The Design and Price of Information

## 6 Optimal Menu with Many Actions

We now extend the analysis of the optimal menu to environments with many actions and many states. In order to make progress in this richer environment, we restrict our attention to the case of binary types and matching payoffs defined in (2). We then provide a suitable generalization of the optimal menu derived in Section 4. We continue to define the low

[^14]and the high type $\theta \in\left\{\theta^{L}, \theta^{H}\right\}$ such that the high type values the completely informative experiment more than the low type, as in (16).

The construction of the optimal menu now proceeds differently than in Section 4. We first solve a relaxed problem wherein we require the high type $\theta^{H}$ to take action $a_{i}$ upon observing signal $s_{i}$ even when buying the experiment destined for the low type $\theta^{L}$. We then show that the solution to the relaxed problem-which disregards the obedience constraints-indeed satisfies the original constraints; hence, it also solves the full problem. In other words, we first guess and then verify that the optimal mechanism is obedient on and off the equilibrium path. In Example 3, we show that this relaxed approach fails to deliver a valid solution with more than two types. Thus, a different approach is required if we want to consider problems with arbitrary cardinality in both the action/state space and the type space.

In the relaxed problem, we replace the high type's incentive-compatibility constraint

$$
V\left(\theta^{H}\right) \geq V\left(E\left(\theta^{L}\right), \theta^{H}\right)-t\left(\theta^{L}\right)=\sum_{j=1}^{I} \max _{i}\left\{\theta_{i}^{H} u_{i} \pi_{i j}\left(\theta^{L}\right)\right\}-\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}-t\left(\theta^{L}\right),
$$

with the weaker constraint

$$
V\left(\theta^{H}\right) \geq \sum_{i=1}^{I} \theta_{i}^{H} u_{i} \pi_{i i}\left(\theta^{L}\right)-\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}-t\left(\theta^{L}\right) .
$$

The relaxed version of the constraint drops the max operator and simply asks type $\theta^{H}$ to accept the recommendation $a_{i}$ implicit in signal $s_{i}$.

Proposition 3 established that both the low type's participation constraint and the high type's incentive constraint bind. This reduces the seller's problem to choosing the diagonal entries of the low-value type's experiment $\pi_{i i}\left(\theta^{L}\right) \in[0,1]$. Substituting $t\left(\theta^{L}\right)=V\left(E\left(\theta^{L}\right), \theta^{L}\right)$ in the right-hand side of (26) above, the seller maximizes

$$
(1-\gamma) \underbrace{\left(\sum_{i=1}^{I} \theta_{i}^{L} u_{i} \pi_{i i}\left(\theta^{L}\right)-\max _{i} \theta_{i}^{L} u_{i}\right)}_{t\left(\theta^{L}\right)=V\left(E\left(\theta^{L}\right), \theta^{L}\right)}+\gamma \underbrace{\left(\sum_{i=1}^{I} \theta_{i}^{H} u_{i}-\max _{i} \theta_{i}^{H} u_{i}-V\left(\theta^{H}\right)\right)}_{t\left(\theta^{H}\right)=V\left(\bar{E}, \theta^{H}\right)-V\left(\theta^{H}\right)}
$$

subject to the high type's participation constraint

$$
V\left(\theta^{H}\right)=\sum_{i=1}^{I}\left(\theta_{i}^{H}-\theta_{i}^{L}\right) u_{i} \pi_{i i}\left(\theta^{L}\right)-\max _{i}\left\{\theta_{i}^{H} u_{i}\right\}+\max _{i}\left\{\theta_{i}^{L} u_{i}\right\} \geq 0 .
$$

The relaxation of the incentive constraints-fixing a mapping from signals to actions for both types and experiments-turns the seller's problem into a linear program.

Intuitively, the seller's choice of a partially informative experiment $E\left(\theta^{L}\right)$ is guided by the disagreement in the beliefs over states of the two types. The seller is most willing to
introduce noise into signals about states that the low type considers relatively less likely than the high type. The resulting distortions in the decisions facilitate screening without sacrificing too much of the surplus. To formalize this intuition, we re-order the states $\omega_{j}$ by the likelihood ratios of the two types' beliefs. In particular, let

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

A basic application of the Lagrange multiplier method yields the following characterization result, which generalizes the optimal experiment in Section 4.2.


