$$
\begin{aligned}
& \bar{u}_{c} \triangleq \frac{1-\delta}{1-\delta^{T_{c}}} \sum_{t=1}^{T_{c}} \delta^{t-1} u\left(a_{t}, \omega_{t}\right) \\
& \bar{u}_{a} \triangleq \frac{1-\delta}{1-\delta^{T_{c}+T_{a}}} \sum_{t=1}^{T_{c}+T_{a}} \delta^{t-1} u\left(a_{t}, \omega_{t}\right)
\end{aligned}
$$

so that $\bar{u}_{c}, \bar{u}_{a} \in \operatorname{proj}_{n+1} \mathscr{F}^{*}$ and $U=\mathbb{E}_{p}\left[\bar{u}_{c}\right]$.
We can bound the difference between $\bar{u}_{a}$ and $\bar{u}_{c}$ as follows:

$$
\begin{aligned}
\bar{u}_{a}-\bar{u}_{c} & =\frac{\delta^{T_{c}}\left(1-\delta^{T_{a}}\right)}{1-\delta^{T_{c}+T_{a}}}\left(\frac{1-\delta}{1-\delta^{T_{a}}} \sum_{t=T_{c}+1}^{T_{c}+T_{a}} \delta^{t-T_{c}-1} u\left(a_{t}, \omega_{t}\right)-\bar{u}_{c}\right), \\
\left|\bar{u}_{a}-\bar{u}_{c}\right| & \leq \frac{\delta^{T_{c}}\left(1-\delta^{T_{a}}\right)}{1-\delta^{T_{c}+T_{a}}} \bar{u} \leq \frac{\delta^{T_{c}}\left(1-\delta^{\hat{T}_{a}+1}\right)}{1-\delta^{T_{c}+\hat{T}_{a}+1}} \bar{u}=\frac{\delta\left|\bar{v}_{c}-V\right|+o(1) \eta}{\delta\left|\bar{v}_{c}-V\right|+\eta+o(1) \eta} \bar{u}
\end{aligned}
$$

where $o(1)$ is a function converging to 0 as $\delta \rightarrow 1$, and the second inequality follows by (3) and the fact that $\delta^{T_{c}} \rightarrow 1$ as $\delta \rightarrow 1$.

By construction $\mathbb{E}_{p}\left[\bar{v}_{c}\right]=V^{*}=V$. We now argue that as $\delta$ goes to 1, the difference $\left|\bar{v}_{c}-V\right|$ converges in probability to 0 independently of the starting distribution. Fix a
sender $i \in N$. First, by the ergodic theorem, for any starting distribution $\pi \in \Delta(\Omega)$,

$$
\operatorname{Pr}_{\pi}\left[\lim _{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} v_{i}\left(\mathrm{a}\left(\omega_{t}\right)\right)=\sum_{\omega \in \Omega} p(\omega) v_{i}(\mathrm{a}(\omega))\right]=1,
$$

Second, for any sequence of states $\left\{\omega_{t}\right\}_{t=0}^{T-1}$

$$
\left|\frac{1-\delta}{1-\delta^{T}} \sum_{t=1}^{T} \delta^{t-1} v_{i}\left(\mathrm{a}\left(\omega_{t}\right)\right)-\frac{1}{T} \sum_{t=1}^{T} v_{i}\left(\mathrm{a}\left(\omega_{t}\right)\right)\right|<\max \left\{\left|T \frac{1-\delta}{1-\delta^{T}}-1\right|,\left|T \frac{1-\delta}{1-\delta^{T}} \delta^{T-1}-1\right|\right\} \bar{v}_{i} .
$$

Using the fact $\delta^{T_{c}(\delta)} \rightarrow 1$ and $T_{c}(\delta) \rightarrow \infty$ as $\delta \rightarrow 1$, one can show that the right-hand side of (5) converges to 0 as $\delta \rightarrow 1$. Combining these two observations, by (4) it follows that for any $\varepsilon$, for $\delta$ sufficiently large, $\operatorname{Pr}_{\pi}\left[\left|\bar{u}_{c}-\bar{u}_{a}\right|>\varepsilon\right]<1-\varepsilon$.

Consequently, because the receiver's stage payoffs are bounded, and $T_{c}(\delta) \rightarrow \infty$ as $\delta \rightarrow 1, \mathbb{E}_{\pi}\left[\left|\bar{u}_{a}-\bar{u}_{c}\right|\right] \rightarrow 0$ as $\delta \rightarrow 1$, where $\pi \in \Delta(\Omega)$ is the belief held by the receiver at the beginning of the block. On the other hand, by the ergodic theorem $\mathbb{E}_{\pi}\left[\bar{u}_{c}\right] \rightarrow U$.

We conclude the section by showing the receiver's expected continuation payoff after any public history is close to $U$. Since the equilibrium play is reset in every block, it suffices to show that (i) the contribution of a single block to the total payoff is negligible and (ii) the impact of current beliefs on payoffs in future blocks is negligible.

For (i), observe that by (3)

$$
\left(1-\delta^{T_{a}-1}\right) \eta \leq \frac{1-\delta^{T_{c}}}{\delta^{T_{c}}}\left|\bar{v}_{c}-V\right| \leq \frac{1-\delta^{T_{c}}}{\delta^{T_{c}}} N \max _{i \in N} \bar{v}_{i} .
$$

Consequently, $\delta^{T_{a}-1} \rightarrow 1$ as $\delta \rightarrow 1$ and $\delta^{T_{c}} \rightarrow 1$, so the contribution of a single block, weighted by $1-\delta^{T_{c}+T_{a}+1}$, goes to 0 as $\delta \rightarrow 1$.

For (ii), observe that from before, as $\delta \rightarrow 1, \mathbb{E}_{\pi}\left[\bar{u}_{a}\right]$ converges in expectation to $U$ for any starting belief $\pi$.

Incentives We now check players' incentives to follow the suggested strategies $\left(\pi^{*}, \alpha^{*}\right)$; that is, that they are sequentially rational. Consider senders' incentives. Because babbling is an equilibrium of a stage game, the off-path behavior $\left(\pi_{\varnothing}, \alpha_{\varnothing}\right)$ is sequentially rational. Also, since the senders' expected continuation payoffs are independent of their messages, they have no incentives to deviate. Thus, $\pi_{1}$ is sequentially rational.

We consider now the receiver's incentives. Let $a: \Delta(\Omega) \rightarrow A$ be the action that maximizes the receiver's expected payoff in the stage game for any given distribution over $\Omega$, i.e.,

$$
a(\pi) \in \arg \max _{a \in A} \sum_{\omega \in \Omega} \pi(\omega) u(a, \omega) .
$$

Define the map $\phi: \Delta(\Omega) \rightarrow \Delta(\Omega)$ as $\phi(\pi)(\omega):=\sum_{\omega^{\prime} \in \Omega} \pi\left(\omega^{\prime}\right) k\left(\omega \mid \omega^{\prime}\right)$. The map describes how the belief of the receiver evolves in the absence of information.

The receiver's continuation payoff following a deviation at time $t$ equals

$$
\mathbb{E}_{\pi_{t}}\left[(1-\delta) \sum_{\tau=1}^{\infty} \delta^{\tau} u\left(a\left(\phi^{(\tau)}\left(\pi_{t}\right)\right), \omega_{t+\tau}\right)\right]
$$

where $\pi_{t}$ is the receiver's belief at the end of period $t$.
Since the Markov chain is aperiodic, for any $\pi_{t}, \lim _{\tau \rightarrow \infty} \phi^{(\tau)}\left(\pi_{t}\right)=p$. Hence, for any $\varepsilon$ and any $\pi_{t} \in \Delta(\Omega)$

$$
\left|\mathbb{E}_{\pi_{t}}\left[(1-\delta) \sum_{\tau=1}^{\infty} \delta^{\tau} u\left(a\left(\phi^{(\tau)}\left(\pi_{t}\right)\right), \omega_{t+\tau}\right)\right]-\underline{U}\right|<\varepsilon
$$

provided $\delta$ is high enough.
In words, the receiver can benefit from the current information only in the short run. In particular, the receiver's continuation payoff following a deviation can be taken to be arbitrarily close to $\underline{U}$ as $\delta \rightarrow 1$.


Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.

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

As we showed before, the equilibrium expected continuation payoff after any history is in the neighborhood of $U$. Since $U \in \mathscr{F}^{* *}, U>\underline{U}$. Hence, the receiver is willing to obey if sufficiently patient facing the threat of switching to babbling. It follows that $\alpha_{1}$ is sequentially rational.

To complete the proof, we specify the system of beliefs. Bayes rule uniquely pins down players' belief, unless a sequence of inconsistent messages are reported. Recall that when the current messages $m_{t}$ are inconsistent with the messages from the previous period $m_{t-1}$, the receiver takes action a $(\hat{\omega})$, where $\hat{\omega}$ is an artificially generated state. At these histories, the receiver updates his belief according to Bayes rule as if the reported state was $\hat{\omega}$. The belief of the deviator is computed by Bayes rule. The belief of any other sender $i$ equals the posterior belief based only on her private information $\omega_{i}^{t}$.


## References

Abreu, Dilip, David Pearce, and Ennio Stacchetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring," Econometrica, 58, 1041-1063.

Chakraborty, Archishman and Rick Harbaugh (2010) "Persuasion by Cheap Talk," American Economic Review, 100, 2361-2382.

Crawford, Vincent P. and Joel Sobel (1982) "Strategic Information Transmission," Econometrica, 50, 1431-1451.

Ely, Jeffrey C., Johannes Hörner, and Wojciech Olszewski (2005) "Belief-Free Equilibria in Repeated Games," Econometrica, 73, 377-415.

Escobar, Juan F. and Juuso Toikka (2013) "Efficiency in games with Markovian private information," Econometrica, 81, 1887-1934.

Frankel, Alexander (2016) "Discounted quotas," Journal of Economic Theory, 166, 396-444.
Fudenberg, Drew, David Levine, and Eric Maskin (1994) "The Folk Theorem with Imperfect Public Information," Econometrica, 62, 997-1039.

Fudenberg, Drew and Jean Tirole (1991) Game Theory, Cambridge, MA: MIT Press.
Jackson, Matthew O. and Hugo F. Sonnenschein (2007) "Overcoming Incentive Constraints by Linking Decisions," Econometrica, 75, 241-257.

Mailath, George J. and Larry Samuelson (2006) Repeated Games and Reputations: Long-Run Relationships, New York: Oxford University Press.

Myerson, Roger B. (1986) "Multistage Games with Communication," Econometrica, 54, 323-358.

Renault, Jérôme, Eilon Solan, and Nicolas Vieille (2013) "Dynamic sender-receiver games," Journal of Economic Theory, 148, 502-534.

