The Design and Price of Information

## 7 Conclusion

We have studied a monopolist who sells supplemental information to privately informed buyers. The resulting screening problem reflects several key properties of information goods that set it apart from traditional models of price discrimination.

First, the Bayesian nature of the buyers' decision making is fundamental to the seller's problem. Differences in the buyers' private beliefs introduce a novel aspect of horizontal differentiation that widens the seller's scope for price discrimination.

Second, information is inherently rich and can be modified in many ways. Even onedimensional data, such as a consumer's credit score, can be turned into a rich set of information products, as some data buyers may want to identify consumers with excellent scores, while other data buyers may be concerned with avoiding consumers with very low scores.

Third, ultimately, instrumental information is useful as long as it guides the decision of the data buyer. Information therefore enters as an input into the buyer's decision problem. Thus, buyers with different private beliefs may act differently upon receiving the same supplemental information. We have shown that the buyer's ability to adjust his behavior in response to new information complicates the seller's screening problem by introducing the possibility of double deviations. In this respect, selling information is akin to selling multiple inputs
to heterogeneous buyers who can combine them in different ways, according to a privately known production technology.

In this paper, we have deliberately relied solely on belief heterogeneity to motivate sales of supplemental information. In practice, however, buyers of information may differ along several alternative or additional dimensions, including their cost of choosing specific actions, their ability to process data, or their preferences for timely information. ${ }^{25}$ Each of these extensions can be implemented within the framework we have outlined. Combining different sources of heterogeneity appears more challenging but promises to yield additional insights. Finally, we have focused on the packaging or versioning problem of a seller who is free to acquire and degrade information. Thus, our results represent only a first pass at understanding the trade-offs involved in selling information products. A richer model would distinguish the fixed cost of acquiring the information (e.g., building a database) from the variable cost of duplicating, distributing, and potentially degrading the available information.

[^15]


The Design and Price of Information

Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.

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

## A Appendix

Proof of Proposition 1. Consider any type $\theta$ and experiment $E=(S, \pi)$. Without loss of generality, let the type choose a single action after each signal. ${ }^{26}$ Let $S^{a}$ denote the set of signals in experiment $E$ that induces type $\theta$ to choose action $a$. Thus, $\cup_{a \in A} S^{a}=S$. Construct experiment $E^{\prime}=\left(S^{\prime}, \pi^{\prime}\right)$ as a recommendation for type $\theta$ based on experiment $E$, with signal space $S^{\prime}=\left\{s_{a}\right\}_{a \in A}$ and

$$
\pi^{\prime}\left(s_{a} \mid \omega\right)=\int_{S^{a}} \pi(s \mid \omega) \mathrm{d} s \quad \omega \in \Omega, a \in A .
$$

By construction, $E^{\prime}$ and $E$ induce the same outcome distribution for type $\theta$; hence, $V\left(E^{\prime}, \theta\right)=$ $V(E, \theta)$. Moreover, $E^{\prime}$ is a garbling of $E$. By Blackwell's theorem, we have $V\left(E^{\prime}, \theta^{\prime}\right) \leq$ $V\left(E, \theta^{\prime}\right)$ for all $\theta^{\prime}$. Therefore, for any incentive-compatible and individually rational direct mechanism $\{E(\theta), t(\theta)\}$, we can construct another direct mechanism $\left\{E^{\prime}(\theta), t(\theta)\right\}$ whose experiments lead type $\theta$ to take action $a$ after observing signal $s_{a} \in S^{\prime}(\theta)$ that is also incentive compatible and individually rational, thus yielding weakly larger profits. ■

Proof of Proposition 2. (1.) The argument is given in the text.
(2.) Let $\mathcal{M}=\{E(\theta), t(\theta)\}$ be an individually rational and incentive-compatible direct mechanism. Fix an experiment $E \in \mathcal{M}$, let $\pi_{i j}$ denote the conditional probability of signal $s_{j}$ in state $\omega_{i}$, and suppose $\pi_{i j}>0$ for all $i$ and $j$. We argue the seller can improve her profits by replacing $E$ with a non-dispersed experiment. By Proposition 1, we can restrict attention to responsive experiments with $J$ signals. Hence, the value of experiment $E$ for any obedient type $\theta$ is given by

$$
V_{\mathrm{ob}}(E, \theta)=\sum_{i=1}^{I} \theta_{i} \sum_{j=1}^{J} \pi_{i j} u_{i j}-u(\theta) .
$$

For each state $\omega_{i}$, define the worst action $a_{j(i)}$ and the corresponding signal $s_{j(i)}$, where $j(i) \in \arg \min _{j} u_{i j}$. Now let

$$
\varepsilon_{i} \triangleq \frac{\eta}{u_{i i}-u_{i j(i)}} \text { with } \eta \triangleq \min _{i}\left\{\left(u_{i i}-u_{i j(i)}\right) \pi_{i j(i)}\right\},
$$

and construct a new experiment $E^{\prime}$ where $\pi_{i i}^{\prime}=\pi_{i i}+\varepsilon_{i}$ and $\pi_{i j(i)}^{\prime}=\pi_{i j(i)}-\varepsilon_{i}$ for all $i$.

[^16]Experiment $E^{\prime}$ is non-dispersed by construction, i.e., $\pi_{i^{*} j\left(i^{*}\right)}=0$, where

$$
i^{*} \in \arg \min _{i}\left\{\left(u_{i i}-u_{i j(i)}\right) \pi_{i j(i)}\right\} .
$$

We now argue that the seller can improve his profits and relax the incentive constraints. Using (34), we write the incremental value of experiment $E^{\prime}$ for an obedient type $\theta$ as follows:

$$
V_{\mathrm{ob}}\left(E^{\prime}, \theta\right)-V_{\mathrm{ob}}(E, \theta)=\sum_{i=1}^{I} \theta_{i}\left(u_{i i}-u_{i j(i)}\right) \varepsilon_{i}=\sum_{i=1}^{I} \theta_{i} \eta=\eta .
$$

The seller can therefore increase the price of experiment $E$ by exactly $\eta$ and leave the net utility of any truth-telling type $\theta$ unchanged.

Now consider any other type $\theta^{\prime}$ who chooses a different action $a_{j} \neq a_{j(i)}$ after signal $s_{j(i)}$ from experiment $E$. The marginal benefit to type $\theta^{\prime}$ in state $\omega_{i}$ from the shift in probability from $\pi_{i j(i)}$ to $\pi_{i i}$ is given by

$$
u_{i i}-u_{i j} \leq u_{i i}-u_{i j(i)}
$$

by definition of action $j(i)$. Furthermore, if the discrete shift in probabilities causes type $\theta^{\prime}$ to change his action in response to a given signal, the average benefit of such a shift will be a convex combination of $u_{i i}-u_{i j}$ and $u_{i i}-u_{i j(i)}$. Therefore, the value of experiment $E^{\prime}$ for type $\theta^{\prime}$ exceeds the value of $E$ by at most by $\eta$.