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.

## Proposition 8 (Optimal Menu with Two Types)

There exists a critical state $i^{*}<I$ such that an optimal experiment $E\left(\theta^{L}\right)$ has $\pi_{i i}=0$ for all $i<i^{*}$ and $\pi_{i i}=1$ for all $i>i^{*}$.

Because, without loss of generality, the low type chooses action $a_{i}$ when observing signal $s_{i}$, a nil diagonal entry $\pi_{i i}=0$ means signal $s_{i}$ is never sent by experiment $E\left(\theta^{L}\right)$. Unlike in the binary action case, the partially informative experiment $E\left(\theta^{L}\right)$ may thus contain fewer signals than the available actions, as the seller drops some signals to reduce the information rent of the high type. Furthermore, Proposition 8 implies that the optimal experiment $E\left(\theta^{L}\right)$ has a lower triangular shape, with at most one strictly interior diagonal entry $\pi_{i^{*} i^{*}} \in(0,1)$ :

| $E\left(\theta^{L}\right)$ | $s_{1}$ | ⋯ |  | $s_{i^{*}}$ | ⋯ |  | $s_{I}$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 0 | ⋯ | 0 | $\pi_{1 i^{*}}$ | ⋯ |  | $\pi_{1 I}$ |
| ⋮ |  |  |  | ⋮ |  |  | ⋮ |
| $\omega_{i^{*}}$ | ⋮ |  | ⋮ | $\pi_{i^{*} i^{*}}$ | ... |  | $\pi_{i^{*} I}$ |
|  |  |  |  | 0 | 1 |  | 0 |
| ⋮ |  |  |  | ⋮ |  | ⋱ |  |
| $\omega_{I}$ | 0 | ... | 0 | 0 | 0 |  | 1 |

In other words, the seller chooses a subset of "targeted" states $i \in\left\{i^{*}, \ldots, I\right\}$ in which the buyer takes the correct action with positive probability. Conversely, when a "residual" state $i \in\left\{1, \ldots, i^{*}-1\right\}$ is realized, the buyer never takes the correct action. Residual states are used to degrade the information revealed about the targeted states. The seller chooses how to partition states based on two factors: the two types' relative beliefs; and the relative frequency of each type.

In order to construct the optimal experiment $E\left(\theta^{L}\right)$, we substitute the expression for the information rent $V\left(\theta^{H}\right)$ from (28) into the seller's objective (27). Up to an additive constant,
the seller's profits are given by

$$
\sum_{i=1}^{I}\left(\theta_{i}^{L}-\gamma \theta_{i}^{H}\right) u_{i} \pi_{i i}\left(\theta^{L}\right) .
$$

The shape of the optimal experiment then depends on whether the participation constraint of the high type binds. In particular, if (28) is slack, the seller's profits (31) are maximized by assigning only extreme values to the diagonal, $\pi_{i i} \in\{0,1\}$. Because the likelihood ratios $\theta_{i}^{L} / \theta_{i}^{H}$ are increasing, we have $\pi_{i i}=1$ for all $i \geq i_{s}$, where

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

To determine whether the participation constraint of the high type is satisfied by this solution, define the following function

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

This expression corresponds to the rent of the high type if the seller drops all signals $s_{i}$ (and, hence, recommended actions $a_{i}$ ) with $i<j$ and sets $\pi_{i i}=1$ for all $i \geq j$. In particular, $R(1)$ is the difference in the value of full information between the high and the low type. By construction, it is positive. We further note that $R(j)$ is strictly decreasing in $j$ if $\theta_{j}^{H}>\theta_{j}^{L}$ and increasing otherwise. Furthermore, because of the likelihood ratio order on states $i$, it attains its minimum at $i_{c} \triangleq \min \left\{i: 1 \leq \theta_{i}^{L} / \theta_{i}^{H}\right\}$, where $i_{c} \geq i_{s}$. In the proof of Proposition 9, we show that $R\left(i_{c}\right)<0$. Therefore, if $R\left(i_{s}\right) \geq 0$, the constraint (28) is slack at the optimum.

