2017) for comprehensive treatments, and Georgiadis (2022) for a review. More recently, a strand of this literature has focused on models where the agent can shape the entire distribution of output under different assumptions about the cost of distributions (Hebert, 2018; Bonham and Riggs-Cragun, 2021, Georgiadis et al. , 2022, and Mattsson and Weibull, 2022). The focus of the moral hazard literature, then, has been on contract design taking the agent's technology as given. Our paper departs from this approach by viewing the production technology as a choice of the agent, raising the problem of technology design. We are unaware of this kind of problem being posed elsewhere in the moral hazard literature.

The question of project design is also related to work on how the primitive contractual environment affects payoffs in moral hazard problems. A relevant example in our context is the development of the Informativeness Principle by Holmstrom (1979), which was later refined for instance by Chaigneau et al. (2019). These papers clarify how additional information about the agent's action can reduce agency costs for the principal.

Another related paper is Condorelli and Szentes (2020) who study the problem of optimally generating information rents in the context of a bilateral trade model. Before interacting with the seller, the buyer can choose the distribution of her valuation for the seller's good. This choice is observed by the seller before she makes a take-it-or-leave-it offer. It turns out that the equilibrium distribution generates a unit-elastic demand, that is, it makes the seller indifferent between setting any price on its support. This is reminiscent of our optimal binary project which makes the principal indifferent across a range of bonuses. ${ }^{11}$ This similarity might explain why, when the buyer is restricted to choose distributions with support in the interval [0, 1], the equilibrium payoffs of the buyer and the seller are also $1 / e$. We further elaborate on the relationship between these two results in Section 4. ${ }^{12}$

[^5]


Optimal Technology Design

## 2 Model

We consider a game between a principal (she) and an agent (he), which proceeds as follows. In the first stage, the agent chooses a cost function $c: \mathcal{F} \rightarrow \mathbb{R}_{+} \cup\{+\infty\}$, where $\mathcal{F}$ denotes the set of CDFs with support on $[0,1]$. We refer to such a function $c$ as a project. Then, after observing $c$, the principal offers a payment scheme $w:[0,1] \rightarrow \mathbb{R}_{+}$, which is restricted to be Borel-measurable. ${ }^{13}$ Finally, after observing the offered payment scheme, the agent chooses a distribution $F \in \mathcal{F}$, and output is realized according to $F .{ }^{14}$ If the realized output is $x$ then the agent's and principal's payoffs are $w(x)-c(F)$ and $x-w(x)$, respectively. Both parties are expected payoff maximizers.

Notation.- For each $F \in \mathcal{F}$, let $\mu_{F}$ denote the expected value of $F$, that is, $\mu_{F}=\int_{0}^{1} x d F(x)$. The set of projects and the set of Borel-measurable payment schemes are denoted by $\mathcal{C}$ and $\mathcal{W}$, respectively. We refer to a triple $(c, w, F) \in \mathcal{C} \times \mathcal{W} \times \mathcal{F}$ as an outcome. Let $U$ and $\Pi$ denote the expected payoffs of the agent and the principal defined on the outcomes; that is,

$$
U(c, w, F)=\int_{0}^{1} w(x) d F(x)-c(F), \text { and } \Pi(w, F)=\int_{0}^{1}[x-w(x)] d F(x)
$$

Equilibrium in a Project.-Loosely speaking, we wish to call a pair $(w, F) \in(\mathcal{W}, \mathcal{F})$ an equilibrium in project $c \in \mathcal{C}$ if it satisfies the following two requirements. First, the distribution $F$ is incentive compatible (or a best response) for the agent given project $c$ and the payment schedule $w$. Second, the payment scheme $w$ is incentive compatible for the principal given project $c$. Formal statements of the incentive compatibility constraints and the equilibrium definition are provided below.

Optimal Projects. - We define the project $c$ to be optimal if there is an equilibrium $(w, F)$ in $c$ such that the agent's payoff in outcome $(c, w, F)$ is larger than in any outcome $\left(c^{\prime}, w^{\prime}, F^{\prime}\right)$ such that $\left(w^{\prime}, F^{\prime}\right)$ is an equilibrium in $c^{\prime}$. Thus, we assess the optimality of a project $c$ that induces multiple equilibria by considering those which give the highest payoff to the agent. This is in line with the approach prevalent in mechanism design, where the designer is permitted to pick the most favorable equilibrium.