Finally, suppose the original experiment was intended for type $\theta$, i.e., $E=E(\theta)$. The direct mechanism $\mathcal{M}^{\prime}$, where $E^{\prime}(\theta)$ replaces $E(\theta)$ and $t(\theta)+\eta$ replaces $t(\theta)$, is individually rational and incentive compatible. Moreover, experiment $E^{\prime}(\theta)$ is non-dispersed by construction, and all transfers are weakly greater than in the original mechanism $\mathcal{M}$.
(3.) For some utility functions $u_{i j}$, multiple actions can be critical for any given state. The uniform-improvement procedure can then be applied to an experiment as long as it assigns positive probabilities to critical actions in every state. As a result, every optimal experiment will contain one row with a number of zero entries greater than or equal to $\min _{i}\left|\arg \min _{j} u_{i j}\right|$. In the case of matching utility, $\left|\arg \min _{j} u_{i j}\right|=J-1$ for all $i$; hence, every optimal experiment is concentrated. ■


The Design and Price of Information

Proof of Proposition 3. (1.) We know from Proposition 2 that at least one type must buy the fully informative experiment $\bar{E}$. Suppose only type $\theta^{L}$ buys $\bar{E}$ as part of the optimal menu. Then the price of $\bar{E}$ is at most $V\left(\bar{E}, \theta^{L}\right)$. By incentive compatibility, if the high type $\theta^{H}$ purchases $E \neq \bar{E}$, it must be that $t\left(\theta^{H}\right)<V\left(\bar{E}, \theta^{L}\right)$. Therefore, eliminating experiment $E\left(\theta^{H}\right)$ from the menu strictly improves the seller's profits, yielding a contradiction.
(2.) The participation constraint of $\theta^{L}$ must bind. Indeed, some participation constraint must bind, otherwise the seller could increase both prices. Suppose the constraint of $\theta^{H}$ is
binding and that of $\theta^{L}$ is not. Since $\theta^{H}$ is served by $\bar{E}$, then $t\left(\theta^{H}\right)=V\left(\bar{E}, \theta^{H}\right) \geq V\left(\bar{E}, \theta^{L}\right)$. Hence, we can increase the price $t\left(\theta^{L}\right)$ without violating incentive compatibility.
(3.) The incentive constraint of type $\theta^{H}$ must bind. Suppose not and then consider two cases: if the participation constraint of type $\theta^{H}$ does not bind, we can increase $t\left(\theta^{H}\right)$; if the participation constraint of type $\theta^{H}$ does bind, then it must be that $E\left(\theta^{L}\right)$ is not equal to $\bar{E}$. Since payoffs are continuous in $\pi_{i j}$ we can increase both the informativeness of $E\left(\theta^{L}\right)$ and the price $t\left(\theta^{L}\right)$. ■

Proof of Proposition 4. (1.) Recall the definition $\theta^{*}=u_{2} /\left(u_{1}+u_{2}\right)$, and consider the case of congruent priors $\theta^{L}>\theta^{H}>\theta^{*}$. It follows from Proposition 9 that an optimal menu contains only the fully informative experiment $\bar{E}$. The value of experiment $\bar{E}$ is given by $\left(1-\theta^{H}\right) u_{2}$ and $\left(1-\theta^{L}\right) u_{2}$ for the high type and the low type, respectively. The profits from selling to one or both types are given by $\gamma\left(1-\theta^{H}\right) u_{2}$ and $\left(1-\theta^{L}\right) u_{2}$, respectively. Thus, it is optimal to serve both types if and only if $\gamma \leq\left(1-\theta^{L}\right) /\left(1-\theta^{H}\right)$.
(2.) Consider the case of noncongruent priors, $\theta^{L}<\theta^{*}<\theta^{H}$. Let $q \triangleq \pi_{11} u_{1}-\pi_{22} u_{2}$ and denote $q^{L}=q\left(\theta^{L}\right), q^{H}=q\left(\theta^{H}\right)$. It follows from Proposition 9 that in an optimal menu, we have $q^{L} \leq q^{H}=u_{1}-u_{2}$. If $\gamma<\theta^{L} / \theta^{H}$, we wish to show that flat pricing is optimal, i.e., $q^{L}=q^{H}=u_{1}-u_{2}$ and $t_{1}=t_{2}=\left(1-\theta^{L}\right) u_{1}$. Fix an incentive-compatible menu $\left(q^{H}, q^{L}, t^{H}, t^{L}\right)$ with $q^{L}<u_{1}-u_{2}$ and define the following modification:

$$
\left(q^{H \prime}, t^{H \prime}, q^{L \prime}, t^{L \prime}\right)=\left(q^{H}, t^{H}-\varepsilon\left(\theta^{H}-\theta^{L}\right), q^{L}+\varepsilon, t^{L}+\varepsilon \theta^{L}\right) .
$$

If $q^{L}<u_{1}-u_{2}$, modification (35) with $\varepsilon \in\left(0, u_{1}-u_{2}-q_{L}\right)$ preserves incentive compatibility. Furthermore, because $\gamma<\theta^{L} / \theta^{H}$, this improves profits by at least

$$
\varepsilon \theta^{L}(1-\gamma)-\varepsilon\left(\theta^{H}-\theta^{L}\right) \gamma>0,
$$

which yields a contradiction. Finally, we argue that if $\gamma>\theta^{L} / \theta^{H}$, then discriminatory pricing is optimal. First, notice that the individual rationality of the high type must bind, i.e., $q^{H}=u_{1}-u_{2}$ and $t^{H}=\left(1-\theta^{H}\right) u_{2}$; otherwise, modification (35) would be profitable for some $\varepsilon<0$. Second, $q^{L}$ and $t^{L}$ maximize the payment of the low type, subject to his individual-rationality constraint and to the high type's incentive-compatibility constraint. At the optimum, both constraints bind, and the solution is given by

$$
q^{L}=\frac{\theta^{H} u_{1}-\left(1-\theta^{L}\right) u_{2}}{\theta^{H}-\theta^{L}}, t^{L}=\frac{\left(\theta^{H} u_{1}-\left(1-\theta^{H}\right) u_{2}\right) \theta^{L}}{\theta^{H}-\theta^{L}} .
$$

Substituting the definition of $q$ yields expression (20) in the text. ■

Proof of Proposition 5. If priors are noncongruent and the optimal menu is discriminatory, experiment $E\left(\theta^{L}\right)$ has

$$
\hat{\pi}_{1}=\frac{u_{1} \theta^{H}-u_{2}\left(1-\theta^{H}\right)}{u_{1}\left(\theta^{H}-\theta^{L}\right)} \leq 1 .
$$

The informativeness $\pi_{1}$ is increasing in types: $\partial \hat{\pi}_{1} / \partial \theta^{H} \geq 0$ and $\partial \hat{\pi}_{1} / \partial \theta^{L} \geq 0$. The proposition then follows directly from the optimal menu characterization presented in the text.
(1.) If priors are congruent then $\pi_{1}=1$ for $\gamma \leq\left(1-\theta^{L}\right) /\left(1-\theta^{H}\right)$ and $\pi_{1}=0$ otherwise. If priors are noncongruent then $\pi_{1}=1$ for $\gamma \leq \theta^{L} / \theta^{H}$ and $\pi_{1}=\hat{\pi}_{1}$ otherwise.
(2.) If priors are congruent then $\left|\theta^{L}-\theta^{*}\right|=\theta^{L}-\theta^{*}$. The optimal menu has $\pi_{1}=1$ if $\theta^{L} \leq 1-\gamma\left(1-\theta^{H}\right)$ and $\pi_{1}=0$ otherwise. If priors are noncongruent then $\left|\theta^{L}-\theta^{*}\right|=\theta^{*}-\theta^{L}$. The optimal menu has $\pi_{1}=0$ if $\theta^{L}<\gamma \theta^{H}$ and $\pi_{1}=\hat{\pi}_{1}$, which is increasing in $\theta^{L}$, otherwise. (3.) By the normalization of a high type $\left|\theta^{H}-\theta^{*}\right|=\theta^{H}-\theta^{*}$. If priors are congruent then the optimal menu is $\pi_{1}=0$ for $\theta^{H}<1-\left(1-\theta^{L}\right) / \gamma$ and $\pi_{1}=1$ otherwise. If priors are noncongruent and the menu is discriminatory, then $\pi_{1}=\hat{\pi}_{1}$, which is increasing in $\theta^{H}$. ■

Proof of Lemma 1. We begin with necessity. Consider a responsive menu $\{q(\theta)\}_{\theta \in \Theta}$ and any two types $\theta_{2}>\theta_{1}$ who follow the recommendations of signals in both experiments $q_{1}$ and $q_{2}$. We can then rewrite the net utility of experiment $q$ in (21) as

$$
V(q, \theta)=\theta q+u_{2}+\min \left\{u_{1}-u_{2}-q, 0\right\}-\max \left\{\theta u_{1},(1-\theta) u_{2}\right\} .
$$