Conversely, if $R\left(i_{s}\right)<0$, then the solution to unconstrained problem (31) violates the high type's participation constraint. Therefore, constraint (28) must bind at the optimum. In particular, the seller chooses a state $i_{b}$ and a diagonal entry $\pi_{i_{b} i_{b}} \in[0,1]$ to satisfy (28) with equality when, in addition, $\pi_{i i}=1$ for all states $i>i_{b}$ and $\pi_{i i}=0$ for all $i<i_{b}$. By the properties of $R(j)$, this critical state $i_{b}$ is given by the unique solution to

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

Intuitively, state $i_{b}$ corresponds to the minimum number of signals (and corresponding action recommendations) that must be eliminated in order to satisfy the high type's incentive constraint while extracting all the rent. Proposition 9 establishes that the participation constraint of the high type binds if and only if $i_{b}<i_{s}$.


The Design and Price of Information

## Proposition 9 (Information Rents)

1. The critical state $\omega_{i^{*}}$ in experiment (30) is given by $i^{*}=\min \left\{i_{s}, i_{b}\right\}$.
2. If $i_{b} \geq i_{s}$, then $E\left(\theta^{L}\right)$ has $\pi_{i^{*} i^{*}}=1$ and $V\left(\theta^{H}\right)>0$.
3. If $i_{b}<i_{s}$, then $E\left(\theta^{L}\right)$ has $\pi_{i^{*} i^{*}}=\pi_{i^{*}}^{B}$ and $V\left(\theta^{H}\right)=0$.

To grasp the intuition, note that the definition of state $i_{b}$ does not depend on the distribution of types, while state $i_{s}$ is increasing in the fraction of high types $\gamma$. The fraction of high types represents the shadow cost of providing information to the low types. When this opportunity cost is low, the monopolist prefers to limit distortions and concede rents to the high type. As $\gamma$ increases, the informativeness of the low type's experiment decreases. For example, if $\gamma<\theta_{1}^{L} / \theta_{1}^{H}$ (i.e., $i_{s}=1$ ), Proposition 9 implies that both types purchase the fully informative experiment $\bar{E}$, and that the high type obtains positive rents. Conversely, if $\gamma>\theta_{i}^{L} / \theta_{i}^{H}$ for all $i$ such that $\theta_{i}^{L} / \theta_{i}^{H}<1$, then $i_{s}>i_{b}$ and the high type obtains no rent.

To complete the description of the optimal experiment $E\left(\theta^{L}\right)$, we need to specify the distribution of signals $\pi_{i j}$ for $i \leq i^{*}$. We construct an off-diagonal assignment procedure that induces both types to follow the recommendation of every signal. In particular, for every state $i \leq i^{*}$, we begin with the last signal $s_{I}$ and assign the off-diagonal probabilities $\pi_{i I}$ such that the high type is indifferent between actions $a_{i}$ and $a_{I}$. We then proceed backward to signal $s_{I-1}$ preserving indifference and placing the residual probability, if any, on $\pi_{i i^{*}}$. We show that the high type prefers action $a_{i^{*}}$ to any action $a_{i}$ with $i<i^{*}$. Therefore, the solution to the relaxed problem satisfies the original constraints. We illustrate the construction of the optimal experiment and the implications for information rents in Example 1 in the Supplemental Appendix.

Earlier, we defined any two types $\theta^{L}$ and $\theta^{H}$ as congruent if they shared the same optimal action $a_{i}$ for some $i$ given their interim beliefs:

$$
\underset{a_{i} \in A}{\arg \max }\left\{\sum_{j=1}^{J} \theta_{j}^{L} u_{i j}\right\}=\underset{a_{i} \in A}{\arg \max }\left\{\sum_{j=1}^{J} \theta_{j}^{H} u_{i j}\right\}=a_{i} .
$$

We now define two types to be strongly congruent if the posterior belief of $\theta^{L}$ can also be represented as a convex combination of $\theta^{H}$ and the vertex of the probability simplex identifying state $\omega_{i}$, i.e., if (33) holds and there exists $\lambda \in[0,1]$ such that

$$
\theta^{L}=\lambda \theta^{H}+(1-\lambda)(0, \ldots, 0,1,0, \ldots, 0) .
$$

In other words, if $\theta^{L}$ and $\theta^{H}$ are strongly congruent, then $\theta^{L}$ lies on a ray that goes through the vertex $(0, \ldots, 0,1,0, \ldots, 0)$ and $\theta^{H}$. An implication of this geometric condition is that the types $\theta^{L}$ and $\theta^{H}$ have a constant likelihood ratio $\theta_{j}^{L} / \theta_{j}^{H}=\lambda$ across all states $\omega_{j} \neq \omega_{i}$.