Incentive Compatibility.- We say that choosing $F$ is incentive compatible for the agent in the

[^6]subgame $(c, w)$ if
$$
U(c, w, F) \geq U\left(c, w, F^{\prime}\right) \text { for all } F^{\prime} \in \mathcal{F} \text {. }
$$
To describe the principal's incentive constraint is harder because the agent may not have a best response in a subgame generated by a pair $(c, w)$. In turn, this can make it difficult to assess the profitability of certain deviations. To circumvent this problem, we define the value of the agent, $u(c, w)$, in each subgame $(c, w)$, by
$$
u(c, w) \equiv \sup _{F \in \mathcal{F}} U(c, w, F)
$$
We aim to define the value of the principal in a subgame $(c, w)$ by reference to sequences of distributions along which the agent's payoff converges to his value. In general, there may be many such sequences, potentially generating different limit payoffs to the principal. Let $\mathbf{F}^{c, w}$ denote the set of sequences of distributions $\left(F_{n}\right)$ along which the agent's payoff converges to $u(c, w)$. Formally, $\left(F_{n}\right) \in \mathbf{F}^{c, w}$ if and only if $\lim _{n \rightarrow \infty} U\left(c, w, F_{n}\right)=u(c, w)$. Then, the principal's value in subgame $(c, w)$ is given as
$$
\pi(c, w) \equiv \sup \left\{\limsup _{n \rightarrow \infty} \Pi\left(w, F_{n}\right):\left(F_{n}\right) \in \mathbf{F}^{c, w}\right\} .
$$
Evaluating the principal's value by reference to the supremum is again in the spirit of the approach prevalent in mechanism design, where the principal is permitted to pick the most favorable best response of the agent. The principal's incentive compatibility constraint guaranteeing that she offers payment schedule $w$ in project $c$ can be stated as follows: for all $w^{\prime} \in \mathcal{W}$,
$$
\pi(c, w) \geq \pi\left(c, w^{\prime}\right) .
$$
That is, the principal cannot gain by deviating to a payment schedule $w^{\prime}$, whether or not the agent has a best response to $w^{\prime}$.

Equipped with the incentive compatibility constraints, we are ready to define equilibria formally.
Definition. The pair $(w, F)$ is said to be an equilibrium in project $c$ if

(i) the distribution $F$ satisfies (1),
(ii) the payment scheme $w$ satisfies (2), and
(iii) $\Pi(w, F)=\pi(c, w)$.

Observe that Part (iii) requires that the distribution $F$ indeed generates the principal's value in
subgame $(c, w)$.
Binary Projects and Linear Contracts.- As mentioned in the Introduction, binary projects play an important role in our analysis. Next, we formally define these projects. We call a distribution in $\mathcal{F}$ binary if its support is contained in $\{0,1\}$. For each $\mu \in[0,1]$, let $B_{\mu}$ denote the binary CDF which specifies an atom of size $\mu$ at one. Note that the mean of $B_{\mu}$ is also $\mu$. Let $\mathcal{B}$ denote the set of binary distributions; that is, $\mathcal{B}=\left\{B_{\mu}: \mu \in[0,1]\right\}$. We call a project $c$ binary if $c(F)=+\infty$ whenever $F \notin \mathcal{B}$.

In each binary project, the principal always finds it optimal to offer a compensation scheme which pays zero if output is zero. So, the optimal payment scheme can be summarized by a single bonus, $b$, which is paid to the agent if output is one. If the project is binary, the wage at output $x \notin\{0,1\}$ is irrelevant, so without loss of generality, such a wage contract can be assumed to be linear, denoted by $w_{b}$, that is, $w_{b}(x)=b x$ for all $x$.

If the principal offers a linear contract, $w_{b}$, the output distribution affects the payoffs of the agent and the principal only through its mean. That is, whether or not the project is binary,

$$
U\left(c, w_{b}, F\right)=\mu_{F} b-c(F), \text { and } \Pi(w, F)=\mu_{F}(1-b) .
$$

This implies that, if the principal offers a linear contract to which the agent has a best response, this best response must involve a distribution which is the least costly among those with the same mean.


Optimal Technology Design

## 3 Main Results

This section is devoted to our two main results. In the next section, we show that it suffices to restrict attention to binary projects and, in Section 3.2, we fully characterize an optimal binary project.