Because the menu is implementable, the incentive-compatibility constraints imply


The Design and Price of Information

$$
\begin{aligned}
V\left(q_{1}, \theta_{1}\right)-t_{1} & \geq V\left(q_{2}, \theta_{1}\right)-t_{2}, \\
V\left(q_{2}, \theta_{2}\right)-t_{2} & \geq V\left(q_{1}, \theta_{2}\right)-t_{1}, \\
V\left(q_{2}, \theta_{2}\right)-V\left(q_{1}, \theta_{2}\right) & \geq t_{2}-t_{1} \geq V\left(q_{2}, \theta_{1}\right)-V\left(q_{1}, \theta_{1}\right) .
\end{aligned}
$$

The strict single-crossing property of $V(q, \theta)$ in (36) implies $q_{2} \geq q_{1}$; hence, $q(\theta)$ is increasing. Because the buyer's rent is differentiable with respect to $\theta$ on $\left[0, \theta^{*}\right]$ and $\left[\theta^{*}, 1\right]$ respectively, we can compute the function $V(\theta)$ on these two intervals separately. By inspection of (36), the rent is increasing on the former interval and decreasing on the latter. Furthermore, because the utility function $V(q, \theta)$ is continuous in $\theta$, the rent function $V$ is continuous by the Maximum Theorem. We thus obtain the expression in the text:

$$
V\left(\theta^{*}\right)=V(0)+\int_{0}^{\theta^{*}} V_{\theta}(q, \theta) \mathrm{d} \theta=V(1)-\int_{\theta^{*}}^{1} V_{\theta}(q, \theta) \mathrm{d} \theta .
$$

By the envelope theorem, $V_{\theta}(q, \theta)=q+u_{2}$ for $\theta<\theta^{*}$, and $V_{\theta}(q, \theta)=q-u_{1}$ for $\theta>\theta^{*}$. Finally, because $V(0)=V(1)$, we can simplify the equation above and obtain

$$
\int_{0}^{1} q(\theta) \mathrm{d} \theta=u_{1}-u_{2}
$$

We now turn to sufficiency. Suppose the menu $\{q(\theta)\}_{\theta \in \Theta}$ is increasing and satisfies the integral constraint (24). Then, construct the following transfers:

$$
t(\theta)= \begin{cases}\theta q(\theta)+u_{2}+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}-\int_{0}^{\theta}\left(q(x)+u_{2}\right) \mathrm{d} x-(1-\theta) u_{2} & \text { if } \quad \theta<\theta^{*}, \\ \theta q(\theta)+u_{2}+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}+\int_{\theta}^{1}\left(q(x)-u_{1}\right) \mathrm{d} x-\theta u_{1} & \text { if } \quad \theta \geq \theta^{*} .\end{cases}
$$

Because the menu satisfies the integral constraint, we have

$$
\int_{0}^{\theta} q(x) \mathrm{d} x=u_{1}-u_{2}-\int_{\theta}^{1} q(x) \mathrm{d} x,
$$

and we can express all transfers $t(\theta)$ in (37) as

$$
t(\theta)=\theta q(\theta)+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}-\int_{0}^{\theta} q(x) \mathrm{d} x .
$$

Under these transfers, the expected utility of type $\theta$ from reporting type $\theta^{\prime}$ is given by

$$
V\left(q\left(\theta^{\prime}\right), \theta\right)-t\left(\theta^{\prime}\right)=\left(\theta-\theta^{\prime}\right) q\left(\theta^{\prime}\right)+\int_{0}^{\theta^{\prime}} q(x) \mathrm{d} x+u_{2}-\max \left\{\theta u_{1},(1-\theta) u_{2}\right\}
$$

Because $q$ is monotone, the expression on the right-hand side is maximized at $\theta^{\prime}=\theta$, and hence, the incentive constraints are satisfied. Because the rent $V(\theta) \triangleq V(q(\theta), \theta)-t(\theta)$ is non-negative for all $\theta \in[0,1]$, the participation constraints are also satisfied.

Finally, note that the integral constraint (24) and the monotonicity condition imply the responsiveness condition (22). The set $Q(\theta)$ of responsive experiments is described in Figure 9 below. Suppose to the contrary, that $q(\theta) \notin Q(\theta)$ and, in particular, that $q(\theta)=\hat{q}<u_{1}-u_{2} / \theta$ for some $\theta>\theta^{*}$. Then, by monotonicity, we would have

$$
\int_{0}^{1} q(x) \mathrm{d} x \leq \hat{q} \theta+u_{1}(1-\theta)<u_{1}-u_{2}
$$

which yields a contradiction. ■

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 9: Bounds on Responsive Experiments $Q(\theta)$

Proof of Proposition 6. The Fundamental Theorem of Linear Programming applies to settings with $b \in \mathbb{R}^{m}$ and $c \in \mathbb{R}^{n}$, with $m<n$. Let $A$ be an $m \times n$ matrix with the $m$ rows being linearly independent. If the linear problem

$$
\begin{aligned}
& \max _{x \in \mathbb{R}^{n}} c \cdot x \\
& \text { s.t. } A x=b \text { and } x \geq 0
\end{aligned}
$$

has a solution, then it has a solution with all but $m$ entries being zero.
In this form, the Fundamental Theorem cannot be directly applied to the monopolist's problem (25): in this problem, (i) the number of choice variables is a continuum; (ii) the objective function is nonlinear; and (iii) monotonicity constraints are absent in the canonical representation of the theorem. To address (i), we discretize the state space into a fine grid with a radius $\varepsilon,[0, \varepsilon, 2 \varepsilon, \ldots, 1]$. To address (ii), we recall from Proposition 2 that an optimal menu contains the fully informative experiment. Hence, for any optimal menu $\{q(\theta)\}_{\theta \in \Theta}$, there is some type $\theta^{I}$ such that $q\left(\theta^{I}\right)=u_{1}-u_{2}$. Consequently, any optimal $q$ must solve the problem with the additional constraint $q\left(\theta^{I}\right)=u_{1}-u_{2}$ for an appropriately chosen $\theta^{I}$. Because the menu is monotone, the objective can be written linearly with $\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}=0$ for all $\theta<\theta^{I}$ and $\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}=u_{1}-u_{2}-q(\theta)$ for all $\theta>\theta^{I}$. Finally, to address (iii), we change variables from $q(\theta)$ to its increment $\hat{q}(\theta) \triangleq q(\theta)-q(\theta-\varepsilon)$, where $\hat{q}(0) \triangleq 0$. Finally, we substitute $q(\theta)=-u_{2}+\sum_{x=0}^{\theta} \hat{q}(x)$ and rewrite the monotonicity constraint as $\hat{q}(\theta) \geq 0$.

The monopolist's problem (25) can be restated as

$$
\begin{aligned}
& \max _{\hat{q}(\theta)} \sum_{\theta=0}^{1} \hat{q}(\theta)\left(\sum_{x=\theta}^{1}(x f(x)+F(x))-\sum_{x=\theta^{I}}^{1} f(x)\right) \\
& \text { s.t. } \sum_{\theta=0}^{\theta^{I}} \hat{q}(\theta)=u_{1}, \\
& \quad \sum_{\theta=\theta^{I}}^{1} \hat{q}(\theta)=u_{2}, \\
& \quad \sum_{\theta=0}^{1}\left(-u_{2}+\sum_{x=0}^{\theta} \hat{q}(x)\right)=u_{1}-u_{2}, \\
& \quad \hat{q}(\theta) \geq 0 .
\end{aligned}
$$


The Design and Price of Information

This is a canonical linear programming problem with three linearly independent equality constraints. The first two correspond to the total change from the experiment $q(0)=-u_{2}$ to the fully informative experiment $q\left(\theta^{I}\right)=u_{1}-u_{2}$ to $q(1)=u_{1}$. The last equality is integral constraint (24). Hence, by the Fundamental Theorem, for any $\theta^{I}$ there is an optimal menu of experiments with at most three positive entries. Since $q(1)=u_{1}$ corresponds to an uninformative experiment, this implies there are at most two informative experiments in any optimal menu.