With more than two states, discriminatory pricing can now be profitable even if the two types are congruent, as long as they are not strongly congruent.


The Design and Price of Information

## Proposition 10 (Partial Information)

1. With strongly congruent priors, type $\theta^{L}$ receives zero or full information.
2. Without strongly congruent priors, type $\theta^{L}$ receives zero, partial or full information.

Proposition 10 thus strengthens and generalizes Proposition 4. With more than two states, the seller can exploit disagreement along any dimension and extract all the surplus through discriminatory pricing. Example 2 in the Supplemental Appendix illustrates how congruent, but not strongly congruent beliefs, allow for surplus extraction.

With more than two types (and more than two actions and states), our relaxed approach is not always valid, i.e., the optimal menu leads different types to choose different actions in response to the same signal realizations. In particular, type $\theta$ need not follow the recommendation of every signal in all experiments $E\left(\theta^{\prime}\right)$, for $\theta \neq \theta^{\prime}$. We provide an instance of this additional issue in Example 3 of the Supplemental Appendix.


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

## 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

## Citation and provenance

**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.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/the-design-and-price-of-information.md

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The Optimality of Upgrade Pricing

> Machine-readable author manuscript.
> Authors: Dirk Bergemann; Alessandro Bonatti; Andreas Haupt; Alex Smolin.
> Canonical citation: Bergemann, Dirk, Alessandro Bonatti, Andreas Haupt, and Alex Smolin. “The Optimality of Upgrade Pricing.” In Web and Internet Economics, 41–58. Springer, 2021.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


The Optimality of Upgrade Pricing

# The Optimality of Upgrade Pricing

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

**Manuscript date:** 2021-12-02

#### Abstract

We consider a multiproduct monopoly pricing model. We provide sufficient conditions under which the optimal mechanism can be implemented via upgrade pricing-a menu of product bundles that are nested in the strong set order. Our approach exploits duality methods to identify conditions on the distribution of consumer types under which (a) each product is purchased by the same set of buyers as under separate monopoly pricing (though the transfers can be different), and (b) these sets are nested. We exhibit two distinct sets of sufficient conditions. The first set of conditions weakens the monotonicity requirement of types and virtual values but maintains a regularity assumption, i.e., that the product-by-product revenue curves are single-peaked. The second set of conditions establishes the optimality of upgrade pricing for type spaces with monotone marginal rates of substitution (MRS)-the relative preference ratios for any two products are monotone across types. The monotone MRS condition allows us to relax the earlier regularity assumption. Under both sets of conditions, we fully characterize the product bundles and prices that form the optimal upgrade pricing menu. Finally, we show that, if the consumer's types are monotone, the seller can equivalently post a vector of single-item prices: upgrade pricing and separate pricing are equivalent.

Keywords: Revenue maximization; mechanism design; strong duality; upgrade pricing.


The Optimality of Upgrade Pricing

## 1 Introduction


The Optimality of Upgrade Pricing

### 1.1 Motivation and Results

Pricing multiple goods with market power is a canonical problem in the theory of mechanism design. It is also a challenge of growing importance and complexity for online retailers and service providers, such as Amazon and Netflix. Both in theory and in practice, designing the optimal mixed bundling mechanism, (i.e., pricing every subset of products) becomes exceedingly complex in the presence of a large number of goods.

A natural question is then whether simpler pricing schemes are optimal under suitable demand conditions. A simple, commonly used mechanism consists of upgrade pricing, whereby the available options are ranked by set inclusion, i.e., some goods are only available as add-ons, Ellison (2005). For example, many online streaming services use a tiered subscription model, whereby users can pay to upgrade to a "premium package"-a subscription with a larger selection of the provider's content relative to the "basic package", Philips (2017).