Optimal Technology Design

### 3.1 Binary Projects

In this section, we fix a project $c^{*}$ and an equilibrium $\left(w^{*}, F^{*}\right)$ in $c^{*}$. Our aim is to construct a binary project $\widetilde{c}$ and an equilibrium $(\widetilde{w}, \widetilde{F})$ in $\widetilde{c}$ so that the outcome $(\widetilde{c}, \widetilde{w}, \widetilde{F})$ Pareto dominates the outcome $\left(c^{*}, w^{*}, F^{*}\right)$. Since $c^{*}$ can be an optimal project, this result implies that there exists an optimal project in the class of binary projects.

As explained in the Introduction, the key observation for this result is that an output realization not only determines the principal's payoff, but also serves as an informative signal about the agent's action. If this signal is made less informative in the sense of Blackwell, incentivizing the agent becomes harder for the principal. To see how an output distribution can be made less informative, consider the following garbling: instead of observing output $x$, the principal observes output one with probability $x$ and output zero otherwise. That is, the garbling of each $F \in \mathcal{F}$ is $B_{\mu_{F}}$, so the expected output is unaffected. In fact, $B_{\mu_{F}}$ is the least informative garbling of $F$, as the same transformation can be applied to any other garbling of $F$ which would again result in $B_{\mu_{F}}$. So, if the principal could contract only on the realization of $B_{\mu_{F}}$ but not on that of $F$, her wage cost of implementing $F$ would increase. We next explain how this observation can be used to transform the project $c^{*}$ to a binary one which is more beneficial for the agent.

The idea behind the construction of the binary project, $\widetilde{c}$, is as follows. First, define the agent's cost of any binary distribution to be the cost of the cheapest distribution in project $c^{*}$ with the same mean. In this binary project, the principal's wage cost of attaining any level of expected output is higher than in project $c^{*}$. In fact, the wage cost of generating $\mu_{F^{*}}$ may be so high that the principal prefers to implement a distribution with a lower mean, thus saving on payments to the agent. In this case, the payoffs of both parties can be lower. Therefore, we further modify the binary project by reducing the agent's cost of $B_{\mu_{F^{*}}}$ so that the principal can implement it at exactly the same wage cost as that of $F^{*}$ in $c^{*}$.

Before stating the main result of this section, let us introduce an additional piece of notation. Note that the expected payment in outcome $\left(c^{*}, w^{*}, F^{*}\right)$ is $\mathbb{E}_{F^{*}}\left[w^{*}\right]$. If $\mu_{F^{*}}>0$ (as must be the case if the outcome is optimal for the agent), we can define $b^{*}$ to equal $\mathbb{E}_{F^{*}}\left[w^{*}\right] / \mu_{F^{*}}$. We can then observe that ${ }^{15}$

$$
\mathbb{E}_{F^{*}}\left[w^{*}\right]=\mu_{F^{*}} b^{*}=\mathbb{E}_{F^{*}}\left[w_{b^{*}}\right]=\mathbb{E}_{B_{\mu_{F^{*}}}}\left[w_{b^{*}}\right] .
$$

That is, the expected payment induced by the pair $\left(w^{*}, F^{*}\right)$ is the same as that induced by $\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$.

Proposition 1. Suppose that $\left(w^{*}, F^{*}\right)$ is an equilibrium in project $c^{*}$ with $\mu_{F^{*}}>0$. Then there exists a binary project, $\widetilde{c}$, such that

(i) $\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ is an equilibrium in $\widetilde{c}$,
(ii) $U\left(c^{*}, w^{*}, F^{*}\right) \leq U\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right)$, and

[^7](iii) $\Pi\left(w^{*}, F^{*}\right)=\Pi\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$.