Finally, consider an arbitrary discrete distribution that converges in distribution to a continuous distribution. The profits converge for any fixed menu, and hence, optimal profits converge as well. Furthermore, the set of profits that can be achieved by two-item menus is compact. Therefore, there exists an optimal menu with continuous types that has at most two items. ■

Proof of Proposition 7. We first derive the seller's objective in the usual way. Using (38) to write the expected transfers and integrating by parts, we obtain

$$
\int_{0}^{1} t(\theta) \mathrm{d} F(\theta)=\int_{0}^{1}\left(\theta q(\theta)+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\}-\frac{1-F(\theta)}{f(\theta)} q(\theta)\right) \mathrm{d} F(\theta) .
$$

Using the integral constraint, we obtain (up to an additive constant)

$$
\int_{0}^{1} t(\theta) \mathrm{d} F(\theta)=\int_{0}^{1}\left[(\theta f(\theta)+F(\theta)) q(\theta)+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\} f(\theta)\right] \mathrm{d} \theta .
$$

We now establish that the solution to the seller's problem (25) can be characterized through Lagrangian methods. For necessity, note that the objective is concave in the experiment; the set of non-decreasing functions is convex, and the integral constraint can be
weakened to the real-valued inequality constraint

$$
\int_{0}^{1} q(\theta) \mathrm{d} \theta \leq u_{1}-u_{2} .
$$

Necessity of the Lagrangian then follows from Theorem 8.3.1 in Luenberger (1969). Sufficiency follows from Theorem 8.4.1 in Luenberger (1969). In particular, any solution maximizer of the Lagrangian $q(\theta)$ with

$$
\int_{0}^{1} q(\theta) \mathrm{d} \theta=\bar{q}
$$

maximizes the original objective subject to the inequality constraint

$$
\int_{0}^{1} q(\theta) \mathrm{d} \theta \leq \bar{q} .
$$

Thus, any solution to the Lagrangian that satisfies the constraint solves the original problem.
Because the Lagrangian approach is valid, we apply the results of Toikka (2011) to solve the seller's problem for a given value of the multiplier $\lambda$ on the integral constraint. Write the Lagrangian as

$$
\int_{0}^{1}\left[(\theta f(\theta)+F(\theta)) q(\theta)+\min \left\{u_{1}-u_{2}-q(\theta), 0\right\} f(\theta)+\lambda\left(u_{1}-u_{2}-q(\theta)\right)\right] \mathrm{d} \theta .
$$

In order to maximize the Lagrangian subject to the monotonicity constraint, consider the generalized virtual surplus

$$
\Phi(\theta, q):=\int_{-u_{2}}^{q}\left(\bar{\phi}(\theta, x)-\lambda^{*}\right) \mathrm{d} x
$$

where $\bar{\phi}(\theta, x)$ denotes the ironed virtual value for experiment $x$. (Up to a constant, $\Phi(\theta, q)$ is a general formulation of the virtual value in the statement of the proposition.) Note that the virtual surplus $\Phi(\theta, q)$ is weakly concave in $q$. Because the multiplier $\lambda$ shifts all virtual values by a constant, the result in Proposition 7 follows from Theorem 4.4 in Toikka (2011). Finally, note that the optimal value of $\lambda^{*}$ is strictly positive. If not, the pointwise maximizer of $\bar{\phi}(\theta, q)$ would be weakly above $u_{1}-u_{2}$ and strictly so for some $\theta$, leading to a violation of the integral constraint. Therefore, the inequality constraint (39) must bind. ■

Proof of Proposition 9. We know from Proposition 3 that the high type $\theta^{H}$ purchases the fully informative experiment. We now derive the optimal experiment $E\left(\theta^{L}\right)$. Suppose (as we later verify) that both types $\theta^{H}$ and $\theta^{L}$ choose action $a_{i}$ after observing signal $s_{i}$ from
experiment $E\left(\theta^{L}\right)$. The seller's relaxed problem can be written as

$$
\begin{aligned}
\max _{0 \leq \pi_{i i} \leq 1, V\left(\theta^{H}\right)} & (1-\gamma) \sum_{i=1}^{I} \theta_{i}^{L} u_{i} \pi_{i i}-\gamma V\left(\theta^{H}\right), \\
& \text { s.t. } V\left(\theta^{H}\right) \geq \sum_{i=1}^{I} \pi_{i i} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)-\max _{i} \theta_{i}^{H} u_{i}+\max _{i} \theta_{i}^{L} u_{i} \geq 0 .
\end{aligned}
$$

where the latter inequality ensures that the high type $\theta^{H}$ achieves a non-negative payoff when misreporting his type and following every signal's recommendation.

We now simplify the problem as follows. By Proposition 3, the incentive-compatibility constraint of type $\theta^{H}$ binds and the relaxed problem can be rewritten as

$$
\begin{aligned}
& \max _{0 \leq \pi_{i i} \leq 1} \sum_{i=1}^{I} \pi_{i i} u_{i}\left(\theta_{i}^{L}-\gamma \theta_{i}^{H}\right) \\
& \quad \text { s.t. } \sum_{i=1}^{I} \pi_{i i} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)-\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}+\max _{i}\left\{\theta_{i}^{L} u_{i}\right\} \geq 0
\end{aligned}
$$

Now, arrange the states such that

$$
\frac{\theta_{1}^{L}}{\theta_{1}^{H}} \leq \cdots \leq \frac{\theta_{I}^{L}}{\theta_{I}^{H}},
$$

and consider the function $R(i)$ defined in the text

$$
R(j) \triangleq \sum_{i=j}^{I} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)-\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}+\max _{i}\left\{\theta_{i}^{L} u_{i}\right\} .
$$

The function $R(j)$ is decreasing in $j$ for $j<i_{c}$ and increasing in $j$ for $j>i_{c}$, where

$$
i_{c} \triangleq \min \left\{i: 1 \leq \theta_{i}^{L} / \theta_{i}^{H}\right\} .
$$

We now show that $\min _{j} R(j)=R\left(i_{c}\right)<0$. This is immediate if $\max _{i}\left\{\theta_{i}^{H} u_{i}\right\} \geq \max _{i}\left\{\theta_{i}^{L} u_{i}\right\}$, as we have $\sum_{i=i_{c}}^{I} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)<0$ by construction. Conversely, if $\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}<\max _{i}\left\{\theta_{i}^{L} u_{i}\right\}$, then we let $i_{L} \triangleq \arg \max _{i}\left\{\theta_{i}^{L} u_{i}\right\}$, and we derive the following bound:


The Design and Price of Information

$$
R\left(i_{c}\right) \leq R\left(i_{L}\right) \leq \sum_{i=i_{L}}^{I} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)-\theta_{i_{L}}^{H} u_{i_{L}}+\theta_{i_{L}}^{L} u_{i_{L}}=\sum_{i=i_{L}+1}^{I} u_{i}\left(\theta_{i}^{H}-\theta_{i}^{L}\right)<0
$$

where the latter inequality follows from the fact that $\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}<\max _{i}\left\{\theta_{i}^{L} u_{i}\right\}$ implies $\theta_{i_{L}}^{L}>\theta_{i_{L}}^{H}$, which means $i_{L} \geq i_{c}$.

We now turn to the solution to the seller's problem. Notice that (40)-(41) is a linear problem that attains a monotone solution. In particular, any solution must be a monotone sequence: $\pi_{i i}=0$ for $i<\hat{\imath}$ and $\pi_{i i}=1$ for $i>\hat{\imath}$, for some $\hat{\imath} \in\{1, \ldots, I\}$. (This structure can
be seen immediately from the Lagrange multiplier method, as the solution must maximize some linear combination of the objective function and the constraint.)