In this paper, we obtain sufficient conditions under which upgrade pricing maximizes the seller's revenue. Our approach consists of first identifying conditions under which the consumer's types can be ordered in terms of their absolute or relative willingness to pay for the seller's goods, and then ranking the goods themselves by the profitability of selling them to larger sets of consumer types. Our sufficient conditions not only establish the optimality of some upgrade pricing menu: they also show that the optimal bundles are deterministic, and they reveal the order in which they are ranked in the menu. That is, we identify all the nested bundles that appear in the seller's menu, and the profit-maximizing price for each one.

Our results consist of two distinct sets of conditions. The first set of conditions (Theorem 1) illustrates the essence upgrade pricing optimality in what we label as "regular" settings. While these conditions are reminiscent of regularity in one dimension, they are in fact weaker than the monotonicity of the buyer's multidimensional types and of the (item by item) Myersonian virtual values. What we require is for the consumer's types to be ranked in such a way that the virtual values for each item are negative over an initial and positive over a final segment. Furthermore, we require any consumer with a positive virtual value for an item to also have a larger value for that item, relative to any type with a negative virtual value. At the optimal prices, the lowest type buying each good is indifferent between buying it and not buying it. Finally, the sets of types buying each item are nested under the weak monotonicity property, which implies the optimal allocation can be implemented via upgrade pricing.

The second set of conditions (Theorem 2) describes our best attempt at extending our approach to non-regular distribution of types. In order to further weaken the regularity requirement, we restrict attention to type spaces for which the relative preference ratios for any two goods are monotone across types. An example of ordered relative preferences is if higher types have a stronger prefer-
ence for good 2 over good 1 . We refer to such a condition as "monotone marginal rates of substitution" (monotone MRS).

The intuition for our two results can be grasped by considering the demand functions for each good separately. Under monotonicity and monotone MRS, the optimal monopoly prices for each of the goods are ranked. In the special case where the Myersonian virtual values for our ordered types

$$
\phi_{i}^{k}=\theta_{i}^{k}-\frac{1-F_{i}}{f_{i}}\left(\theta_{i+1}^{k}-\theta_{i}^{k}\right)
$$

are also monotone for each item $k$, the first set of conditions applies.
When virtual values are not monotone, however, they can cross zero more than once. In that case, the result still holds, but the proof requires the right ironing procedure. Our ironing procedure relaxes the standard approach of ? and the literature up to Haghpanah and Hartline (2020). Specifically, we do not iron with the goal of monotone virtual values, which corresponds to a concave revenue curve. Rather we iron towards single-crossing virtual values which leads to a quasiconcave revenue curve. We then use the structure implied by monotone MRS to derive a dual certificate of optimality.

Under either set of conditions, each good is purchased by the same set of buyers that would buy it if that were the seller's only product. We further show (Theorem 3) that, if the consumer's types are (not weakly) monotone, the seller can equivalently post the vector of single-item monopoly prices-i.e., bundling is redundant. For example, in the case of two goods sold separately, monotone type spaces mean that no consumer type will buy good 2 without also buying good 1. More generally, the seller benefits from restricting the set of bundles the consumer can purchase through a proper menu of options with the upgrade property. However, examples also show that implementability through separate pricing is neither necessary nor sufficient for the optimality of upgrade pricing.


The Optimality of Upgrade Pricing

### 1.2 Related Literature

First and foremost, our paper contributes to the economics literature on product bundling. The profitability of mixed bundling relative to separate pricing was first examined by Adams and Yellen (1976), and further generalized by McAfee et al. (1989). More recently, a number of contributions have studied the optimal selling mechanisms in the case of two or three goods, and derived conditions for the optimality of pure bundling (see, for example, Manelli and Vincent (2006) and Pavlov (2011)). Daskalakis et al. (2017) use duality methods to characterize the solution of the multiproduct monopolist's problem, and show how the optimal mechanism may involve a continuum of lotteries over items. Bikhchandani and Mishra (2020) derive conditions under which the optimal mechanism is deterministic when the buyer's utility is not necessarily additive. Finally, Ghili (2021) establishes conditions for the optimality of pure bundling when buyers' values are interdependent. Relative to all these papers, we focus on a specific class of simple mechanisms, which includes pure bundling as a special case.