[^0]:    *We are grateful to Larry Samuelson for support and encouragement throughout this project. We also thank Dirk Bergemann, Johannes Hörner and seminar participants at Yale and the 2015 World Congress of the Econometric Society for helpful discussions and suggestions. Finally, we wish to thank the editor and the anonymous referees for many valuable comments.
    ${ }^{\dagger}$ Department of Economics, Boston University, margaria@bu.edu
    ${ }^{\ddagger}$ Institute for Microeconomics, University of Bonn, alexey.v.smolin@gmail.com

[^1]:    ${ }^{1}$ Chakraborty and Harbaugh (2010) provide many more examples of such strong biases.
    ${ }^{2}$ There is no standard formulation of a "folk theorem" in repeated games with incomplete information.

[^2]:    ${ }^{3}$ In the equilibria we construct, after every history each sender is indifferent between sending any message, independently of the private histories observed by the other senders. Nevertheless, these equilibria are not "belief-free" in the sense of Ely et al. (2005) as the same property does not hold for the receiver.
    ${ }^{4}$ Escobar and Toikka (2013) also studied limit payoffs with dynamic communication but their setting and techniques are significantly different from ours. They focus on the case of independent private values, so the players know their own payoffs. Their proof relies on statistical tests similar to those used by Renault et al. (2013) which are of limited use in our setting.
    ${ }^{5}$ The single action of the receiver affects all senders' payoffs at once. Hence, the problem with many senders is not simply a collection of separate single-sender problems.
    ${ }^{6}$ We can dispense with the assumption that the messages are public if we allow the receiver to announce the messages at the end of each period.

[^3]:    ${ }^{7}$ This implies that the receiver does not observe his payoffs. This assumption is standard in repeated games with incomplete information and can be justified in at least two ways. First, it can approximate the situation when payoffs are observed far into the future. Second, the receiver may want to make informed decisions even if she cannot ever confirm their accuracy just like a judge wants to acquit innocents.
    ${ }^{8}$ This assumption is still a priori restrictive because the players cannot commit to their strategies and the revelation principle does not apply. However, it suffices to obtain a folk theorem.
    ${ }^{9}$ We adopt a convention that for any stochastic process $x$ its time- $t$ realization is denoted by subscript $x_{t}$ and the history up to time $t$ is denoted by superscript $x^{t} \triangleq\left\{x_{s}\right\}_{s=1}^{t}$.

[^4]:    ${ }^{10}$ We imposed the standard restriction (B) in Fudenberg and Tirole (1991); because types are correlated, condition (B) needs to be adjusted (see Fudenberg and Tirole, 1991, pp. 349-350).
    ${ }^{11}$ The relative interior of a set $\mathscr{F}^{*}$, relint $\left(\mathscr{F}^{*}\right)$, is its interior under the topology induced on the affine hull of $\mathscr{F}^{*}$. Hence the statement is meaningful even if $\mathscr{F}^{*}$ has a dimension lower than the number of players.

[^5]:    ${ }^{12}$ For example, $T_{c}(\delta)=\left\lceil(1-\delta)^{-1 / 2}\right\rceil$. In what follows we will often omit dependence of $T_{c}$ on $\delta$.


## Citation and provenance

**Authors:** Chiara Margaria; Alex Smolin

**Canonical citation:** Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/dynamic-communication-with-biased-senders.md

**Source record:** https://alexsmolin.com/files/dynamic-communication-with-biased-senders-working-paper.pdf

**Published record:** https://doi.org/10.1016/j.geb.2017.10.017

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

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


The Design and Price of Information

> Machine-readable author manuscript.
> 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.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


The Design and Price of Information

# The Design and Price of Information

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

**Manuscript date:** 2017-06-03

#### Abstract

A data buyer faces a decision problem under uncertainty. He can augment his initial private information with supplemental data from a data seller. His willingness to pay for supplemental data is determined by the quality of his initial private information. The data seller optimally offers a menu of statistical experiments. We establish the properties that any revenue-maximizing menu of experiments must satisfy. Every experiment is a non-dispersed stochastic matrix, and every menu contains a fully informative experiment. In the cases of binary states and actions, or binary types, we provide an explicit construction of the optimal menu of experiments.

Keywords: information design, price of information, statistical experiments, mechanism design, price discrimination, hypothesis testing.

JEL Codes: D42, D82, D83.

[^0]


The Design and Price of Information

## 1 Introduction


The Design and Price of Information

### 1.1 Motivation

The mechanisms by which information is traded can shape the creation and the distribution of surplus in many important markets. Information about individual borrowers guides banks' lending decisions, information about consumers' characteristics facilitates targeted online advertising, and information about a patient's genome enhances health care delivery. In all these settings, information buyers (i.e., lenders, advertisers, and health care providers) have private knowledge relevant to their decision problem at the time of contracting (e.g., independent knowledge of a borrower, prior interactions with specific consumers, access to a patient's family history). Thus, potential data buyers seek to acquire supplemental information to improve the quality of their decision making.

In this paper, we develop a framework to analyze the sale of supplemental information. We consider a data buyer who faces a decision problem under uncertainty. A monopolist data seller owns a database containing information about a "state" variable that is relevant to the buyer's decision. Initially, the data buyer has only partial information about the state. This information is private to the data buyer and unknown to the data seller. The precision of the buyer's private information determines his willingness to pay for any supplemental information. Thus, from the perspective of the data seller, there are many possible types of the data buyer. We investigate the revenue-maximizing information policy, i.e., how much information the data seller should provide and how she should price access to the data.

In order to screen the heterogeneous data buyer types, the seller offers a menu of information products. In our context, these products are statistical experiments-signals that reveal information about the payoff-relevant state. Only the information product itself is assumed to be contractible. By contrast, payments cannot be made contingent on either the buyer's action or the realized state and signal. Consequently, the value of an experiment to a buyer is determined by the buyer's private belief and can be computed independently of the price of the experiment. We can recast the resulting screening problem as a nonlinear pricing framework wherein the buyer's type is given by his prior belief. In other words, the seller's problem is to design and price different versions of experiments, that is, different information products from the same underlying database. Because the design of information can be rephrased in terms of hypothesis testing, the present analysis can also be interpreted as a pricing model for statistical tests.

A large body of literature studies the problem of versioning information goods, emphasizing that digital production allows sellers to easily customize (or degrade) the attributes of such products (Shapiro and Varian, 1999). This argument applies even more forcefully
to information products (i.e., statistical experiments). In a nutshell, the data seller's problem consists of degrading the quality of the information sold to some buyers in order to charge higher prices to higher-value buyers. We show that the very nature of information products enriches the scope of price discrimination. Because information is valuable to the extent to which it affects decision making, buyers with different beliefs do not simply value experiments differently: they may even disagree on their ranking. In this sense, the value of information naturally has both a vertical element (the quality of the information), and a horizontal element (the position of the information).

We show that the optimal menu contains, in general, both the fully informative experiment and partially informative, "distorted" experiments. The distorted information products are not simply noisy versions of the same data. Instead, optimality imposes considerable structure on the distortions in the information provided. In particular, every experiment offered as part of the optimal menu is non-dispersed, i.e., it contains a signal realization that rules out one of the states. Moreover, if the buyer's decision problem is to match his action with a state, every experiment is concentrated, i.e., it induces the buyer to take the correct action with probability one, conditional on at least one realized state.

We provide a full characterization of the optimal menu in the case of binary states and actions. This setting yields sharp insights into the profitability of discriminatory pricing for selling information. In the binary-state environment, the buyer's types are one dimensional and the utilities are piecewise linear with a kink at the belief at which the buyer would switch his optimal action. If all buyer types are congruent, i.e., they take identical actions without additional information, an intuition analogous to the "no-haggling" result for monopoly pricing (Myerson, 1981; Riley and Zeckhauser, 1983) applies: the seller simply offers the fully informative experiment at a fixed price. In general, however, the seller's problem consists of screening types both within and across classes of congruent types.

The beneficial use of partial information can be seen with two types that are ranked according to their valuation of the fully informative experiment. The "high" type is ex ante less informed, while the "low" type is ex ante more informed. Suppose that types would pursue distinct actions in the absence of additional information. A feasible policy for the seller is to offer the high type the fully informative experiment and the low type a partially informative experiment that generates one of two signals: with small but positive probability, the signal informs the low type without noise about the state that he considers less likely ex ante; with the remaining probability, it sends a second noisy signal. This allows the low type to improve the quality of his decision making; thus, he would be willing to pay a positive amount for the experiment. By contrast, the high type would not attach a positive value to this partial information. After all, he would have chosen the action suggested by the
noiseless signal under his prior anyway, and given his prior, the noisy signal is too weak to modify his action.


The Design and Price of Information

The optimal menu for two types exploits the horizontal element of information to extract value from the low type without conceding any rents to the high type. Such profitable screening by providing partial information is the novel element that distinguishes the pricing of information from other monopoly problems, such as designing insurance or goods of differentiated quality. With a continuum of types, the optimal menu still contains at most two experiments: one is fully informative, and the other contains two signals, one of which perfectly reveals the true state. In particular, the optimal menu involves discriminatory pricing (i.e., two different experiments are offered) only if "ironing" is required (Myerson, 1981). Intuitively, the second experiment intends to serve buyers in one group, while charging higher prices to the other group.

Our findings have concrete implications for the sale of information. In Section 5, we illustrate our results in the context of the information being sold by online data brokers, focusing on a broad class of products ("data appends") that are used for marketing and risk-mitigation purposes. We use the language of hypothesis testing and statistical errors to demonstrate how the design of data products can be informed by the structural properties of the optimal experiments we identified in our analysis. In particular, we argue that no experiment in an optimal menu should add unbiased noise to the seller's information, and we discuss whether enabling the buyer to access only a portion of the seller's data is equivalent to introducing noise.

In Section 6, we take the first step toward more general results. We fully characterize the menu with two types that face a matching decision problem. In an optimal menu, the high type purchases the fully informative experiment, while the low type purchases an experiment that is at least partially informative. This can occur even if the types are congruent. The latter experiment provides "directional" information about the states the low type perceives to be relatively more likely and induces him to take the corresponding actions more often. In this way, the seller optimally reduces the information rents of the less informed type, possibly to zero, without needing to exclude the more informed type.


The Design and Price of Information

### 1.2 Related Literature

Our paper is part of the body of literature on selling information to imperfectly informed decision makers. In seminal papers, Admati and Pfleiderer $(1986,1990)$ analyze the sale of information to a continuum of ex ante homogeneous agents, all with the same prior information. After the purchase of supplemental information, the agents trade an asset
with a common value. They show that it is optimal to provide noisy, idiosyncratic and, hence, heterogeneous information. This idiosyncratic information guarantees the traders a local monopoly, which preserves the value of acquiring information even in an informative, rational-expectations equilibrium. Thus, Admati and Pfleiderer $(1986,1990)$ explicitly consider interactions among data buyers that we do not pursue here. By contrast, we focus on ex ante heterogeneous types of a single buyer who value information differently due to their different prior beliefs. The data seller in our setting offers noisy versions of the data to screen the buyer's initial information and to extract more surplus, leading to profound differences in the optimal experiments. A second contribution of this paper relative to Admati and Pfleiderer $(1986,1990)$ is that we consider all feasible statistical experiments, whereas they restrict their attention to normally distributed priors and signals. We shall see that the optimal experiment is outside of the normal class, even if the priors are normally distributed.