Suppose that the constraint does not bind at the optimum. Then, the solution assigns $\pi_{i i}=1$ if and only if $i \geq i_{s} \triangleq \min \left\{i: \gamma \leq \theta_{i}^{L} / \theta_{i}^{H}\right\}$. Note that $i_{s} \leq i_{c}$, which was defined in (42). Therefore, if $R\left(i_{s}\right) \geq 0$ the constraint is satisfied at the unconstrained optimum. If, instead, we have $R\left(i_{s}\right)<0$, then by the definition of the critical state $i_{b}$,

$$
R\left(i_{b}\right)>0>R\left(i_{b}+1\right),
$$

we have $i_{b}<i_{s}$. In this case, the solution is given by $\pi_{i i}=0$ for $i<i_{b}, \pi_{i i}=1$ for $i>i_{b}$ and by setting $\pi_{i_{b} i_{b}}$ to satisfy (41) with equality, i.e.,

$$
\pi_{i_{b} i_{b}}=\frac{\sum_{j=i_{b}+1}^{I}\left(\theta_{j}^{L}-\theta_{j}^{H}\right) u_{j}-\max _{j} \theta_{j}^{L} u_{j}+\max _{j} \theta_{j}^{H} u_{j}}{\left(\theta_{i_{b}}^{H}-\theta_{i_{b}}^{L}\right) u_{i_{b}}} .
$$

Combining these two arguments, we conclude that the optimal cutoff state $\hat{\imath}$ is given by $i^{*}=\min \left\{i_{b}, i_{s}\right\}$ and that $\pi_{i^{*} i^{*}} \in\left\{\pi_{i_{b} i_{b}}, 1\right\}$ depending on whether $i_{b} \lessgtr i_{s}$.

To complete the menu, we now need to specify the off-diagonal entries $\pi_{i j}$ to ensure both types $\theta^{H}$ and $\theta^{L}$ choose action $a_{i}$ when observing signal $s_{i}$. This requires

$$
\pi_{i i} \theta_{i} u_{i} \geq \pi_{j i} \theta_{j} u_{j}
$$

for both types and for all $j<i$, because the signal matrix can be taken to be lower triangular. In particular, we need to ensure that, for all $j<i^{*}$,

$$
\begin{aligned}
& \pi_{i^{*} i^{*}} u_{i^{*}} \theta i_{i^{*}}^{H} \geq \pi_{j i^{*}} u_{j} \theta_{j}^{H} \\
& \pi_{i^{*} i^{*}} u_{i^{*}} \theta_{i^{*}}^{L} \geq \pi_{j i^{*}} u_{j} \theta_{j}^{L}
\end{aligned}
$$

Because $\theta_{j}^{L} / \theta_{j}^{H} \leq \theta_{i}^{L} / \theta_{i}^{H}$ for $j<i$, it suffices to satisfy the constraint of type $\theta^{H}$.
We proceed as follows. Fix an alternative action $a_{j}$ with $j<i^{*}$. For any $i>i^{*}$, we make type $\theta^{H}$ indifferent between following the recommendation of signal $i$ and choosing action $a_{j}$; we do so beginning with $\pi_{j I}$ and proceeding backward as long as required. If this procedure assigns positive weight to $\pi_{j i^{*}}$, then it must be that

$$
\pi_{j i^{*}}=1-\sum_{i=i^{*}+1}^{I} \frac{\theta_{i}^{H} u_{i}}{\theta_{j}^{H} u_{j}} .
$$

We argue that type $\theta^{H}$ has strict incentives to follow the recommendation of signal $i^{*}$ when
$\pi_{i^{*} i^{*}}=\pi_{i^{*}}^{B}$. (A fortiori, type $\theta^{H}$ has strict incentives to choose action $a_{i}$ following any signal $s_{i}$ with $i \geq i^{*}$ if $\pi_{i i}=1$.) Recall the definition

$$
\pi_{i^{*} i^{*}} \theta_{i^{*}}^{H} u_{i^{*}}=\frac{\sum_{i=i^{*}+1}^{I} u_{i}\left(\theta_{i}^{L}-\theta_{i}^{H}\right)-\max _{i} \theta_{i}^{L} u_{i}+\max _{i} \theta_{i}^{H} u_{i}}{\theta_{i^{*}}^{H}-\theta_{i^{*}}^{L}} \theta_{i^{*}}^{H} .
$$

Let $i_{L}=\arg \max _{i} \theta_{i}^{L} u_{i}$ and $i_{H}=\arg \max _{i} \theta_{i}^{H} u_{i}$, and consider the following two cases.
(1.) If $i_{L}>i^{*}$, then we know that

$$
\sum_{i=i^{*}+1}^{I} \theta_{i}^{L} u_{i}>\max _{i} \theta_{i}^{L} u_{i},
$$

and we bound $\pi_{i^{*} i^{*}} \theta_{i^{*}}^{H} u_{i^{*}}$ by

$$
\pi_{i^{*} i^{*}} \theta_{i^{*}}^{H} u_{i^{*}}>\frac{\theta_{i^{*}}^{H}}{\theta_{i^{*}}^{H}-\theta_{i^{*}}^{L}}\left(\theta_{j}^{H} u_{j}-\Sigma_{i=i^{*}+1}^{I} \theta_{i}^{H} u_{i}\right)>\theta_{j}^{H} u_{j}-\Sigma_{i=i^{*}+1}^{I} \theta_{i}^{H} u_{i}=\pi_{j i^{*}} \theta_{j}^{H} u_{j} .
$$

(2.) If $i_{L} \leq i^{*}$, then from (43) and (44), we know that the difference $\pi_{i^{*} i^{*}} \theta_{i^{*}}^{H} u_{i^{*}}-\pi_{j i^{*}} \theta_{j}^{H} u_{j}$ is proportional to

$$
\begin{aligned}
& \sum_{i=i^{*}+1}^{I} u_{i}\left(\theta_{i}^{L}-\theta_{i}^{H}\right)-\max _{i} \theta_{i}^{L} u_{i}+\max _{i} \theta_{i}^{H} u_{i}-\left(1-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}}\right)\left(\theta_{j}^{H} u_{j}-\Sigma_{i=i^{*}+1}^{I} \theta_{i}^{H} u_{i}\right) \\
= & \sum_{i=i^{*}+1}^{I} u_{i}\left(\theta_{i}^{L}-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}} \theta_{i}^{H}\right)-\max _{i} \theta_{i}^{L} u_{i}+\max _{i} \theta_{i}^{H} u_{i}-\left(1-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}}\right) \theta_{j}^{H} u_{j} .
\end{aligned}
$$

Notice that every term in the sum is positive because the likelihood ratio is increasing in $i$. The remaining terms can be written as

$$
\begin{aligned}
-\max _{i} \theta_{i}^{L} u_{i}+\max _{i} \theta_{i}^{H} u_{i}-\left(1-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}}\right) \theta_{j}^{H} u_{j} & =-\frac{\theta_{i_{L}}^{L}}{\theta_{i_{L}}^{H}} \theta_{i_{L}}^{H} u_{i_{L}}+\theta_{i_{H}}^{H} u_{i_{H}}-\left(1-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}}\right) \theta_{j}^{H} u_{j} \\
& \geq \theta_{i_{H}}^{H} u_{i_{H}}\left(1-\frac{\theta_{i_{L}}^{L}}{\theta_{i_{L}}^{H}}\right)-\left(1-\frac{\theta_{i^{*}}^{L}}{\theta_{i^{*}}^{H}}\right) \theta_{j}^{H} u_{j} .
\end{aligned}
$$

Here, the likelihood ratios are ranked (because $i_{L} \leq i^{*}$ in this case), and $\theta_{i_{H}}^{H} u_{i_{H}} \geq \theta_{j}^{H} u_{j}$ by the definition of $i_{H}$. ■


The Design and Price of Information

## B Supplemental (Online) Appendix


The Design and Price of Information

## B. 1 Many Actions and States

We illustrate the construction of the optimal experiment as determined by Proposition 8 and the implications for information rents in the following example.