Hart and Nisan (2017) and Babaioff et al. (2014) also study the properties of simpler schemes. The former derives a lower bound on the revenue obtained from separate item pricing. The latter obtains an upper bound on the revenue of the optimal mechanism, relative to the better of pure bundling and separate pricing.

In the context of nonlinear pricing, ? suggested a "demand profile" approach that determines the price of each incremental unit by treating it as separate market. This approach is particularly attractive in settings where there is a natural ordering over the items. This in particular is the case when there is a homogeneous good that is offered in various quantities, such as in energy markets for electricity or water. This approach naturally generates a sequence of upgrade prices. The demand profile approach, and in particular the incremental pricing rule implied by it, does not always yield an optimal mechanism as consumers may wish to obtain earlier units in order to obtain the later units. Thus, a contribution of the current paper is to determine when upgrade pricing is exactly optimal and then to find the upgrade prices as solutions to the global revenue maximization problem rather than the incremental item problem. Other papers make assumptions that make sure that a demand profile-type approach yields an optimal mechanism. In Johnson and Myatt (2003), buyers have unit demand and sellers offer different varieties of a single good. The approach in their paper is to assume a quality ranking on the varieties and to solve for the upgrade prices-the additional payments required to buy a better variety. The survey of the nonlinear pricing literature by Armstrong (2016) covers related approaches that optimize upgrades separately.

Our formulation of the dual problem follows Cai et al. (2016), who present a general duality approach to Bayesian mechanism design. Cai et al. (2016) formulate virtual valuations in terms of dual variables, state the weak and the strong duality results, and use them to establish lower bounds for relative performance of simple mechanisms. An important contribution by Haghpanah and Hartline (2020) exploits the duality machinery to provide sufficient conditions for the exact optimality of a specific, simple mechanism-pure bundling-consisting of offering a maximal bundle at a posted price. Under their sufficient conditions, the dual variables can be recovered from a single-dimensional problem in which the seller is restricted to bundle all items together.

We follow the approach of Haghpanah and Hartline (2020) by leveraging the duality approach to provide sufficient conditions for the optimality of a particular class of mechanisms. Haghpanah and Hartline (2020) gave a characterization of the optimality of the grand bundle, we provide a characterization for upgrade pricing. As upgrade pricing allows multiple bundles to be present in the menu, we cannot assign the dual variables by solving a one-dimensional problem. Instead, we develop a novel ironing algorithm that generates these variables by ironing different item's revenue curves for different types. Under our sufficient conditions, the so-constructed virtual surplus is maximized by an element-wise monotone allocation that can be implemented by upgrade pricing; by complementary slackness, this certifies the optimality of upgrade pricing. Because pure
bundling is one instance of upgrade pricing, our conditions differ from those of Haghpanah and Hartline (2020).

Our ironing differs from existing ironing approaches using duality and tackles a more general problem. In comparison to Haghpanah and Hartline (2020), we prove optimality for mechanisms with menu size surpassing two. Fiat et al. (2016) studies a two-parameter model, and uses an ironing approach that leads from the revenue curves to their concave closure. Devanur et al. (2020) generalizes Fiat et al. (2016) to more general orders on the second parameter. Our approach tackles optimality for an arbitrary finite number of items and varies the ironing procedure. On a technical level, our ironing procedure yields quasi-concave ironed revenue curves, whereas the ironed revenue curves in Haghpanah and Hartline (2020); Fiat et al. (2016); Devanur et al. (2020) are concave.

Our results also feed into a literature specifying optimal finite mechanisms for multi-dimensional types. (Daskalakis et al., 2017, section 7) for example characterizes the optimal mechanisms for the two-good monopolist problem if the optimal mechanism has a particular structure. While Daskalakis et al. (2017) requires that the region of the type space that is not allocated any item is not adjacent to all regions getting specific constant allocations, upgrade pricing mechanisms consistently break this requirement.


The Optimality of Upgrade Pricing

### 1.3 Structure of the Paper

The model is introduced in section 2. The first set of sufficient condition is presented in section 3. In section 4, we present our results for monotone MRS type spaces. In section 5, we discuss the relationship between separate pricing and upgrade pricing. We conclude in section 6.