In recent work, Babaioff, Kleinberg, and Paes Leme (2012) also analyze the optimal mechanisms for selling information. While we consider the same general question, the details of the model, the contracting environment, and the nature of the results differ substantially. In their model, the ex post payoff function of the data buyer depends on two state variables. The seller has private information about one state variable, and the buyer has private information about the other. Their contracting environment differs from ours in that the seller is allowed to make the information disclosure and the price dependent on his privately observed signal. By contrast, we ask the data seller to commit to a selling mechanism before the realization of any state variable. The central results of their paper are statements of the revelation principle and algorithms for the optimal mechanism using surplus extraction arguments, as in Cremer and McLean (1988).

Within the mechanism design literature, our approach is related to, yet conceptually distinct from, models of discriminatory information disclosure in which the seller of a good discloses match-value information and sets a price. Several papers, including Lizzeri (1999), Ottaviani and Prat (2001), Johnson and Myatt (2006), Bergemann and Pesendorfer (2007), Eső and Szentes (2007a), Krähmer and Strausz (2015), and Li and Shi (2015), analyze this problem from an ex ante perspective, where the seller commits (simultaneously or sequentially) to a disclosure rule and a pricing policy. ${ }^{1}$ Eső and Szentes (2007b) consider a related model of selling advice. Their model is distinct from our analysis in two dimensions. First, the private information of the agent is the expected value difference between two possible actions. Thus, the private information is one dimensional rather than multidimensional.

[^1]Second, the seller can make the payment contingent on both the statistical experiment and the buyer's action. By contrast, our seller can price the information but not the action itself. Commitment to a disclosure policy is present in the literature on Bayesian persuasion, e.g., Rayo and Segal (2010), Kamenica and Gentzkow (2011), and Kolotilin, Li, Mylovanov, and Zapechelnyuk (2015). In contrast to this line of work, our model admits monetary transfers and rules out any direct effect of the buyer's ex post action on the seller's utility.

Our previous work (Bergemann and Bonatti, 2015) considered the information-acquisition policy of a data buyer who then decided on the placement of display advertising. This earlier model was simpler in many respects. First, the price of information was given or determined by a competitive market. Second, the data buyer did not have any private information. Third, despite allowing for a continuum of matching values (states) and advertising levels (actions), the available information structures were restricted to simple queries that perfectly revealed individual state realizations. The analysis focused on the nature of the buyer's optimal queries given the distribution of match values and the cost of advertising.

Hörner and Skrzypacz (2016) share a similar title but consider a very different setting. They consider a dynamic hold-up game, except that information rather than a physical object is sold. At the beginning of the game, the buyer has no private information and wants to hire a competent data seller. The data seller knows whether she is competent and can prove her competence by sequentially undertaking tests within a fixed subclass of statistical experiments. Hörner and Skrzypacz (2016) allow for sequential monetary transfers and characterize an equilibrium that is most favorable to the competent seller.

Finally, our seller's problem bears some resemblance to a bundling problem. With more than two states, the buyer types are multidimensional and it is well-known-see, for example, Pavlov (2011b)-that the single-price result of Myerson (1981) and Riley and Zeckhauser (1983) does not hold. Indeed, the optimal menu involves stochastic bundling quite generally, and the structure of the bundles offered can be quite rich. ${ }^{2}$ Stochastic bundles are analogous to the partially informative experiments in our model. To further distinguish from these classic multidimensional problems, stochastic bundling can arise in our setting even when buyer types are one dimensional.

[^2]


The Design and Price of Information

## 2 Model

A single decision maker, the data buyer, faces a decision problem under uncertainty. The state of nature $\omega$ is drawn from a finite set $\Omega=\left\{\omega_{1}, \ldots, \omega_{i}, \ldots, \omega_{I}\right\}$. The data buyer chooses an action $a$ from a finite set $A=\left\{a_{1}, \ldots, a_{j}, \ldots, a_{J}\right\}$. The ex post utility is denoted by

$$
u\left(\omega_{i}, a_{j}\right) \triangleq u_{i j} \in \mathbb{R}_{+} .
$$

The ex post payoffs can thus be represented by an $I \times J$ matrix:

| $u$ | $a_{1}$ | ⋯ | $a_{J}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | $u_{11}$ | ⋯ | $u_{1 J}$ |
| ⋮ | ⋮ |  | ⋮ |
| $\omega_{I}$ | $u_{I 1}$ | ... | $u_{I J}$ |

We impose the following weak assumptions on the matrix: (i) $I \leq J$, and (ii) $u_{i i}>u_{i j}$ for all $j \neq i$. These two assumptions capture the idea that the action space is at least as rich as the state space and that for every state $\omega_{i}$, there is a unique action (labeled $a_{i}$ ) that maximizes the decision maker's utility in that state.

Matching Utility A useful special case is one in which the data buyer faces binary ex post payoffs in each state, i.e., he seeks to match the state and the action. In that case, we frequently drop the second subscript for the ex post utility on the diagonal. The utility function $u\left(\omega_{i}, a_{j}\right)$ is then given by

$$
u\left(\omega_{i}, a_{j}\right) \triangleq \mathbb{I}_{[i=j]} \cdot u_{i}, \text { and } u_{i} \triangleq u_{i i} ;
$$

or in matrix form,

| $u$ | $a_{1}$ | ... | $a_{I}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | $u_{1}$ |  | 0 |
| ⋮ |  | ⋱ |  |
| $\omega_{I}$ | 0 |  | $u_{I}$ |

This formulation assumes that, in each state, the data buyer assigns the same value (normalized to zero) to each wrong action. This assumption has no bite across states because adding a state-dependent translation to the utility function does not affect preferences over actions. Under this assumption, it is without loss of generality to assume that the sets of actions and states have the same cardinality: $|A|=|\Omega|=I=J$.

Prior Information The interim belief $\theta$ about the state is the type of the data buyer

$$
\theta \in \Theta \triangleq \Delta \Omega,
$$

where $\theta_{i}$ denotes the interim probability that type $\theta$ assigns to state $\omega_{i}$, with $i=1, \ldots, I$. The interim beliefs of the data buyer are his private information. From the perspective of the data seller, these beliefs are distributed according to a distribution

$$
F \in \Delta \Theta,
$$

which we take as a primitive of our model. ${ }^{3}$
In line with the interpretation of selling supplemental information, however, we note that the beliefs $\theta \in \Theta$ can be generated from a common prior and privately observed signals. Thus, suppose there is a common prior $\mu \in \Delta \Omega$. The decision maker privately observes a signal $r \in R$ according to a commonly known experiment

$$
\lambda: \Omega \rightarrow \Delta R .
$$

The decision maker then forms his interim belief via Bayes' rule

$$
\theta(\omega \mid r) \triangleq \frac{\lambda(r \mid \omega) \mu(\omega)}{\sum_{\omega^{\prime} \in \Omega} \lambda\left(r \mid \omega^{\prime}\right) \mu\left(\omega^{\prime}\right)} .
$$

The interim beliefs $\theta(\omega \mid r)$, simply denoted by $\theta$, are thus the private information of the data buyer. From the data seller's perspective, the common prior $\mu \in \Delta \Omega$ and the distribution of signals $\lambda: \Omega \rightarrow \Delta R$ induce a distribution $F \in \Delta \Theta$ of interim beliefs.

Supplemental Information The data buyer seeks to augment his initial private information by obtaining additional information from the data seller in order to improve the quality of his decision making. A statistical experiment (equivalently, an information structure) $E=(S, \pi)$ consists of a set $S$ of signals $s$ and a likelihood function:

$$
\pi: \Omega \rightarrow \Delta S .
$$

We assume throughout that the realization of the buyer's private signal $r \in R$ and that of the signal $s \in S$ from any experiment $E$ are independent, conditional on the state $\omega$. In other words, the buyer and the seller draw their information from independent sources.

For a given experiment $E=(S, \pi)$, let $S$ denote the (finite) set of signals that are in the

[^3]support of the experiment and $\pi_{i k}$ the conditional probability of signal $s_{k} \in S$ in state $\omega_{i}$. Letting $K \triangleq|S|$, we have
$$
\pi_{i k} \triangleq \operatorname{Pr}\left(s_{k} \mid \omega_{i}\right),
$$
where $\pi_{i k} \geq 0$ and $\sum_{k=1}^{K} \pi_{i k}=1$ for all $i$. We then obtain the stochastic matrix

| $E$ | $s_{1}$ | ⋯ | $s_{k}$ | ⋯ | $s_{K}$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\omega_{1}$ | $\pi_{11}$ | ⋯ | $\pi_{1 k}$ | ⋯ | $\pi_{1 K}$ |
| ⋮ | ⋮ |  |  |  | ⋮ |
| $\omega_{i}$ | $\pi_{i 1}$ |  | $\pi_{i k}$ |  | $\pi_{i K}$ |
| ⋮ | ⋮ |  |  |  | ⋮ |
| $\omega_{I}$ | $\pi_{I 1}$ | ... | $\pi_{I k}$ | ⋯ | $\pi_{I K}$ |

The following experiments are of particular interest:

1. a non-dispersed experiment that contains at least one nil entry $\pi_{i k}=0$ for some $i, k$;
2. a concentrated experiment that contains a standard basis vector $\pi_{i i}=1$ for some $i$;
3. the fully informative experiment $\bar{E}$, with $\pi_{i i}=1$ for all $i$.

In non-dispersed experiment, one signal $s_{k}$ allows the decision maker to rule out some state $\omega_{i}$. In a concentrated experiment, there exists a state $\omega_{i}$ that is ruled out by all signals $s_{k} \neq s_{i}$. The fully informative experiment perfectly reveals the true state.

A menu of experiments $\mathcal{M}=\{\mathcal{E}, t\}$ (or an information policy) consists of a collection $\mathcal{E}$ of experiments $E$ and an associated tariff function

$$
t: \mathcal{E} \rightarrow \mathbb{R}_{+} .
$$

Our goal is to characterize the revenue-maximizing menu for the seller. The timing of the game is as follows:

1. the seller posts a menu $\mathcal{M}$;
2. the true state $\omega$ and the buyer's type $\theta$ are realized;
3. the buyer chooses an experiment $E \in \mathcal{E}$ and pays the corresponding price $t(E)$;
4. the buyer observes a signal $s$ from experiment $E$ and chooses an action $a$.


The Design and Price of Information

We emphasize that the data seller is unrestricted in her choice of statistical experiment (i.e., the seller can improve upon the buyer's original information with arbitrarily accurate
signals), and that the marginal cost of providing the information is nil. These assumptions capture settings in which sellers hold very precise data and distribution costs are negligible.