Example 1 (Noncongruent Types) Consider uniform match values ( $u_{i}=1$ for all $i=$ 1,2,3) and two types, $\theta^{L}=(1 / 10,1 / 10,8 / 10)$ and $\theta^{H}=(4 / 10,3 / 10,3 / 10)$. These types are noncongruent: without additional information, $\theta^{H}$ would choose action $a_{1}$ and $\theta^{L}$ would choose $a_{3}$. The likelihood ratios $\theta_{i}^{L} / \theta_{i}^{H}$ are (1/4, 1/3, 8/3). This implies $i_{b}=2$, whereas $i_{s} \in\{1,2\}$ depending on the prior probability $\gamma$ of the high type. For $\gamma \in[0,1 / 4]$ and $\gamma \in[1 / 4,1 / 3]$, Proposition 9,2 applies, and the high type obtains positive rents. Furthermore, for $\gamma \geq 1 / 4$, the partially informative experiment $E\left(\theta^{L}\right)$ involves dropping signal $s_{1}$. The optimal experiment $E\left(\theta^{L}\right)$ as a function of $\gamma$ is given by

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 | 0 |
| $\omega_{2}$ | 0 | 1 | 0 |
| $\omega_{3}$ | 0 | 0 | 1 |
|  | if | < | 1/4, |

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 0 | 1/4 | 3/4 |
| $\omega_{2}$ | 0 | 1 | 0 |
| $\omega_{3}$ | 0 | 0 | 1 |
|  | if | $\in[1$ | 1/4, 1/3], |

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 0 | 1/4 | 3/4 |
| $\omega_{2}$ | 0 | 1/2 | 1/2 |
| $\omega_{3}$ | 0 | 0 | 1 |
|  | if | > 1 | 1/3. |

Example 2 illustrates how congruent, but not strongly congruent beliefs, allow for surplus extraction. In the example, the two types deem state $\omega_{2}$ the most likely. Thus the types are congruent but not strongly congruent, as they disagree on the relative likelihood of states $\omega_{1}$ and $\omega_{3}$.

Example 2 (Congruent Priors) Consider uniform match values ( $u_{i}=1$ for all $i=$ $1,2,3)$ and two types, $\theta^{L}=(5 / 10,1 / 10,4 / 10)$ and $\theta^{H}=(4 / 10,3 / 10,3 / 10)$. Because $i_{b}=1$, the optimal experiment $E\left(\theta^{L}\right)$ as a function of $\gamma$ is given by

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 | 0 |
| $\omega_{2}$ | 0 | 1 | 0 |
| $\omega_{3}$ | 0 | 0 | 1 |
|  | if | $\leq$ | 1/3, |

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 | 0 |
| $\omega_{2}$ | 0 | 1/2 | 1/2 |
| $\omega_{3}$ | 0 | 0 | 1 |
|  | if | > 1 | 1/3, |

and the high type obtains positive rents only if $\gamma<1 / 3$.

In Example 3 we illustrate that the relaxed approach is not valid with many types. In the example below with three types, no experiment $E\left(\theta^{1}\right)$ can lead both types $\theta^{2}$ and $\theta^{3}$ to follow the action recommended by every signal. Thus, the profits in the relaxed problem are strictly greater than those in original problem.

Example 3 (Many Types and Actions) Consider uniform match values ( $u_{i}=1$ for all $i=1,2,3)$ and three types, $\theta^{1}=(1 / 6,1 / 6,4 / 6), \theta^{2}=(1 / 2,1 / 2,0)$, and $\theta^{3}=(1 / 2,0,1 / 2)$, which are all equally likely. In the relaxed problem, the monopolist sells the fully informative experiment to types $\theta^{2}$ and $\theta^{3}$. Type $\theta^{1}$ is offered the partially informative experiment

| $E\left(\theta^{1}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1/2 | 0 | 1/2 |
| $\omega_{2}$ | 0 | 1 | 0 |
| $\omega_{3}$ | 0 | 0 | 1 |

and the seller's revenues are equal to 5/12. However, if type $\theta^{2}$ purchased experiment $E\left(\theta^{1}\right)$, he would choose action $a_{1}$ when observing signal $s_{3}$. In the solution to the full problem, which we can construct by a guess-and-verify approach, the optimal experiment $E\left(\theta^{1}\right)$ consists of

| $E\left(\theta^{1}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1/2 | 0 | 1/2 |
| $\omega_{2}$ | 1/2 | 1/2 | 0 |
| $\omega_{3}$ | 0 | 0 | 1 |

which yields revenues of 1/3, i.e., revenues are strictly lower in the relaxed program.


The Design and Price of Information

## B. 2 More Actions than States

We consider a setting with two types, and we relax the assumption of matching state-action payoffs. In particular, we consider the following example with two types, two states, and three actions. The data buyer's payoff is given by

| $u(\omega, a)$ | $a_{1}$ | $a_{2}$ | $a_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 | 4/5 |
| $\omega_{2}$ | 0 | 1 | 4/5 |

Thus, action $a_{i}$ is the optimal action in state $\omega_{i}$, but action $a_{3}$ provides a lower bound on the payoffs that is uniform across states-an insurance action. Let the two types be given by $\theta^{L}=(1 / 10,9 / 10)$ and $\theta^{H}=(6 / 10,4 / 10)$. As the results of Proposition 3 do not rely
on matching payoffs, we know type $\theta^{H}$ receives full information, $E\left(\theta^{H}\right)=\bar{E}$, his incentive constraint binds, and the participation constraint of type $\theta^{L}$ binds.

For the case $\gamma \triangleq \operatorname{Pr}\left(\theta=\theta^{H}\right)=3 / 4$, an optimal menu contains the experiment

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1/3 | 2/15 | 8/15 |
| $\omega_{2}$ | 0 | 4/5 | 1/5 |

at a price $t\left(\theta^{L}\right)=1 / 25$ and the experiment $\bar{E}$ at a price $t\left(\theta^{H}\right)=1 / 5$. The optimal menu is illustrated in Figure 10.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 10: Optimal Menu

This menu has the following notable properties: (i) the seller extracts all the surplus from both types; (ii) type $\theta^{L}$ follows the recommendation of every signal in $E\left(\theta^{L}\right)$; (iii) type $\theta^{H}$, if purchasing experiment $E\left(\theta^{L}\right)$, is indifferent between action $a_{2}$ and action $a_{3}$ when observing signal $s_{2}$ as well as between $a_{1}$ and $a_{3}$ when observing signal $s_{3}$; and (iv) the optimal profits are strictly lower than those in the relaxed problem.

Indeed, ignoring the off-path obedience constraints, the optimal menu is discriminatory, and the seller extracts all the rents by offering the experiment

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | 2/5 | 3/5 |
| $\omega_{2}$ | 0 | 1 |

at a price $t\left(\theta^{L}\right)=1 / 25$ and the experiment $\bar{E}$ at a price $t\left(\theta^{H}\right)=2 / 5$.
In the full problem, the seller cannot turn experiment $E\left(\theta^{L}\right)$ in (46) into the more informative one in (47). If she did, buyer type $\theta^{H}$ would choose action $a_{3}$ after deviating
and observing signal $s_{2}$. ${ }^{27}$ In other words, the seller extracts the surplus, but at the cost of additional distortion-notably, there is no "1" entry in (46).

Interestingly, action $a_{3}$ may not be induced in an optimal menu yet still restrict the seller. Indeed, if we modify the above example by setting $\gamma=2 / 3$, an optimal menu can be calculated to contain the experiment

| $E\left(\theta^{L}\right)$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | 5/6 | 1/6 |
| $\omega_{2}$ | 0 | 1 |

at a price $t\left(\theta^{L}\right)=1 / 12$ and the experiment $\bar{E}$ at a price of $t\left(\theta^{H}\right)=11 / 60<1 / 5=V\left(\theta^{H}, \bar{E}\right)$. As before, type $\theta^{H}$ is indifferent between $a_{2}$ and $a_{3}$ after deviating to $E\left(\theta^{L}\right)$ and observing $s_{2}$. Contrary to our earlier examples, the high type makes positive rents despite the seller's discriminatory menu offering.