The Optimality of Upgrade Pricing

## 2 Model

We consider a standard multiple-good monopoly setting. There is a single seller of $d \geq 1$ goods and a single buyer. The seller's marginal costs of production are normalized to zero. The buyer's utility function is additive across goods. We refer to the vector of marginal utilities $\theta_{i} \in \mathbb{R}^{d}$ as the buyer's type. Therefore, the utility of buyer type $\theta_{i}$ from the consumption vector $q \in[0,1]^{d}$ is given by

$$
U\left(\theta_{i}, q\right)=\sum_{k=1}^{d} \theta_{i}^{k} q^{k}
$$

We also adopt the shorthand notation $\left\langle\theta_{i}, q\right\rangle:=\sum_{k=1}^{d} \theta_{i}^{k} q^{k}$. As a convention, we denote types by subscripts and items by superscripts. The buyer's utility is quasi-linear in transfers and her outside option is also normalized to zero.

The buyer knows her type. From the seller's perspective, the buyer's type is distributed over a finite set $\Theta \subseteq \mathbb{R}_{+}^{d}$, with $|\Theta|=n$, according to the distribution $f \in \Delta(\Theta)$. For any positive integer $n$, we adopt the convention that $[n]:=$
$\{1,2, \ldots, n\}$, and we index types by $i \in[n]$. We let $f_{i}:=f\left(\theta_{i}\right)$ and denote the cumulative distribution sequence by $F_{i}=\sum_{j=1}^{i} f_{j}, i \in[n]$.

The seller aims to maximize revenue. By the revelation principle, we can focus on direct mechanisms $(q, t)=\left(q_{i}, t_{i}\right)_{i \in\{0\} \cup[n]}$. These mechanisms can be interpreted as menus with $n+1$ items so that item $i$ delivers consumption vector $q_{i}$ at price $t_{i}$ and item $\left(q_{0}, t_{0}\right):=(0,0)$ captures the buyer's outside option.

We call a menu upgrade pricing if $\left\{q_{0}, q_{1}, \ldots, q_{n}\right\}$ can be ordered in the component-wise partial order on $\mathbb{R}^{d}$ given by $q \leq q^{\prime} \Leftrightarrow \forall k \in[d]: q^{k} \leq\left(q^{\prime}\right)^{k}$. Our main goal is to provide conditions under which upgrade pricing maximizes the seller's revenue among all direct mechanisms.


The Optimality of Upgrade Pricing

## 3 Optimal Mechanisms for Regular Distributions

We will make prominent use of the (partial) Lagrangian duality-based certificate of optimality used by Cai et al. (2016). We state the underlying duality result to fix notation.


The Optimality of Upgrade Pricing

### 3.1 Duality

In what follows, we will associate with $\lambda_{j i}$ the Lagrange multiplier of the incentive compatibility constraint of type $\theta_{j}$ deviating to type $\theta_{i}, j \in[n], i \in\{0\} \cup[n]$ :

$$
\left\langle q_{j}, \theta_{j}\right\rangle-t_{j} \geq\left\langle q_{i}, \theta_{j}\right\rangle-t_{i} .
$$

We note that the incentive constraints corresponding to $\lambda_{j 0}, j \in[n]$ are type $j$ 's individual rationality constraints. As a main tool in our analysis, we define the multi-dimensional virtual values associated with Lagrange multipliers $\lambda \in$ $\mathbb{R}^{n} \times \mathbb{R}^{n+1}$ as

$$
\phi_{i}^{\lambda}:=\theta_{i}-\frac{1}{f_{i}} \sum_{j=1}^{n} \lambda_{j i}\left(\theta_{j}-\theta_{i}\right) .
$$

Lemma 1. A mechanism $\left(q_{i}, t_{i}\right)_{i \in\{0\} \cup[n]}$ maximizes revenue if and only if there exist multipliers $\lambda_{j i}, j \in[n], i \in\{0\} \cup[n]$ such that