We deliberately focus on the pure problem of the design and pricing of statistical experiments. We thus assume that the seller commits to a menu of experiments before the realization of the state $\omega$ and the type $\theta$. We further assume that none of the following are contractible: the buyer's action $a$, the realized state $\omega$, or the signal $s$. Thus, scoring rules and other belief-elicitation schemes that compare the elicited beliefs with the realization of some random variable are not available to the seller. Finally, we consider only static mechanisms and do not investigate sequential protocols. We expect that the sequential sale of experiments would allow the data seller to extract additional surplus relative to static mechanisms, as this would allow the seller to correlate individual payments with the realized states. However, the nature of the incentive constraints would not be affected (at least in the last round of communication), and we expect the qualitative results to remain unchanged.


The Design and Price of Information

## 3 Information Design


The Design and Price of Information

### 3.1 Value of Information

We first describe the value of the buyer's initial information and then determine the incremental value of an experiment $E=(S, \pi)$. The value of the data buyer's problem under prior information only is given by choosing the action $a_{j}$ that maximizes the expected utility given the interim belief $\theta$ :

$$
a(\theta) \triangleq \underset{a_{j} \in A}{\arg \max }\left\{\sum_{i=1}^{I} \theta_{i} u_{i j}\right\} .
$$

The expected utility of type $\theta$ is therefore given by

$$
u(\theta) \triangleq \max _{a_{j} \in A}\left\{\sum_{i=1}^{I} \theta_{i} u_{i j}\right\} .
$$

By contrast, if the data buyer has access to an experiment $E=(S, \pi)$, he first observes the signal realization $s_{k} \in S$, updates his beliefs and then chooses an appropriate action. The marginal distribution of signals $s_{k}$ from the perspective of type $\theta$ is given by

$$
\operatorname{Pr}\left[s_{k} \mid \theta\right]=\sum_{i=1}^{I} \theta_{i} \pi_{i k} .
$$

Consequently, for any signal $s_{k}$ that occurs with strictly positive probability $\operatorname{Pr}\left[s_{k} \mid \theta\right]$, the
action that maximizes the expected utility of type $\theta$ is given by

$$
a\left(s_{k} \mid \theta\right) \triangleq \underset{a_{j} \in A}{\arg \max }\left\{\sum_{i=1}^{I}\left(\frac{\theta_{i} \pi_{i k}}{\sum_{i^{\prime}=1}^{I} \theta_{i^{\prime}} \pi_{i^{\prime} k}}\right) u_{i j}\right\},
$$

which leads to the following conditional expected utility:

$$
u\left(s_{k} \mid \theta\right) \triangleq \max _{a_{j} \in A}\left\{\sum_{i=1}^{I}\left(\frac{\theta_{i} \pi_{i k}}{\sum_{i^{\prime}=1}^{I} \theta_{i^{\prime}} \pi_{i^{\prime} k}}\right) u_{i j}\right\} .
$$

Integrating over all signal realizations $s_{k}$ and subtracting the value of prior information, the (net) value of an experiment $E$ for type $\theta$ is given by

$$
V(E, \theta) \triangleq \mathbb{E}[u(s \mid \theta)]-u(\theta)=\sum_{k=1}^{K} \max _{j}\left\{\sum_{i=1}^{I} \theta_{i} \pi_{i k} u_{i j}\right\}-u(\theta) .
$$

In the case of the previously defined matching utility (2), the value of information takes the simpler form

$$
V(E, \theta)=\sum_{k=1}^{K} \max _{i}\left\{\theta_{i} \pi_{i k} u_{i}\right\}-\max _{i}\left\{\theta_{i} u_{i}\right\} .
$$

The value of the prior information, given by $\max _{i} \theta_{i} u_{i}$, is generated by the action that has the highest value-weighted probability of matching the state. The value of an experiment $E$ is generated by choosing an action on the basis of the posterior belief $\theta_{i} \pi_{i k}$ induced by each signal $s_{k}$. Under matching utility, the value of information is given by the (value-weighted) incremental probability of choosing the correct action. ${ }^{4}$

In Figure 1, we illustrate the value of an experiment in a model with three actions and three states $\omega_{i} \in\left\{\omega_{1}, \omega_{2}, \omega_{3}\right\}$. The prior belief of each agent is therefore an element of the two-dimensional simplex, $\theta=\left(\theta_{1}, \theta_{2}, 1-\theta_{1}-\theta_{2}\right)$. The utility function is given by state-action matching with uniform weights, i.e., $u_{i} \triangleq 1$. We display the value of the fully informative experiment $\bar{E}$ and of a partially informative experiment $E$ as a function of the buyer's prior. ${ }^{5}$

[^4]| $E$ | $s_{1}$ | $s_{2}$ | $s_{3}$ |
| :--- | :--- | :--- | :--- |
| $\omega_{1}$ | 1/2 | 1/4 | 1/4 |
| $\omega_{2}$ | 0 | 3/4 | 1/4 |
| $\omega_{3}$ | 0 | 1/4 | 3/4 |

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: Value of Full and Partial Information, $I=J=3$

Viewed as a function of the types, the value of an experiment $V(E, \theta)$ is piecewise linear in $\theta$ with a finite number of linear components. The linearity of the value function is a consequence of the Bayesian nature of our problem, where types are prior probabilities of states. The downward kinks are due to the max operator in the buyer's reservation utility $u(\theta)$. They correspond to changes in the buyer's action without supplemental information. The upward kinks are generated by the max operator in (5) and reflect changes in the buyer's preferred action upon observing a signal. Finally, the experiment $E$ is only valuable if at least two signals lead to different actions. If the buyer chooses a constant action following every signal $s_{k}$, then $V(E, \theta)=0$, as can be seen in formulations (6) and (7), as well as in Figure 1 (right).


The Design and Price of Information

### 3.2 The Seller's Problem

The seller's choice of a profit-maximizing menu of experiments may involve, in principle, designing one experiment per buyer type. In turn, each experiment entails a type-dependent mapping from signals into actions. The seller's problem can, however, be simplified by reducing the set of menus of experiments to a smaller and very tractable class.

First, by the revelation principle, we can restrict our attention to direct mechanisms $\mathcal{M}=\{E(\theta), t(\theta)\}$ that assign an experiment $E(\theta)=(S(\theta), \pi(\theta))$ and a price $t(\theta)$ to each type $\theta$ of data buyer. We denote the indirect (net) utility for the truth-telling agent by

$$
V(\theta) \triangleq V(E(\theta), \theta)-t(\theta) .
$$

The seller's problem consists of maximizing the expected transfers

$$
\max _{\{E(\theta), t(\theta)\}} \int_{\theta \in \Theta} t(\theta) \mathrm{d} F(\theta)
$$

subject to incentive-compatibility constraints

$$
V(\theta) \geq V\left(E\left(\theta^{\prime}\right), \theta\right)-t\left(\theta^{\prime}\right), \quad \forall \theta, \theta^{\prime} \in \Theta,
$$

and individual-rationality constraints

$$
V(\theta) \geq 0, \quad \forall \theta \in \Theta .
$$

Second, given any direct mechanism $\mathcal{M}=\{E(\theta), t(\theta)\}$, we say that experiment $E(\theta)$ is responsive if every signal $s \in S(\theta)$ leads type $\theta$ to a different optimal choice of action and, in particular,

$$
a\left(s_{k} \mid \theta\right)=a_{k} \text { for all } s_{k} \in S(\theta) \text {. }
$$

Importantly, condition (10) is only required for every experiment $E(\theta)$ and for the corresponding type $\theta$. In other words, we do not require this condition to hold if signals $s_{k} \in S(\theta)$ are evaluated by a different type $\theta^{\prime} \neq \theta$. Finally, we define an outcome of a menu as the joint distribution of states, actions, and monetary transfers resulting from every type's optimal choice of experiment and subsequent choice of action.

Proposition 1 establishes that it is without loss of generality to restrict attention to responsive menus-direct revelation mechanisms in which every experiment $E(\theta)$ is responsive.


The Design and Price of Information

## Proposition 1 (Responsive Menus)

The outcome of every menu $\mathcal{M}$ can be attained by a responsive menu.
Our proof closely follows the argument of the revelation principle for Bayesian games of communication established by Myerson (1982). We show that we can always reduce the size of the signal space to the size of the action space recommended in equilibrium. The intuition is straightforward. Consider an incentive-compatible menu that contains an experiment $E(\theta)$ with more signals than actions. We can combine all signals in $E(\theta)$ that lead to the same choice of action for type $\theta$. The value of this experiment remains constant for type $\theta$, who does not modify his behavior. However, because the new experiment is (weakly) less informative than the experiment we started with, $V\left(E(\theta), \theta^{\prime}\right)$ decreases (weakly) for all $\theta^{\prime} \neq \theta$, relaxing the incentive constraints. Finally, because every signal sent with positive probability under experiment $E(\theta)$ leads type $\theta$ to a different action, we can order the signals such that each $s_{k} \in S(\theta)$ recommends the corresponding action $a_{k}$.

The language of the Bayesian games of communication, as suggested by Myerson (1982), is helpful for understanding the nature of the seller's problem more generally. The solution to the data seller's problem has to satisfy two different constraints, the truth-telling (or honesty) constraint given by (9) and the obedience constraint given by (4). Thus, the buyer must be jointly honest and obedient. In particular, double deviations (lying and disobeying) must not be profitable for the buyer.

An immediate implication of Proposition 1 is that, without loss of generality, we can restrict our attention to experiments in which the signal space has the cardinality of the action space, i.e., $K=J$. This insight allows us to write the likelihood function of every experiment (3) as a matrix with the same dimensions as the payoff matrix, i.e.,

| $E(\theta)$ | $s_{1}\left(=a_{1}\right)$ | ... | $s_{i}\left(=a_{j}\right)$ | ⋯ | $s_{J}\left(=a_{J}\right)$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $\omega_{1}$ | $\pi_{11}$ | ⋯ | $\pi_{1 j}$ | ⋯ | $\pi_{1 J}$ |
| ⋮ | ⋮ |  |  |  | ⋮ |
| $\omega_{i}$ | $\pi_{i 1}$ |  | $\pi_{i j}$ |  | $\pi_{i J}$ |
| ⋮ | ⋮ |  |  |  | ⋮ |
| $\omega_{I}$ | $\pi_{I 1}$ | ⋯ | $\pi_{I j}$ | ⋯ | $\pi_{I J}$ |

Thus, we can replace the signal $s_{j}$ with the action recommendation $a_{j}$. This property does not require that every action $a_{j}$ is recommended with strictly positive probability. For instance, some signals $s_{j}$ may never be sent, corresponding to a column vector of zeros at the $j$-th position. The resulting value of experiment $E(\theta)$ for type $\theta$ can be written as

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

Under the restriction to responsive experiments, formulation (11) removes the first max operator from the value of experiment $E(\theta)$ for the truth-telling type $\theta$. Because a misreporting type need not always take the recommended action, we must still use the original formulation (6) when computing the value of experiment $E(\theta)$ for type $\theta^{\prime}$.