To emphasize, action $a_{3}$ is not chosen in an optimal menu by either type, yet it looms large and prevents the seller from extracting the full surplus. In particular, the seller would like to reduce $\pi_{11}$ in order to relax the high type's incentive constraint and increase $t^{H}$. However, by doing so she would induce type $\theta^{H}$ to choose action $a_{3}$ after $s_{2}$. This means that the high type's marginal benefit from a probability shift from $\pi_{11}$ to $\pi_{12}$ is $\theta_{1}^{H}(-1+4 / 5)=-3 / 25$, while the corresponding marginal change in price $t^{L}$ is $-\theta_{1}^{L}=-1 / 10$. Therefore, $t^{H}$ can only increase at rate 1/50, which is not profitable for the seller when the fraction of high types is $\gamma=2 / 3$. If both types were instead required to follow the signals' recommendations, the high type's misreporting value would change at rate $-\theta_{1}^{H}=-3 / 5$, allowing the seller to increase $t^{H}$ at the profitable rate of 1/2. The kink in the "exchange rate" when the high type is indifferent among several actions prevents the seller from making the modification.

To reinforce the point, maintain the assumption $\gamma=2 / 3$ but exclude action $a_{3}$ from the set of available actions. The optimal menu is again given by the experiment in (47), which is now less informative than (48), but allows the seller to extract all the rents.


The Design and Price of Information

## B. 3 Sequential Design

We show that sequential design of experiments can increase the seller's revenues in our leading binary-type example. We focus on the simplest instance of a dynamic protocol, whereby the seller first releases a free informative experiment to the buyer and then, without observing

[^17]the realized signal, offers a menu of (experiment, price) pairs from which to choose.
Let $\Omega=\left\{\omega_{1}, \omega_{2}\right\}, A=\left\{a_{1}, a_{2}\right\}$, and assume uniform match values, i.e.,
$$
u\left(\omega_{i}, a_{j}\right)=\mathbb{I}_{[i=j]} .
$$
Consider two equally likely types with interim beliefs $\theta^{L}=1 / 8$ and $\theta^{H}=1 / 4$,respectively, where $\theta \triangleq \operatorname{Pr}\left[\omega_{1}\right]$.

Because the two types are congruent, an optimal static mechanism (Proposition 4) contains only the fully informative experiment. In the current example, the seller is indifferent between charging prices $t=1 / 8$ and $t=1 / 4$. In either case, the monopoly profits are

$$
\pi_{\text {static }}^{*}=1 / 8 .
$$

Consider the following sequential scheme. First, the seller reveals an outcome of the following experiment $E_{0}$ at no cost to the buyer

| $E_{0}$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 |
| $\omega_{2}$ | 1/3 | 2/3 |

After observing signal $s_{2}$, the buyer is convinced that the state is $\omega=\omega_{2}$, which confirms his prior, and does not buy further information. After realization $s_{1}$, however, the buyer's beliefs are updated to

$$
\theta^{L}\left(s_{1}\right)=3 / 10, \quad \theta^{H}\left(s_{1}\right)=1 / 2 .
$$

At this point, the seller offers the fully informative experiment at a price $\bar{t}=3 / 10$.
The key observation is that signal $s_{1}$ under experiment $E_{0}$ is more likely be realized for the high type $\theta^{H}=1 / 4$ than for the low type $\theta^{L}=1 / 8$. In particular, the signal distribution is given by

$$
\operatorname{Pr}\left[s_{1} \mid \theta^{L}\right]=\frac{5}{12}, \operatorname{Pr}\left[s_{1} \mid \theta^{H}\right]=\frac{1}{2} .
$$

As a consequence, the monopolist's profit is given by

$$
\pi_{\mathrm{dyn}}^{*} \triangleq \bar{t}\left(\gamma \operatorname{Pr}\left[s_{1} \mid \theta^{H}\right]+(1-\gamma) \operatorname{Pr}\left[s_{1} \mid \theta^{L}\right]\right)=11 / 80 .
$$

Thus, the sequential sale outperforms the static sale in this example, i.e.,

$$
\pi_{\mathrm{dyn}}^{*}=11 / 80>1 / 8=\pi_{\mathrm{static}}^{*} .
$$

Taking a step back, it is clear that the seller would ideally like to condition payments on the realized states. In this case, she could charge a payment of 1 upon realization of state $\omega_{1}$, which is the state less likely for either type. Both types would accept such a contract, and the seller achieves the first-best profits. As we do not allow for the payments to be made contingent on the realization of the state, a sequential mechanism essential represents a costly instrument to (partially) circumvent this restriction.

In essence, the proposed sequential scheme charges a constant price $\bar{t}=3 / 10$ upon realization of signal $s_{1}$. Because the signal is correlated with the state under experiment $E_{0}$, it occurs more frequently for the higher type, allowing the seller to effectively price discriminate without ever giving the buyer a choice of experiment.

Finally, note that the seller could do better within the simple class of mechanisms that initially release a free experiment, followed by a menu.

Intuitively, as the correlation between state and signal $s_{1}$ becomes more precise (i.e., as $s_{1}$ becomes more informative), the seller's ability to condition payments on states improves. Ultimately, however, the seller must balance the ability to correlate payments with the willingness to pay for supplemental information after observing signal $s_{1}$ (e.g., the signal cannot be arbitrarily precise).

To formalize the intuition, consider offering free experiments of the following form

| $E(x)$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | 1 | 0 |
| $\omega_{2}$ | $1-x$ | $x$ |

These experiments lead to posterior beliefs

$$
\begin{aligned}
\theta^{L}(x) & \triangleq \operatorname{Pr}\left[s_{1} \mid \theta^{L}\right]=\frac{1}{8-7 x}, \\
\theta^{H}(x) & \triangleq \operatorname{Pr}\left[s_{1} \mid \theta^{H}\right]=\frac{1}{4-3 x} .
\end{aligned}
$$

These beliefs satisfy the condition $1 / 2=\gamma \geq \theta^{L}(x) / \theta^{H}(x)$ for all $x$. Therefore, after releasing experiment $E(x)$ the seller optimally offers the fully informative experiment $\bar{E}$ at a price

$$
\bar{t}(x)=\min \left\{\theta^{L}(x), 1-\theta^{H}(x)\right\} .
$$

Finally, a straightforward calculation reveals that the seller's profits are maximized by choosing $x$ such that $\theta^{L}(x)<1 / 2<\theta^{H}(x)$. In particular, it is optimal for the seller to induce
the two types to have identical willingness to pay for the full information, i.e.,

$$
\theta^{L}\left(x^{*}\right)=1-\theta^{H}\left(x^{*}\right) .
$$

The optimal experiment has

$$
x^{*}=1-1 / \sqrt{21} \approx 0.781,
$$

which is larger than $x=2 / 3$, as used in the initial example, and yields profits $\pi^{*}=(7+$ $2 \sqrt{21}) / 112 \approx 0.144$ that exceed 11/80, as computed above.


The Design and Price of Information

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[^0]:    *We thank the co-editor, Jeff Ely, and three anonymous referees for their productive suggestions. We are grateful for conversations with Ben Brooks, Giacomo Calzolari, Gabriel Carroll, Gonzalo Cisternas, Jacques Crémer, Andrei Hagiu, Teck Ho, Bruno Jullien, Emir Kamenica, Alessandro Lizzeri, Alessandro Pavan, Mike Powell, Phil Reny, Mike Riordan, Maher Said, Jean Tirole, Juuso Toikka, Alex Wolitzky, and to seminar participants at Berkeley, Bocconi, Bologna, Carnegie Mellon, Chicago, Eief, Harvard, Mannheim, Northwestern, NYU, Oxford, QMUL, Toulouse, UCL, Vienna, Warwick, Yale, and the 2015 World Congress of the Econometric Society. We also thank Ian Ball for excellent research assistance.
    ${ }^{\dagger}$ Yale University, 30 Hillhouse Ave., New Haven, CT 06520, USA, dirk.bergemann@yale.edu.
    ${ }^{\ddagger}$ MIT Sloan School of Management, 100 Main Street, Cambridge, MA 02142, USA bonatti@mit.edu.
    ${ }^{§}$ University of Bonn, Lennéstraße 37, 53113, Bonn, Germany alexey.v.smolin@gmail.com.