Let us describe the binary project $\widetilde{c}$ and the main arguments in the proof of the proposition. It turns out that, in this binary project, the agent's rent can be ensured by making it hard for the principal to dissuade the agent from deviating downwards (i.e., to distributions with lower means). Upwards deviations need not play a role, so we specify the agent's cost of each $B_{\mu}$ with $\mu>\mu_{F^{*}}$ to be infinity throughout the construction. We now explain the two steps of constructing $\widetilde{c}$ from $c^{*}$ in more detail. In the first step, we take the agent's cost of $B_{\mu_{F^{*}}}$ to also be infinite. For each $\mu<\mu_{F^{*}}$, we specify the cost of $B_{\mu}$ to be the cost of the cheapest distribution in project $c^{*}$ with expectation $\mu .^{16}$ We then prove that, in order to achieve any expected output, the principal must make a higher expected payment in this binary project than in $c^{*}$. In the second step, we redefine the agent's cost of $B_{\mu_{F^{*}}}$ so that the principal's wage cost of implementing $B_{\mu_{F^{*}}}$ is exactly $\mathbb{E}_{F^{*}}\left[w^{*}\right]=\mathbb{E}_{B_{\mu_{F^{*}}}}\left[w_{b^{*}}\right]$, thus obtaining the project $\widetilde{c}$. We show that the agent's cost of $B_{\mu_{F^{*}}}$ in $\widetilde{c}$ is less than $c^{*}\left(F^{*}\right)$. This means that, by Equation (4), Parts (ii) and (iii) of the proposition are satisfied.

We now explain how to obtain Part (i). Note that, by our choice of the agent's cost of the distribution $B_{\mu_{F^{*}}}$ in project $\widetilde{c}$, the agent best responds to $w_{b^{*}}$ by choosing $B_{\mu_{F^{*}}}$. Therefore, to prove that $\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ is an equilibrium in this project, we need to demonstrate only that offering $w_{b^{*}}$ is incentive compatible for the principal. By construction, if the principal wants to implement $B_{\mu_{F^{*}}}$, she offers payment schedule $w_{b^{*}}$. She therefore receives a payoff of $\Pi\left(w^{*}, F^{*}\right)$. As explained, attaining any other expected output $\mu\left(\mu \neq \mu_{F^{*}}\right)$ is more expensive for the principal in $\widetilde{c}$ than in $c^{*}$. Therefore, since the principal found it optimal to implement $F^{*}$ in $c^{*}$, she optimally chooses to implement $B_{\mu_{F^{*}}}$ in $\widetilde{c}$ by offering $w_{b^{*}}$; that is, $w_{b^{*}}$ is incentive compatible in $\widetilde{c}$.

Towards the first step described above, let us define the binary project, $\widehat{c}$, as follows:

$$
\widehat{c}\left(B_{\mu}\right)= \begin{cases}\inf \left\{c^{*}(F): \mu_{F}=\mu\right\} & \text { if } \mu<\mu_{F^{*}} \\ \infty & \text { otherwise }\end{cases}
$$


Optimal Technology Design

We next formalize the aforementioned implication of the Informativeness Principle; in particular, we demonstrate that the principal is worse off in $\widehat{c}$ than in $c^{*}$. In fact, we show that, from the principal's point of view, the transformed project $\widehat{c}$ is worse than being restricted to linear contracts in $c^{*}$ in the sense that each contract $w_{b}$ generates weakly more profit to the principal in project $c^{*}$ than in $\widehat{c}$.

[^8]Lemma 1. For all $b \in[0,1], \pi\left(\widehat{c}, w_{b}\right) \leq \pi\left(c^{*}, w_{b}\right)$.
Proof. See the Appendix. QED

Let us illustrate the argument behind the proof of this lemma for the case where the principal's value in subgame $\left(c^{*}, w_{b}\right), \pi\left(c^{*}, w_{b}\right)$, is generated by a best response of the agent. Since the agent's expected payment generated by the linear contract $w_{b}$ depends only on the expected output, he chooses a distribution only if it is the cheapest among those with the same mean. Therefore, the agent's value in subgame ( $c^{*}, w_{b}$ ) is

$$
\sup _{\mu \in[0,1]}\left\{\mu b-\inf \left\{c^{*}(F): F \in \mathcal{F}, \mu_{F}=\mu\right\}\right\} .
$$