When payoffs are given by state-action matching as in (2), the value of experiment $E(\theta)$ for the truth-telling type $\theta$ simplifies to

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

In formulation (12), the value of experiment $E(\theta)$ for type $\theta$ is fully determined by its diagonal entries $\pi_{i i}$. By contrast, the off-diagonal entries $\pi_{i j}$ may enter the value of experiment $E(\theta)$ for different types $\theta^{\prime}$, as in (7).


The Design and Price of Information

### 3.3 Structural Properties

We now leverage the nature of the value of information to impose additional structure on the experiments that are part of an optimal menu.


The Design and Price of Information

## Proposition 2 (Optimal Experiments)

1. The fully informative experiment $\bar{E}$ is part of an optimal menu.
2. Every experiment in any optimal menu is non-dispersed, i.e., $\pi_{i j}=0$ for some $i \neq j$.
3. In the matching case, every experiment in any optimal menu is concentrated, i.e., $\pi_{i i}=1$ for some $i$.

The first part of this result can be established via contradiction. Every type $\theta$ values the fully informative experiment $\bar{E}$ the most among all the experiments. Suppose, then, that $\bar{E}$ is not part of the menu, and denote the most expensive item currently on the menu by $E^{\prime}$. The seller can replace experiment $E^{\prime}$ with the complete experiment $\bar{E}$, keeping all other prices constant and charging a higher price for $\bar{E}$ than for $E^{\prime}$. The new menu weakly increases the seller's revenue without lowering the net utility of any buyer type. ${ }^{6}$

The second and third parts imply that every optimal experiment eliminates the buyer's uncertainty along some dimension. They are also established by an improvement argument. Fix an experiment $E$ and suppose all entries $\pi_{i j}$ are strictly positive. In each state (row) $i$, the seller increases the probability $\pi_{i i}$ of the signal $s_{i}$ that yields the highest payoff $u_{i i}$ for an obedient type in that state. Concurrently, the seller reduces the probability $\pi_{i j}$ of the signal $s_{j}$ that yields the lowest payoff $\min _{j} u_{i j}$ for an obedient type. The seller shifts a probability mass inversely proportional to the difference in payoffs $u_{i i}-u_{i j}$, and hence, the resulting payoff gain is uniform across states $\omega_{i}$. This procedure is applied until the first entry $\pi_{i j}$ reaches 0. Because the beliefs of each type $\theta$ sum to one, this shift is valued uniformly by all obedient types and weakly less by any other type. A commensurate increase in the price of the experiment offsets the value of the additional information provided and, hence, strictly increases profits while (weakly) relaxing the truth-telling and participation constraints.

Thus, with arbitrary payoffs, every experiment $E$ contains a signal $s_{j}$ that allows the buyer to rule out some state $\omega_{i}$. The limits of this result are related to the possibility of double deviations evoked earlier. A misreporting type $\theta^{\prime}$ may not necessarily choose the action recommended by every signal in the experiment $E(\theta)$ intended for type $\theta$. For

[^5]the above improvement argument to discourage double deviations, it is critical that the probability mass is shifted away from the worst action in each state $\omega_{i}$, thus yielding the largest possible marginal benefit $\left(\max _{j} u_{i j}-\min _{j} u_{i j}\right)$ for every obedient type in state $\omega_{i}$. Any other shift may yield a strictly higher benefit to type $\theta^{\prime}$ than to type $\theta$ and lead to the violation of the truth-telling constraints.

In the case of matching utility, the seller can shift the probability mass to the diagonal from any off-diagonal entry until the first entry $\pi_{i i}$ reaches 1. Indeed, when the probability is shifted to the diagonal, any type that does not follow the signal's recommendation obtains a strictly lower benefit relative to an obedient type. As a result, any optimal experiment is concentrated, i.e., there exists at least one state $\omega_{i}$ under which signal $s_{i}$ is sent with probability one, and the buyer takes the correct action $a_{i}$. Conversely, the buyer is able to rule out (at least) one state $\omega_{i}$ after observing any signal $s_{k} \neq s_{i}$.

In Sections 4 and 6, we show that with binary states and actions or binary types, respectively, it is sufficient to consider truth-telling and obedience separately. By contrast, Example 3 in the Supplemental Appendix demonstrates that with three (or more) actions and types, double deviations typically impose additional restrictions on the optimal menu.


The Design and Price of Information

## 4 Optimal Menu with Binary Actions

In this section, we consider an environment with two actions $a \in\left\{a_{1}, a_{2}\right\}$ and two states $\omega \in\left\{\omega_{1}, \omega_{2}\right\}$. In this setting, the restriction to matching utility functions entails no loss of generality relative to a general payoff matrix. To wit, for every state $\omega_{i}$, we can always subtract the (state-dependent) constant $u_{i j}$ with $i \neq j$. This linear transformation normalizes the payoffs of the data buyer by setting $u_{12}=u_{21}=0$ without affecting the optimality conditions of the data buyer's decision problem.

We thus obtain a diagonal payoff matrix as in (2) with positive entries given by

$$
u_{1} \triangleq u_{11}, u_{2} \triangleq u_{22} .
$$

With binary states, the interim belief of the data buyer (his type) is one dimensional. We identify each type with the interim probability of state $\omega_{1}$ :

$$
\theta \triangleq \operatorname{Pr}\left[\omega=\omega_{1}\right] \in[0,1] .
$$

We denote the interim belief type $\theta$ that is indifferent between action $a_{1}$ and $a_{2}$ by $\theta^{*}$ as

$$
\theta^{*} u_{1}=\left(1-\theta^{*}\right) u_{2} \Leftrightarrow \theta^{*}=\frac{u_{2}}{u_{1}+u_{2}} \text {. }
$$


The Design and Price of Information

### 4.1 Binary Experiments

With binary actions, Proposition 1 implies that it is sufficient to consider for every type $\theta$ experiments $E(\theta)$ that generate (at most) two signals:

| $E(\theta)$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | $\pi_{11}(\theta)$ | $1-\pi_{11}(\theta)$. |
| $\omega_{2}$ | $1-\pi_{22}(\theta)$ | $\pi_{22}(\theta)$ |

With binary signals, we can simplify the notation by dropping the second subscript for diagonal entries, as we did for the payoff function:

$$
\pi_{1}(\theta) \triangleq \pi_{11}(\theta), \pi_{2}(\theta) \triangleq \pi_{22}(\theta) .
$$

Without loss of generality, we assume that $\pi_{1}(\theta)+\pi_{2}(\theta) \geq 1$. In other words, signal $s_{1}$ is relatively more likely to occur than signal $s_{2}$ under state $\omega_{1}$ than under state $\omega_{2}$, or

$$
\frac{\pi_{1}(\theta)}{1-\pi_{1}(\theta)} \geq \frac{1-\pi_{2}(\theta)}{\pi_{2}(\theta)} .
$$

With binary states and actions, the general payoff environment conforms to the matching utility environment. We can therefore write the value for an arbitrary experiment $E$ as follows by modifying expression (12):

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

As in the earlier formulation (12), the diagonal entries of the matrix $\pi(\theta)$ generate the probability that experiment $E(\theta)$ allows type $\theta$ to match the realized state with his action. Conversely, the first max operator accounts for the possibility that a misreporting type $\theta^{\prime}$ does not follow the recommendation implicit in one of the signals of $E(\theta)$ and, hence, derives no value from the supplemental information.

In Figure 2, we illustrate how the value of information changes as a function of the type $\theta$ for the case $u_{1}=u_{2}=1$. We compare two experiments with binary signals, namely, the fully informative experiment $E^{\prime}=\left(\pi_{1}^{\prime}, \pi_{2}^{\prime}\right)=(1,1)$ and a partially informative experiment $E^{\prime \prime}=\left(\pi_{1}^{\prime \prime}, \pi_{2}^{\prime \prime}\right)=(1 / 2,1)$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Value of Full and Partial Information $\left(I=J=2, \quad u_{1}=u_{2}=1\right)$

The value of information as a function of the buyer's interim beliefs $\theta$ reflects many intuitive properties that we formally establish in the next subsection:

1. The most valuable type for the seller is the ex ante least informed. In the examples in Figure 2, this is type $\theta^{*}=1 / 2$. Conversely, the most informed types $\theta \in\{0,1\}$ have zero value of information. The linear decline in each direction away from $\theta^{*}$ follows from the linearity of the value of information in the interim probability.
2. When we consider any asymmetric experiment, such as the one displayed in the right panel of Figure 2, the distance from the least informed type $\left|\theta-\theta^{*}\right|$ is not a sufficient statistic for the value of information. The different slopes on each side of $\theta^{*}$ indicate different marginal benefits for matching state $\omega_{1}$ versus state $\omega_{2}$ on the basis of differences in the interim beliefs $\theta$.

Thus, even in an environment where types are clearly one dimensional, information products are inherently multidimensional (in this case, two dimensional). In particular, information always has both vertical (quality) and horizontal (positioning) dimensions. Unlike in models of nonlinear pricing (with respect to either quantity or quality) where all the types agree on the relative ranking of all the products, in the current environment, the types disagree on the very ranking of all partially informative experiments.


The Design and Price of Information

### 4.2 Binary Types

We derive the optimal menu with two data buyer types, $\theta \in \Theta=\left\{\theta^{L}, \theta^{H}\right\}$. The high (value) type $\theta^{H}$ assigns a higher value to receiving the fully informative experiment, i.e.,

$$
V\left(\bar{E}, \theta^{H}\right)>V\left(\bar{E}, \theta^{L}\right) .
$$

With uniform weights $u_{1}=u_{2}>0$, this simply means the high (value) type is less wellinformed ex ante, i.e.,

$$
\left|\theta^{H}-1 / 2\right| \leq\left|\theta^{L}-1 / 2\right| .
$$

We refer to the distance between the interim belief $\theta$ and the indifference belief $\theta^{*}$ as the precision of the type $\theta$. We denote the frequency of a high type as

$$
\gamma \triangleq \operatorname{Pr}\left(\theta=\theta^{H}\right) .
$$

The following distinction proves helpful. The interim beliefs of the two types are said to be congruent if both types would choose the same action without additional information. If we adopt the convention that the high type chooses action $a_{1}$ under his prior information (i.e., $\theta^{H}>\theta^{*}$ ), then beliefs (and corresponding types) are congruent if $\theta^{*}<\theta^{H}<\theta^{L}$ and noncongruent if $\theta^{L}<\theta^{*}<\theta^{H}$.

We first establish two familiar properties: "no distortion at the top," i.e., for the high type and "no rent at the bottom," i.e., for the low type.


The Design and Price of Information

## Proposition 3 (Binding Constraints)

In an optimal menu:

1. type $\theta^{H}$ purchases the fully informative experiment $\bar{E}$;
2. the participation constraint of type $\theta^{L}$ binds;
3. the incentive-compatibility constraint of type $\theta^{H}$ binds.

We note that these properties of the two-type environment hold for any number of states and actions under arbitrary payoffs, and we shall revisit them in Section 6. The description of the optimal menu is then completed by characterizing the experiment $E\left(\theta^{L}\right)$ purchased by the low type. Here, it is productive to distinguish between congruent and noncongruent beliefs.