1. $\lambda_{j i} \geq 0$ (Non-Negativity)
2. $\left(q_{i}\right)_{i \in[n]}$ optimizes $\max _{\left(q_{i}\right)_{i \in[n]} \in[0,1]^{n}} \sum_{i=1}^{n} f_{i}\left\langle q_{i} \cdot \phi_{i}^{\lambda}\right\rangle$ (Virtual Welfare Maximization)
3. $f_{i}=\sum_{j=0}^{n} \lambda_{i j}-\sum_{j=1}^{n} \lambda_{j i}$ for all $i \in[n]$ (Feasibility of Flow)
4. $\lambda_{j i}\left(\left\langle q_{j}, \theta_{j}\right\rangle-t_{j}-\left\langle q_{i}, \theta_{j}\right\rangle-t_{i}\right)=0$ for all $j \in[n], i \in\{0\} \cup[n]$ (Complementary Slackness)
5. There are transfers $t$ such that $(q, t)$ is incentive compatible and individually rational (Implementability)

We call the dual variables $\lambda_{j i}, j \in[n], i \in[n] \cup\{0\}$ flows from type $j$ to type $i$ whenever they are non-negative and satisfy Lemma 1item 3. This name is inspired by flow conservation constraints from the maximum flow and minimum cost flow problem in discrete mathematics (Korte and Vygen, 2011).

Proof of Lemma 1. Slater's condition for affine inequality constraints (Boyd and Vandenberghe, 2004, p. 227) allows us to write revenue maximization subject to the incentive compatibility and individual rationality constraints as an unconstrained optimization problem for $\left(q_{i}, t_{i}\right)_{i \in[n]}$ subject to complementary slackness and non-negativity of dual variables. The Lagrangian reads:

$$
\begin{aligned}
\mathcal{L} & =\sum_{i=1}^{n} f_{i} t_{i}+\sum_{j=1}^{n} \sum_{i=0}^{n} \lambda_{j i}\left(\left\langle q_{j}, \theta_{j}\right\rangle-t_{j}-\left\langle q_{i}, \theta_{j}\right\rangle-t_{i}\right) \\
& =\sum_{i=1}^{n} t_{i}\left(f_{i}-\sum_{j=0}^{n} \lambda_{i j}+\sum_{j=1}^{n} \lambda_{j i}\right)+\sum_{j=1}^{n} \sum_{i=0}^{n} \lambda_{j i}\left\langle q_{j}, \theta_{j}\right\rangle-\sum_{j=1}^{n} \sum_{i=0}^{n} \lambda_{j i}\left\langle q_{i}, \theta_{j}\right\rangle \\
& =\sum_{j=1}^{n} \sum_{i=0}^{n} \lambda_{j i}\left\langle q_{j}, \theta_{j}\right\rangle-\sum_{j=1}^{n} \sum_{i=0}^{n} \lambda_{j i}\left\langle q_{i}, \theta_{j}\right\rangle \\
& =\sum_{j=1}^{n}\left(\left(\sum_{i=1}^{n} \lambda_{i j}-\sum_{i=0}^{n} \lambda_{j i}\right)\left\langle q_{j}, \theta_{j}\right\rangle-\sum_{i=0}^{n} \lambda_{j i}\left(\left\langle q_{i}, \theta_{j}\right\rangle-\left\langle q_{j}, \theta_{j}\right\rangle\right)\right) \\
& =\sum_{j=1}^{n}\left(f_{j}\left\langle q_{j}, \theta_{j}\right\rangle-\sum_{i=0}^{n} \lambda_{j i}\left(\left\langle q_{i}, \theta_{j}\right\rangle-\left\langle q_{j}, \theta_{j}\right\rangle\right)\right)=\sum_{j=1}^{n} f_{j}\left\langle q_{j}, \phi_{j}\right\rangle .
\end{aligned}
$$

Clearly, it is necessary for an optimal mechanism to be implementable. To conclude the proof, we need to show that virtual welfare maximization and feasibility of flow are equivalent to maximizing the Lagrangian. Assume virtual welfare maximization and feasibility of flow. Then the above equalities show that the Lagrangian is maximized and certify the optimality of the mechanism $\left(q_{i}, t_{i}\right)_{i \in[n]}$. Conversely, assume that the Lagrangian is maximized by $\left(q_{i}, t_{i}\right)_{i \in[n]}$. If the flow $\lambda_{j i}$ were not feasible, then choosing $t_{i}$ arbitrarily large or small would lead to a higher value for the Lagrangian, which yields a contradiction. Given that this is zero, the Lagrangian equals virtual welfare, and virtual welfare maximization follows from optimality. $\square$