This is the same problem as the one which determines the agent's value in the subgame $\left(\widehat{c}, w_{b}\right)$, except that in the latter, the domain is effectively restricted to be $\left[0, \mu_{F^{*}}\right)$. Suppose now that $F$ is incentive compatible in $\left(c^{*}, w_{b}\right)$ and generates the principal's value. That is, $\mu_{F}$ solves the problem in (5) and $\pi\left(c^{*}, w_{b}\right)=\Pi\left(w_{b}, F\right)$. If $\mu_{F}<\mu_{F^{*}}$, then $\mu_{F}$ also solves the agent's problem with the restricted domain, implying that $B_{\mu_{F}}$ is incentive compatible in $\left(\widehat{c}, w_{b}\right)$. In this case, $\pi\left(\widehat{c}, w_{b}\right)=\mu_{F}(1-b)=\pi\left(c^{*}, w_{b}\right)$. If $\mu_{F} \geq \mu_{F^{*}}$, then the principal's value is at least $\mu_{F^{*}}(1-b)$ in the subgame $\left(c^{*}, w_{b}\right)$, so $\pi\left(c^{*}, w_{b}\right) \geq \mu_{F^{*}}(1-b) \geq \pi\left(\widehat{c}, w_{b}\right)$, where the second inequality holds because, in project $\widehat{c}$, the agent never chooses a distribution which has mean larger than $\mu_{F^{*}}$.

We are now ready to define project $\widetilde{c}$. Our aim is to modify $\widehat{c}$ at $B_{\mu_{F^{*}}}$ so that $\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ is an equilibrium in project $\widetilde{c}$. On the one hand, this requires the cost of $B_{\mu_{F^{*}}}$ to be sufficiently small to guarantee that $B_{\mu_{F^{*}}}$ is a best response to $w_{b^{*}}$. On the other hand, this cost cannot be too small, for otherwise $B_{\mu_{F^{*}}}$ could be implemented with a bonus smaller than $b^{*}$. Therefore, we specify the cost of $B_{\mu_{F^{*}}}$ to be the largest cost at which the agent still best responds to $w_{b^{*}}$ by choosing $B_{\mu_{F^{*}}}$. This cost, denoted by $\bar{c}$, satisfies $\mu_{F^{*}} b^{*}-\bar{c}=\sup \left\{\mu b^{*}-\widehat{c}\left(B_{\mu}\right)\right\}$. The binary project $\widetilde{c}$ is defined as follows:

$$
\widetilde{c}(F)= \begin{cases}\bar{c} & \text { if } F=B_{\mu_{F^{*}}}, \\ \widehat{c}(F) & \text { if } F \neq B_{\mu_{F^{*}}} .\end{cases}
$$

Next, we demonstrate that the outcome $\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ Pareto dominates $\left(c^{*}, w^{*}, F^{*}\right)$. To this end, we first argue that the cost of $B_{\mu_{F^{*}}}$ in project $\widetilde{c}$ is weakly smaller than $c^{*}\left(F^{*}\right)$. Suppose, for a contradiction, that $\bar{c}>c^{*}\left(F^{*}\right)$. Then,

$$
\mu_{F^{*}} b^{*}-c^{*}\left(F^{*}\right)>\mu_{F^{*}} b^{*}-\bar{c}=\sup \left\{\mu b^{*}-\widehat{c}\left(B_{\mu}\right)\right\}=\sup \left\{\mu_{F} b^{*}-c^{*}(F): F \in \mathcal{F}, \mu_{F}<\mu_{F^{*}}\right\},
$$

where the two equalities follow from the definitions of $\bar{c}$ and $\widehat{c}$, respectively. By continuity, this chain implies the existence of $b<b^{*}$ such that

$$
\mu_{F^{*}} b-c^{*}\left(F^{*}\right)>\sup \left\{\mu_{F} b-c^{*}(F): F \in \mathcal{F}, \mu_{F}<\mu_{F^{*}}\right\} .
$$

This means that offering the linear contract $w_{b}$ in project $c^{*}$ provides the principal with a value at least $\mu_{F^{*}}(1-b)$, which is strictly more than her equilibrium payoff, $\Pi\left(w^{*}, F^{*}\right)=\mu_{F^{*}}\left(1-b^{*}\right)$. This contradicts the incentive compatibility of $w^{*}$ in $c^{*}$.