Congruent Beliefs. In the case of congruent beliefs $\left(\theta^{*}<\theta^{H}<\theta^{L}\right)$, the argument is related to the classic monopoly pricing problem. By Proposition 2, we know that the optimal experiment $E\left(\theta^{L}\right)$ is concentrated. With congruent beliefs, both types would choose action $a_{1}$ absent any additional information. The data seller does not want to reduce the information relative to the outside option and, thus, sets $\pi_{1}\left(\theta^{L}\right)=1$. The issue is then how much information to provide about state $\omega_{2}$, i.e.,

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

Now, any partially informative experiment with $\pi_{2}\left(\theta^{L}\right) \in(0,1)$ geared towards the low type is also valuable to the high type. Using the expression for the value of information given by (15), we can write the incentive constraint for the high type as follows:

$$
\theta^{H} u_{1}+\left(1-\theta^{H}\right) u_{2}-t\left(\theta^{H}\right) \geq \theta^{H} u_{1}+\left(1-\theta^{H}\right) \pi_{2}\left(\theta^{L}\right) u_{2}-t\left(\theta^{L}\right),
$$

since $\pi_{1}\left(\theta^{H}\right)=\pi_{2}\left(\theta^{H}\right)=1$ and $\pi_{1}\left(\theta^{L}\right)=1$. The incentive constraint for the high type thus reduces to:

$$
\underbrace{\left(1-\theta^{H}\right)}_{\operatorname{Pr}\left(\omega_{2}\right)} \cdot \underbrace{\left(1-\pi_{2}\left(\theta^{L}\right)\right)}_{\text {additional precision }} \cdot u_{2} \geq t\left(\theta^{H}\right)-t\left(\theta^{L}\right) .
$$

Hence, we observe that both the objective and the constraints in the seller's problem are linear in the choice variable $\pi_{2}$. We can therefore appeal to the no-haggling result of Riley and Zeckhauser (1983) that establishes the optimality of an extremal policy. Such a policy consists of either allocating the object (here, the information) with probability one or not allocating it at all, hence, $\pi_{2} \in\{0,1\}$. As in the single-good monopolist's problem, the optimal policy depends on the distribution of buyer types. In particular, the low type receives the fully informative experiment if and only if the probability $\gamma$ of the high type is sufficiently small, or

$$
\left(1-\theta^{L}\right) u_{2} \geq \gamma\left(1-\theta^{H}\right) u_{2} \Longleftrightarrow \gamma \leq \frac{1-\theta^{L}}{1-\theta^{H}} .
$$

Noncongruent Beliefs. In the case of noncongruent beliefs $\left(\theta^{L}<\theta^{*}<\theta^{H}\right)$, both the argument and the result are distinct from those of the classic monopoly problem. In the absence of additional information, the two types choose different actions. The seller can then provide information in a format that has positive value to one type but zero value to the other type. For example, suppose that $\pi_{2}\left(\theta^{L}\right)=1$ and $\pi_{1}\left(\theta^{L}\right)$ is chosen such that after receiving signal $s_{2}$, the high type $\theta^{H}$ is indifferent between actions $a_{1}$ and $a_{2}$ (i.e., his posterior belief is $\theta^{*}$ ):

$$
\theta^{H}\left(1-\pi_{1}^{\prime}\right) u_{1}=\left(1-\theta^{H}\right) u_{2} \Longleftrightarrow \pi_{1}^{\prime}=\frac{u_{1} \theta^{H}-u_{2}\left(1-\theta^{H}\right)}{u_{1} \theta^{H}} \in(0,1) .
$$

Because type $\theta^{H}$ chooses action $a_{1}$ without additional information, the experiment

| $E^{\prime}$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | $\pi_{1}^{\prime}$ | $1-\pi_{1}^{\prime}$ |
| $\omega_{2}$ | 0 | 1 |

does not lead to a strict improvement in the decision making (or utility) of the high type $\theta^{H}$. By contrast, the low type $\theta^{L}$ assigns a positive value to experiment $E^{\prime}$. After all, signal $s_{1}$ would lead him to match his action to state $\omega_{1}$, which he would never achieve without additional information. Thus, the seller can offer partial information to the low type without incurring any implicit cost in terms of surplus extraction vis-à-vis the high type. ${ }^{7}$

The argument in support of a partially informative experiment is illustrated for the case $u_{1}=u_{2}=1$ in Figure 3, which depicts the value of two experiments net of the price as a function of the buyer's type $\theta \in[0,1]$. In this example, we set $\theta^{L}=1 / 5<1 / 2<7 / 10=\theta^{H}$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 3: Suboptimal Menu: $\left(\pi_{1}, \pi_{2}\right) \in\{(1,1),(4 / 7,1)\}$

The net value of the fully informative experiment $\bar{E}$ is depicted by the solid line at a price $t\left(\theta^{H}\right)=1-\theta^{H}$ that leaves the type $\theta^{H}$ indifferent between buying and not buying the fully informative experiment. The dashed line depicts the partial information experiment given by $\pi_{1}^{\prime}$, as described by (18). The associated partially informative experiment $E^{\prime}$ leaves the low type $\theta^{L}$ indifferent between buying and not buying. As for the high type $\theta^{H}$, this experiment offers zero value at a positive price, leaving his incentive constraint slack. Correspondingly, for the high type, the net value of the partially informative experiment $E^{\prime}$ is strictly below the net value of the fully informative experiment.


The Design and Price of Information

As the incentive-compatibility constraints are slack given this specific menu of experiments for both types, the seller can offer a more informative experiment $E^{\prime \prime}$ to the low type $\theta^{L}$ while still satisfying the incentive constraint for the high type $\theta^{H}$. As the partially informative experiment $E^{\prime \prime}$ has a positive price tailored to type $\theta^{L}$, i.e., $t\left(\theta^{L}\right)=\theta^{L} \pi_{1}^{\prime \prime} u_{1}$, the high type can be made indifferent between experiments $\bar{E}$ and $E^{\prime \prime}$ by making sure that incentive

[^6]constraint (17) is binding:
$$
\underbrace{\theta^{H}}_{\operatorname{Pr}\left(\omega_{1}\right)} \cdot \underbrace{\left(1-\pi_{1}^{\prime \prime}\right)}_{\text {additional precision }} \cdot u_{1}=\underbrace{\left(1-\theta^{H}\right) u_{2}}_{t\left(\theta^{H}\right)}-\underbrace{\theta^{L} \pi_{1}^{\prime \prime} u_{1}}_{t\left(\theta^{L}\right)} \Longleftrightarrow \pi_{1}^{\prime \prime}=\frac{u_{1} \theta^{H}-u_{2}\left(1-\theta^{H}\right)}{u_{1}\left(\theta^{H}-\theta^{L}\right)} .
$$
The experiment $E^{\prime \prime}$ characterized in (20) is as informative as possible while satisfying the high type's incentive compatibility constraint and both participation constraints with equality. Experiments $\bar{E}$ and $E^{\prime \prime}$ are illustrated in Figure 4, again for the case $u_{1}=u_{2}=1$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 4: Optimal Menu: $\left(\pi_{1}, \pi_{2}\right) \in\{(1,1),(4 / 5,1)\}$

This example highlights the horizontal aspect of selling information that increases the scope of screening: the high type $\theta^{H}$ buys the perfectly informative experiment; the low type $\theta^{L}$ buys a partially informative experiment; and the seller extracts the entire surplus (which, however, falls short of the socially efficient surplus). The relative frequency of each buyer type determines the shape of the optimal menu. In particular, the partially informative experiment $E^{\prime \prime}$ is replaced by the fully informative experiment when there is a high proportion of low types, i.e., when $\gamma<\theta^{L} / \theta^{H}$ for all payoffs $\left(u_{1}, u_{2}\right)$.

We have thus shown the following results for the fully binary model.


The Design and Price of Information

## Proposition 4 (Partial Information)

1. With congruent priors, the low type receives either zero or complete information.
2. With noncongruent priors, the low type receives either partial or complete information.

With congruent priors, both types receive complete information if they are sufficiently similar or if the low type is relatively frequent. With noncongruent priors, the optimal menu offers the fully informative experiment to both types if they are sufficiently similar in their prior information, or if the low type is sufficiently frequent. Offering two experiments
becomes optimal if the two types are sufficiently different in their level of informativeness. Importantly, the low type always receives some information in that case and is not excluded. ${ }^{8}$ Furthermore, the high type receives positive rents only if he is pooled with the low type. Otherwise, the seller extracts the entire surplus that is generated.

We now examine the comparative statics of the optimal menu. Because the high type buys the fully informative experiment $\bar{E}$ and the low type's experiment is concentrated with $\pi_{2}=1$, the optimal experiment $E\left(\theta^{L}\right)$ can be described by its first diagonal entry $\pi_{1}$.


The Design and Price of Information

## Proposition 5 (Comparative Statics)

The informativeness $\pi_{1}$ of the optimal experiment $E\left(\theta^{L}\right)$ is

1. decreasing in the probability $\gamma$ of the high type;
2. decreasing in the precision of the low type's prior belief $\left|\theta^{L}-\theta^{*}\right|$;
3. increasing in the precision of the high type's prior belief $\left|\theta^{H}-\theta^{*}\right|$ when priors are congruent or the menu is discriminatory.

Thus, even though the shape of the optimal menu depends on whether types have congruent or noncongruent priors, the comparative statics of the optimal experiment are robust across the different scenarios. The rent extraction vs. efficiency trade-off is resolved at the expense of the low type as (i) the fraction of high types increases, or (ii) the low type's willingness to pay for the complete experiment decreases.

Finally, as the high type's willingness to pay for information decreases (his prior becomes more precise), the optimal menu may switch from offering full to partial information to the low type. This occurs because separating the two types becomes profitable whenever the gap in their prior beliefs is sufficiently wide. Once partial information is offered, however, the optimal distortions decrease with the precision of the high type. This accounts for the qualifying statement in Proposition 5.3.


The Design and Price of Information

### 4.3 Continuum of Types

We now complete the analysis of the binary-action environment. We denote the interim belief of the data buyer by $\theta \triangleq \operatorname{Pr}\left(\omega=\omega_{1}\right)$ and consider a continuum of types $\theta \in[0,1]$ on the unit interval, with a distribution $F(\theta)$ and associated density $f(\theta)$. We shall show that

[^7]many qualitative properties of the two-type case-including the cardinality of the optimal menu-hold in this setting.