The Design and Price of Information

[^1]:    ${ }^{1}$ In addition, a number of more recent papers, including Balestrieri and Izmalkov (2014), Celik (2014), Koessler and Skreta (2016), and Mylovanov and Tröger (2014), analyze this question from an informed principal perspective.

[^2]:    ${ }^{2}$ For example, see Manelli and Vincent (2006), Pycia (2006), Pavlov (2011a), and Rochet and Thanassoulis (2015). In particular, Daskalakis, Deckelbaum, and Tzamos (2017) construct an example wherein the types follow a Beta distribution, and the optimal menu contains a continuum of stochastic allocations.

[^3]:    ${ }^{3}$ As usual, the model allows for the alternative interpretations of a single buyer and a continuum of buyers.

[^4]:    ${ }^{4}$ The value of information for the data buyer differs from a consumer's value for multiple goods or bundles of characteristics. In particular, the first max operator in (6) and (7) corresponds to the optimality condition for the buyer's action given the available information. The second max operator reflects the type-dependent nature of participation constraints. Both elements are missing from the multiproduct monopolist's problem (Pavlov, 2011a b; Daskalakis, Deckelbaum, and Tzamos, 2017).
    ${ }^{5}$ The information structure in the right panel is given by the following stochastic matrix:

[^5]:    ${ }^{6}$ The fully informative experiment is part of every optimal menu if all types assing positive probability to all states. It need not be part of every optimal menu if some types assign zero probability to some states. In that case, an optimal menu may contain a partially informative experiment that, combined with some type's prior, fully reveals all states that have positive probability, and so remains fully informative in this weaker sense.

[^6]:    ${ }^{7}$ Discriminatory menus that do not shut down the low-value type are optimal in nonlinear screening models, such as the monopolistic insurance market studied in Stiglitz (1977). The novel element of selling information to buyers with noncongruent priors is that the seller can more easily extract (possibly all) the information rents.

[^7]:    ${ }^{8}$ In Section 6, we establish a stronger result: with many actions and states, there always exists a distribution $\gamma$ such that partial information is provided to the low type unless the two types agree on both the most likely state and the relative likelihood of any other two states. In the binary-state model, the latter condition is vacuously satisfied. However, with more than two states, the latter condition fails generically.

[^8]:    ${ }^{9}$ Intuitively, if the rent function had a downward jump at $\theta^{*}$, some nearby type $\theta^{*}+\varepsilon$ could purchase the experiment intended for type $\theta^{*}-\varepsilon$. This yields a payoff that is arbitrarily close to $V\left(\theta^{*}-\varepsilon\right)$, and hence, gives $\theta^{*}+\varepsilon$ an incentive to misreport his type.
    ${ }^{10}$ Our environment with type-dependent participation constraints is an instance of the "high convexity"

[^9]:    case in Jullien (2000). As such, the integral condition for implementability (24) differs from budget, capacity, or enforceability constraints because the distribution $F(\theta)$ does not appear in the integrand. A similar condition, for different reasons and with different implications, appears as a persuasion budget constraint in Kolotilin, Li, Mylovanov, and Zapechelnyuk (2015).
    ${ }^{11}$ This result is related to linear mechanism-design problems with budget constraints, such as Samuelson (1984), Brusco and Hopenhayn (2007), and Fuchs and Skrzypacz (2015). In our setting, the reduction to a linear program is obtained through a change of variable from $q(\theta)$ to its increments, and by imposing the additional constraint that some type $\theta^{*} \in(0,1)$ purchases the fully informative experiment.

[^10]:    ${ }^{12}$ The distribution in Figure 6 (left) represents a "perturbation" of the two-type example in Section 4.2 It is a mixture (with equal weights) of two Beta distributions with parameters $(8,30)$ and (60, 30).

[^11]:    ${ }^{13}$ Marketing and risk-mitigation products generate 46\% and 41\% of data brokers' revenues, respectively (Federal Trade Commission, 2014).
    ${ }^{14}$ Many brokers offer both kinds of products. For example, Nielsen Total Audience identifies two ways of packaging its data: a DMP to "expand, optimize, segment and activate your customer data across all marketing channels and platforms" and Data as a Service (DaaS) to "find your target audience segment, or customize your own based on the characteristics that are most important to you." See http://www.nielsen.com/us/en/solutions/capabilities/total-audience.html for more details.
    ${ }^{15}$ https://www.oracle.com/marketingcloud/products/data-management-platform/id-graph.html
    ${ }^{16}$ http://www.towerdata.com/email-intelligence/pricing
    ${ }^{17}$ https://www.equifax.com/business/undisclosed-debt-monitoring

[^12]:    ${ }^{18}$ For instance, a local bank deciding whether to give a mortgage at the prevailing rate has a coarser action space than a major credit card company deciding on a new account's credit limit and interest rate.
    ${ }^{19}$ Any data broker enjoys some degree of market power to the extent that its data sources are not perfectly correlated with those of other brokers or it has a superior ability to process information. Furthermore, it can be shown that the structural properties of optimal menus (Proposition 2) hold even in an imperfectly competitive setting with differentiated products (e.g., competition in nonlinear prices among sellers with partially correlated databases). In that case, non-dispersed and concentrated experiments are the most profitable instruments through which to provide any given utility level to the data buyer.

[^13]:    ${ }^{20}$ A type I error leads the decision maker to reject the null hypothesis even though it is true. By contrast, a type II error leads the decision maker to accept the null hypothesis even though it is false.
    ${ }^{21}$ Because distributions $H_{0}$ and $H_{1}$ satisfy the monotone likelihood ratio property, the feasibility frontier is spanned by threshold tests. As the informativeness of the statistic increases, the feasible set expands and approaches our unrestricted setting.


The Design and Price of Information

[^14]:    ${ }^{22}$ For example, the online invitation website Evite sells lists of attendees of events at specific locations; the share-buttons provider AddThis sells lists of internet users' tastes for news; and Mailways sells lists of physical mailing addresses at the mail carrier route level (Anderson and Simester, 2013).
    ${ }^{23}$ Babaioff, Kleinberg, and Paes Leme (2012) provide further insight into the role of realization-contingent pricing. In our model, contracting on signal realization leads to the first-best profits for the seller.
    ${ }^{24}$ In our setting, contracting on the buyer's action leads to the first-best profits when the payoff of matching state and action is constant, though the same is not true in more general environments. Data providerpublisher partnerships can be analyzed more comprehensively in the setting of Eső and Szentes (2007b).

[^15]:    ${ }^{25}$ For example, the Consumer Sentiment Index (released by Thomson-Reuters and the University of Michigan) screens buyers of time-sensitive data by offering information products that differ only in the timing of their availability.

[^16]:    ${ }^{26}$ Any signal inducing a mixed action could first be split (independently of $\omega$ ) into subsignals that each induce one of the pure actions in the support the mixed action. The resulting experiment still satisfies obedience and truth-telling and induces the same outcome.

[^17]:    ${ }^{27}$ Similarly, one can show that type $\theta^{H}$ must be indifferent after signal $s_{3}$ in experiment (46). The argument here is by contradiction: if he strictly preferred action $a_{3}$, the seller could make the experiment more valuable for $\theta^{L}$ without changing its value for $\theta^{H}$; and if he strictly preferred $a_{1}$, the seller could rearrange all signals and relax the incentive-compatibility constraint.


The Design and Price of Information

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**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin

**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “The Design and Price of Information.” American Economic Review 108, no. 1 (2018): 1–48.

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