We are now ready to show that the agent is weakly better off in the outcome $\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ than in $\left(c^{*}, w^{*}, F^{*}\right)$. Indeed,

$$
U\left(c^{*}, w^{*}, F^{*}\right)=\mu_{F^{*}} b^{*}-c^{*}\left(F^{*}\right) \leq \mu_{F^{*}} b^{*}-\widetilde{c}\left(B_{\mu_{F^{*}}}\right)=U\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right),
$$

where the equalities follow from (4) and the inequality follows from $\widetilde{c}\left(B_{\mu_{F^{*}}}\right)=\bar{c} \leq c^{*}\left(F^{*}\right)$. Also note that (4) implies that the principal's payoffs are the same in these two outcomes:

$$
\Pi\left(w^{*}, F^{*}\right)=\mu_{F^{*}}-\mathbb{E}_{F^{*}}\left[w^{*}\right]=\mu_{F^{*}}\left(1-b^{*}\right)=\Pi\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right) .
$$

We defined $\widetilde{c}\left(B_{\mu_{F^{*}}}\right)$ so that the payment schedule $w_{b^{*}}$ implements $B_{\mu_{F^{*}}}$ in project $\widetilde{c}$. Next, we confirm that the principal cannot implement $B_{\mu_{F^{*}}}$ in project $\widetilde{c}$ with any payment schedule $w_{b}$ such that $b<b^{*}$. This means that the principal's value from offering $w_{b}$ in project $\widetilde{c}$ is the same as in project $\widehat{c}$.

Lemma 2. For all $b \in\left[0, b^{*}\right), \pi\left(\widetilde{c}, w_{b}\right)=\pi\left(\widehat{c}, w_{b}\right)$.
Proof. See the Appendix. $\square$


Optimal Technology Design

Let us illustrate the argument of the proof for the case where the agent has a best response to $w_{b^{*}}$ in project $\widehat{c}$, say $B_{\mu^{\prime}}$ (with $\mu^{\prime}<\mu_{F^{*}}$ ); that is, $\mu^{\prime} b^{*}-\widehat{c}\left(B_{\mu^{\prime}}\right)=\sup \left\{\mu b^{*}-\widehat{c}\left(B_{\mu}\right)\right\}$. By the definition of $\widetilde{c}$, this means that $\mu^{\prime} b^{*}-\widetilde{c}\left(B_{\mu^{\prime}}\right)=\mu_{F^{*}} b^{*}-\widetilde{c}\left(B_{\mu_{F^{*}}}\right)$. Since $\mu^{\prime}<\mu_{F^{*}}$, this equality implies that, for all $b \in\left[0, b^{*}\right), \mu^{\prime} b-\widetilde{c}\left(B_{\mu^{\prime}}\right)>\mu_{F^{*}} b-\widetilde{c}\left(B_{\mu_{F^{*}}}\right)$, implying that $B_{\mu_{F^{*}}}$ is not incentive compatible in $\left(\widetilde{c}, w_{b}\right)$. Since the projects $\widetilde{c}$ and $\widehat{c}$ are identical on the rest of their domains, the statement of the lemma follows.

Finally, we are ready to prove Proposition 1.
Proof of Proposition 1. Observe that the outcome $\left(\widetilde{c}, w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ satisfies Parts (ii) and (iii)
of the proposition by Equations (6) and (7), respectively. Therefore, we only need to establish Part (i); that is, we need to argue that $\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)$ is an equilibrium in project $\widetilde{c}$. By the definition of $\bar{c}, B_{\mu_{F^{*}}}$ is incentive compatible in the subgame $\left(\widetilde{c}, w_{b^{*}}\right)$. Next, we prove that $w_{b^{*}}$ is incentive compatible in $\widetilde{c}$.

If $b>b^{*}$, then

$$
\pi\left(\widetilde{c}, w_{b}\right) \leq \mu_{F^{*}}(1-b)<\mu_{F^{*}}\left(1-b^{*}\right)=\Pi\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right),
$$

where the first inequality follows from $\widetilde{c}\left(B_{\mu}\right)=\infty$ for all $\mu>\mu_{F^{*}}$, and the second inequality is implied by $b>b^{*}$.

If $b<b^{*}$, then

$$
\pi\left(\widetilde{c}, w_{b}\right) \leq \pi\left(c^{*}, w_{b}\right) \leq \Pi\left(w^{*}, F^{*}\right)=\Pi\left(w_{b^{*}}, B_{\mu_{F^{*}}}\right)
$$

where the first inequality follows from Lemmas 1 and 2, the second one follows from $\left(w^{*}, F^{*}\right)$ being an equilibrium outcome in project $c^{*}$, and the equality is implied by the definition of $b^{*}$. $\square$