Recall the value of information was described in (15):

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

We can rewrite the value of information as

$$
V(E, \theta)=\max \left\{\theta\left(\pi_{1} u_{1}-\pi_{2} u_{2}\right)+\pi_{2} u_{2}-\max \left\{\theta u_{1},(1-\theta) u_{2}\right\}, 0\right\},
$$

and we capture the value of experiment $E$ for type $\theta$ via a one-dimensional variable

$$
q(\theta) \triangleq \pi_{1}(\theta) u_{1}-\pi_{2}(\theta) u_{2} \in\left[-u_{2}, u_{1}\right],
$$

which describes the differential informativeness of the experiment. The endpoints of the interval $\left[-u_{2}, u_{1}\right]$ identify two extreme experiments $q \in\left\{-u_{2}, u_{1}\right\}$ that are attained when either one of the two signals occurs with probability one in both states. In either case, the resulting experiment reveals no information to the data buyer. Conversely, the fully informative experiment is given by $q=u_{1}-u_{2}$. Because Proposition 2 establishes that either $\pi_{1}(\theta)=1$ or $\pi_{2}(\theta)=1$ (or both) for each type $\theta$, we know that $q(\theta)>u_{1}-u_{2}$ implies $\pi_{1}(\theta)=1$, and that $q(\theta)<u_{1}-u_{2}$ implies $\pi_{2}(\theta)=1$. Collecting terms, we can rewrite the value of an experiment in terms of the one-dimensional variable $q$ as follows:

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

The value of information in (21) illustrates the main properties of our screening problem: (i) the buyer has a type-dependent participation constraint; (ii) the ex ante indifferent type $\theta^{*}$ has the highest willingness to pay for any experiment $q$; (iii) the experiment $q=u_{1}-u_{2}$ is the most valuable for all types $\theta$; (iv) different types $\theta$ rank partially informative experiments differently; and $(v)$ the utility function $V(q, \theta)$ has the single-crossing property in $(\theta, q)$.

The single-crossing property indicates that types with a higher $\theta$, those who believe that state $\omega_{1}$ is more likely, assign a higher value to experiments with a higher $q$. In turn, these experiments contain a signal that delivers stronger evidence regarding state $\omega_{2}$, which they deem less likely ex ante. As in the binary-type case, the vertical dimension (quality of the information) and the horizontal dimension (position of the information) cannot be chosen separately by the seller. In particular, it is not possible to change the differential informativeness of the experiment (i.e., to choose a very high or a very low $q$ ) without
reducing its overall informativeness.
We know from Proposition 1 that we can focus on responsive menus. In the case of binary actions, an experiment is responsive if and only if the value of following both signals' recommendations is non-negative. Therefore, equation (21) implies that experiment $q(\theta)$ offered to type $\theta$ is responsive if and only if

$$
q(\theta) \in Q(\theta) \triangleq\left\{\begin{array}{cl}
{\left[-u_{2}, \frac{u_{1}}{1-\theta}-u_{2}\right]} & \text { for } \quad \theta \leq \theta^{*}, \\
{\left[u_{1}-\frac{u_{2}}{\theta}, u_{1}\right]} & \text { for } \quad \theta \geq \theta^{*} .
\end{array}\right.
$$

In other words, the restriction $q(\theta) \in Q(\theta)$ allows us to eliminate the first max operator from (21) when computing the value $V(q(\theta), \theta)$. We thus obtain a characterization of implementable and responsive menus of experiments.


The Design and Price of Information

## Lemma 1 (Implementable and Responsive Menus)

A menu $\{q(\theta)\}_{\theta \in \Theta}$ is implementable and responsive if and only if

$$
q(\theta) \in\left[-u_{2}, u_{1}\right] \text { is non-decreasing }
$$

and

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

To obtain some intuition for constraint (24), observe that the interior type $\theta^{*}$ assigns the highest value to any experiment $q$, and that the utility function $V(q, \theta)$ has a downward kink at $\theta=\theta^{*}$. In order to compute the buyer's rent function $V$, we apply the envelope theorem to the two intervals on $\left[0, \theta^{*}\right]$ and $\left[\theta^{*}, 1\right]$ separately. Because information has zero value for types $\theta \in\{0,1\}$, we know that $V(0)=V(1)=0$. We thus obtain two (potentially different) expressions for $V\left(\theta^{*}\right)$. Finally, the buyer's gross value of an experiment (21) is jointly continuous in $(q, \theta)$, and hence, the rent function $V$ is continuous by the Maximum Theorem. ${ }^{9}$ Incentive compatibility then imposes the following restriction on responsive allocations:

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

Computing the buyer's marginal rent from (21) and using the definition of $\theta^{*}$ to simplify (??) yields the integral condition in the Lemma. ${ }^{10}$ Notably, condition (22) does not appear

[^8]in the statement of Lemma 1 because it is implied by the monotonicity condition (23) and by the integral constraint (24).

With this result in place, the transfer $t(\theta)$ associated with every experiment $q(\theta)$ can be computed from the envelope formula on the $\left[0, \theta^{*}\right]$ and $\left[\theta^{*}, 1\right]$ separately. Using integral constraint (24) to simplify further, the seller's problem can be written as

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

subject to constraints (23) and (24).
Observe that the seller's objective (25) is piecewise linear (and concave) in the experiment $q(\theta)$. Thus, absent the integral constraint, the optimal experiments take values at the kinks, i.e., $q^{*}(\theta) \in\left\{-u_{2}, u_{1}-u_{2}, u_{1}\right\}$ for all $\theta$. This corresponds to a menu containing the fully informative experiment only. While such a menu is, in fact, optimal under some conditions, the seller can sometimes do better by offering (at most) one additional experiment.


The Design and Price of Information

## Proposition 6 (Cardinality of the Optimal Menu)

An optimal menu consists of at most two experiments.
To establish this property of optimal menus, we reduce the seller's problem (25) to a linear program with equality and non-negativity constraints. An application of the Fundamental Theorem of Linear Programming (e.g., Theorem 8.4 in Chvatal (1983)) implies that the solution is an increasing step function with at most three jumps. ${ }^{11}$ Because it is optimal to set $q(0)=-u_{2}$ and $q(1)=u_{1}$, this means that the optimal menu contains at most two informative experiments $q \in\left(-u_{2}, u_{1}\right)$. When offered, the second experiment contains a signal that perfectly reveals one state. However, the linearity of the environment prevents the seller from offering more than one distorted experiment, i.e., no further versioning is optimal. We formalize this intuition in the following subsection.


The Design and Price of Information

### 4.4 Single-Item vs. Discriminatory Pricing

To obtain further insights into the properties of the optimal menu, we refine our approach to the seller's problem. We combine Lagrange methods, as in the type-dependent participation

[^9]constraints model of Jullien (2000), with the ironing procedure developed by Toikka (2011) extending that in Myerson (1981). This approach allows us to overcome two difficulties posed by our problem: (i) integral constraint (24) and objective (25) have generically different weights, $\mathrm{d} \theta$ and $\mathrm{d} F(\theta)$; hence, (ii) the problem is non-separable in the type $\theta$ and the experiment $q(\theta)$, which interact in two different terms. In particular, the "virtual values" $\phi(\theta, q)$-defined as the partial derivative of the integrand in (25) with respect to $q$-are a non-constant function of the experiment.

We derive the solution to the seller's problem by maximizing the virtual values. Because the integrand in objective (25) is piecewise linear in $q$, its partial derivative $\phi(\theta, q)$ takes on only two values. We then define the following two functions

$$
\begin{aligned}
\phi^{-}(\theta) & \triangleq \theta f(\theta)+F(\theta) \\
\phi^{+}(\theta) & \triangleq(\theta-1) f(\theta)+F(\theta)
\end{aligned}
$$

that describe the virtual value $\phi(\theta, q)$ for $q<u_{1}-u_{2}$ and $q \geq u_{1}-u_{2}$, respectively. Heuristically, the two virtual values represent the marginal benefit to the seller of increasing each type's experiment $q$ from $-u_{2}$ to $u_{1}-u_{2}$ and from $u_{1}-u_{2}$ to $u_{1}$. If ironing à la Myerson is required, we denote the ironed virtual values as $\bar{\phi}^{-}(\theta)$ and $\bar{\phi}^{+}(\theta)$.

Finally, we say that a menu satisfies the pooling property if it is constant on any interval where the relevant (ironed) virtual value is constant.


The Design and Price of Information

## Proposition 7 (Optimal Menu)

The menu $\left\{q^{*}(\theta)\right\}_{\theta \in \Theta}$ is optimal if and only if the following conditions hold:

1. there exists $\lambda^{*}>0$ such that, for all $\theta$,
$$
q^{*}(\theta)=\arg \max _{q \in\left[-u_{2}, u_{1}\right]}\left[\bar{\phi}^{-}(\theta) \min \left\{q, u_{1}-u_{2}\right\}+\bar{\phi}^{+}(\theta) \max \left\{0, q-\left(u_{1}-u_{2}\right)\right\}-\lambda^{*} q\right]
$$
2. $\left\{q^{*}(\theta)\right\}_{\theta \in \Theta}$ has the pooling property and satisfies integral constraint (24).

We now illustrate the optimal menu under flat and discriminatory pricing separately.


The Design and Price of Information

## Corollary 1 (Single-Item Menu)

The optimal menu contains a single item whenever any of the following hold:

1. almost all types have congruent priors, i.e., $F\left(\theta^{*}\right) \in\{0,1\}$;
2. the monopoly price for experiment $\bar{E}$ is equal on $\left[0, \theta^{*}\right]$ and $\left[\theta^{*}, 1\right]$;
3. both virtual values $\phi^{-}(\theta)$ and $\phi^{+}(\theta)$ are strictly increasing.

To grasp the intuition behind these results, consider a relaxed problem where the seller contracts separately with two groups of buyers, $\theta<\theta^{*}$ and $\theta \geq \theta^{*}$. Part (1) essentially addresses this case. Because of the linearity of the problem, the optimal mechanism is a cutoff mechanism: the seller offers only the fully informative experiment to each group, at generically different prices. Part (2) states that, if the cutoff types in the two subproblems have the same willingness to pay, then those cutoffs also solve the unrestricted problem. For instance, when $u_{1}=u_{2}$, any distribution that is symmetric around $\theta=1 / 2$ satisfies this condition. Part (3) identifies regularity conditions on the distribution of types that rule out, for example, most bimodal distributions. Under these conditions, the seller prefers to offer the fully informative experiment to all buyers at an intermediate price.

We note that only condition (3) can be stated independently of the matching values $\left(u_{1}, u_{2}\right)$. For example, an optimal menu contains a single item whenever the values are uniformly distributed, irrespective of the payoffs. Figure 5 describes the optimal menu for the case of constant match values, $u_{1}=u_{2}=1$, and uniformly distributed types. Because both virtual values are strictly increasing, they cross the threshold level $\lambda^{*}$ only once. Thus, the optimal menu is a step function $q(\theta) \in\{-1,0,1\}$ assigning the fully informative experiment $q=0$ to types $\theta \in[1 / 4,3 / 4]$ at a price $t=1 / 4$, and no information to all other types.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 5: Optimal Allocation with Uniform Distribution

If the conditions of Corollary 1 fail, however, the seller may choose to offer a second (distorted) experiment to one group in order to maintain a high price for the fully informative experiment. Furthermore, Corollary 1.3 implies that the seller offers a second experiment only if the distribution of types requires ironing of the virtual values. When types correspond to interim beliefs, it is natural to consider bimodal densities that fail the strong regularity conditions and, therefore, introduce the need for ironing. This is the case, for example, if most buyers are well informed ex ante (with most types close to 0 or 1 ). In other words, ironing is not a technical curiosity in our case but rather a technique that becomes unavoidable because of the properties of the information environment. Figure 6 (left) illustrates the ironed virtual
values for a bimodal probability density function (drawn on a different scale). ${ }^{12}$ Figure 6 (right) illustrates the resulting optimal menu, again for the case $u_{1}=u_{2}$.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 6: Probability Density Function and Optimal Allocation

As in the optimal menu in the binary-type setting, the partially informative experiment $q \approx-0.21$ contains one signal that perfectly reveals state $\omega_{1}$. This experiment is relatively unattractive for higher types, and it allows the monopolist to increase the price for the large mass of types located around $\theta \approx 0.7$. Note that the ex ante least informed type $\theta^{*}$ need not purchase the fully informative experiment, despite having the highest value of information and obtaining the highest rent $V\left(\theta^{*}\right)$. Indeed, the type that is indifferent between the two experiments in Figure 6 is $\theta \approx 0.55$. From the seller's perspective, incentivizing type $\theta=1 / 2$ to purchase the fully informative experiment would require further distortions (and, hence, a lower price) for the second experiment. This leads to a loss of revenue from the types around $\theta \approx 0.2$. Because such types are quite frequent, this loss more than offsets the gain in revenue from the types around $\theta=1 / 2$.

Finally, a sufficient condition for the optimality of two-item menus can be obtained by continuity with the binary-type case. In that setting, when the two types are equally likely, symmetric about 1/2 and sufficiently well informed (i.e., $\theta^{H}>2 / 3$ ), the optimal menu is discriminatory for all $u_{1} \neq u_{2}$.


The Design and Price of Information

## 5 Implications for Data Pricing

We discuss how to bring our model's results to bear on the design of real-world information products. We focus on offline and online brokers of big data-firms such as Acxiom, Nielsen, and Oracle-that sell information about individual consumers to business customers.

[^10]Data obtained from brokers is typically used to facilitate marketing efforts and to mitigate risks. ${ }^{13}$ Information used for marketing purposes is typically sold through data appends and marketing lists. Data appends reveal supplemental information about a firm's existing or potential customers, allowing the firm to place them into more precise market segments. For instance, all major data brokers offer data management platforms (DMPs)-customized software that enables websites to track their users and integrate their own data with 3rdparty data. Most risk-mitigation data products offered by credit rating agencies are also of this kind. Conversely, marketing lists facilitate targeted advertising to new consumers. Advertisers can choose whether to acquire standard lists of potential consumers with prespecified sets of characteristics, or to customize their list of desired consumer attributes.

Our model of selling supplemental information is best suited to analyzing data appends. ${ }^{14}$ We therefore describe these information products in greater detail, and we comment on marketing lists and other data services at the end of this section. For concreteness, consider the following examples of data appends.

- Oracle ID-Graph tracks firms' customers across several devices, augmenting the data collected on the firms' websites with behavioral observations from different sources. ${ }^{15}$
- Email Intelligence by TowerData attaches demographic, income, intent, and purchase information to a merchant's own list of email addresses. ${ }^{16}$
- The credit reporting agency Equifax offers its business customers (e.g., banks and credit card companies) a risk-mitigation product called Undisclosed Debt Monitoring. This product tracks an individual borrower to identify new negative information that arrives between the original loan approval and the closing date. ${ }^{17}$

Each of these products is available in several versions, which differ mainly in terms of the number of informative variables the seller discloses to the buyer. For instance, ID-Graph and Email Intelligence allow buyers to customize their queries to the database (e.g., a consumer's age group, income level, interests, and intent). Similarly, the versions of Undisclosed Debt

[^11]Monitoring differ in terms of the number of potentially negative events (e.g., late payments, credit inquiries, bankruptcy filings) that are monitored and disclosed.

The most common buyers of data products are marketing and advertising firms, lenders and financial services firms, and retail companies (Federal Trade Commission, 2014). The data buyers differ along several dimensions, including their ability to process data and the richness of their action space. ${ }^{18}$ Any of these dimensions of heterogeneity can lead to interesting sorting patterns of buyers into product versions. Here we focus on the differences in the strength of the buyers' priors, i.e., in the availability of initial information.

Our main structural results (Propositions 1 and 2) derive the properties of the distribution of states and signals that are associated with monopolistic screening. In particular, our results impose restrictions on the support of the conditional distribution of signals in a given state. In order to leverage these insights to evaluate and inform the design of data appends, it is useful to rephrase the design of a statistical experiment in terms of hypothesis testing. Our structural results identify the types of statistical errors incurred by data buyers as a consequence of market power in the sale of information. ${ }^{19}$

For the present purpose, consider a data buyer, such as an advertiser or a lender. The data buyer wishes to test the null hypothesis "target with an ad" or "grant a loan" for a specific consumer, i.e., to distinguish a null hypothesis $H_{0}$ from an alternative hypothesis $H_{1}$. The data buyer is a Bayesian decision maker with a prior distribution over the hypotheses. He can take one of two actions, each of which is optimal under the respective hypothesis.

The data seller has a test statistic whose distribution, conditional on the true state of the world, is given by $H_{0}$ or $H_{1}$ as in Figure 7. We assume that $H_{0}$ and $H_{1}$ satisfy the monotone likelihood ratio property. While the data seller could potentially disclose arbitrary information about the distribution of the test statistic, suppose she chooses to inform the data buyer if the statistic is above or below a certain threshold. The data buyer then chooses the corresponding action. We can represent this experiment as a statistical test of the null hypothesis $H_{0}=\left\{\omega_{2}\right\}$,

| $E$ | $s_{1}$ | $s_{2}$ |
| :--- | :--- | :--- |
| $\omega_{1}$ | $1-\beta$ | $\beta$ |
| $\omega_{2}$ | $\alpha$ | $1-\alpha$ |

[^12]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 7: Conditional Distributions of the Test Statistic

where $\alpha$ and $\beta$ denote the probability of a type I and type II statistical error, respectively. ${ }^{20}$
Given the information contained in the database of the seller in Figure 7, the set of feasible statistical tests is described by the area between the blue curve and the red line in the left panel of Figure 8. The blue curve identifies the loci of the least type I and type II statistical errors given the data available. As the data seller can always introduce noise into the test statistic, the set of feasible statistical tests is given by the entire area. In our model, we assumed that the data seller has complete information, and thus, the boundary is given by the blues lines in the right panel of Figure 8, which coincide with the $\alpha$ and $\beta$ axes. ${ }^{21}$

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 8: Feasible Information Structures


The Design and Price of Information

The central issue for the data seller is that she does not know the data buyer's prior beliefs and, hence, the buyer's willingness to pay for this information. She must therefore employ a richer mechanism to screen heterogeneous buyers. In particular, the seller offers the buyer a menu of binary ("pass/fail") tests. Each test reports the outcome "pass" when

[^13]the test statistic is below a particular threshold. Each test yields a different combination of type I and type II errors $(\alpha, \beta)$.

Our main structural result (Proposition 2) identifies systematic patterns in the optimal design of partially informative statistical tests. With binary states and actions, and no constraints on $(\alpha, \beta)$, the seller induces some buyers to make either type I or type II errors, but not both. The logic underlying this result extends to the case of a partially informed seller: an identical argument establishes the stronger result that all optimal tests lie on the lower boundary of the feasible set in Figure 8 (left). Separation in the optimal menu is then supported by the differences in the error structure of each test and by the buyers' heterogeneous preferences over statistical errors.

A concrete implication of the optimality of non-dispersed experiments is that no product in an optimal menu should add unbiased noise to the seller's information. Adding idiosyncratic noise can be useful when multiple buyers compete in a downstream market, as in Admati and Pfleiderer (1986), or when it is important to preserve the anonymity of the data, as in the differential privacy literature (Dwork, 2008). In the absence of these concerns, our results suggest that this practice reduces the seller's revenues. Instead, optimal experiments should minimize the type-II error for any level of type-I error (as in Figure 8). For example, if the data buyer faces a binary advertising decision, the optimal menu should lead to excessively broad or to excessively narrow campaigns, i.e., advertising to a subset of high-value consumers but not to low-value consumers, or to all high-value and to some low-value consumers.

Interestingly, in practice, none of the data products described earlier appear to introduce noise into the data. For example, credit rating agencies do not offer both precise and noisy versions of the same information (e.g., computing a consumer's credit score on the basis of more or less detailed data). Instead, information is degraded by revealing only a portion of the available data to the buyer. We now discuss how omitting explanatory variables can be seen as implementing our optimal mechanism under specific conditions on the buyer's decision problem and on the seller's data.

Consider the following stylized description of the above-mentioned Equifax product Undisclosed Debt Monitoring. A credit rating agency collects $K$ binary characteristics $x_{k} \in\{0,1\}$ for each potential borrower. Each realization $x_{k}=1$ corresponds to a "red flag" on the borrower's record. Suppose, for simplicity, that the data buyer's (the lender's) payoff-relevant state is $\omega=\mathbf{1}\left[\Sigma_{k=1}^{K} x_{k}=0\right]$, i.e., it is optimal to grant a loan if and only if there are no red flags. The data seller knows the borrower's vector of characteristics $\left(x_{1}, \ldots, x_{K}\right)$, while the data buyer privately knows the (identical) probability distribution of each binary characteristic $x_{k}$. In this case, hiding the realized value of any characteristic $x_{k}$ induces the data
buyer to grant a loan to an unqualified borrower with positive probability, but no qualified borrower is ever turned down.

In this example, the data buyer incurs only one type of statistical error. Thus, omitting variables can be part of an optimal design if subsets of the seller's data contain conclusive evidence against at least one state. In other settings, withholding data may introduce fullsupport noise into the experiment. This is the case any time a hidden variable could have overturned the signals based on the revealed data and changed the buyer's action in any direction. For example, it is not difficult to construct examples in which a broker (e.g., Oracle or Nielsen) provides demographic, income and interest information about consumers, but using demographic and income data as proxies for interest leads the data buyer to incur both types of statistical errors.

Data brokers offer several other information products in addition to data appends. For instance, marketing lists are queries to a broker's database of individual consumer records. Such queries allow advertisers to identify potential customers with a set of desired characteristics. ${ }^{22}$ Thus, a marketing list can be viewed as a statistical experiment where each signal indicates (possibly with error) a pre-specified consumer segment. From a modeling standpoint, however, selling a list is very different from appending data to the buyer's existing information. In particular, when selling a list, the seller is able to charge a price contingent on the signal realization. ${ }^{23}$ Marketing lists can also be sold contextually to advertising space. This occurs when a data provider (e.g., Bluekai) partners with a publisher of space (Doubleclick) and charges the advertiser a price per impression in addition to the cost of the advertising space. Thus the price of the data augments the cost of the ads. In other words, the buyer's advertising decision is contractible. ${ }^{24}$