Disclosure and Pricing of Attributes

Lemma 2 clarifies that the buyer incentive structure leads the seller to minimize trade probability whenever it maintains a trade surplus. If $(\mathcal{X}(\theta), \mathcal{Q}(\theta))$ lies in the interior of $\mathcal{F}$, then the seller can reduce the total trade probability while keeping the attribute surplus the same. Such change scales up the attribute expectation conditional on the trade recommendation without affecting the trade surplus. If the seller accompanies this change with a revenue-preserving increase in the object price, then the on-path payoff of type $\theta$ remains the same. However, a higher object price renders deviations to this type's item less appealing.

Theorem 1. (Optimal Disclosure)
There exists an optimal responsive menu in which every experiment is a linear disclosure.
The theorem is an immediate corollary of Lemmas 1 and 2 and does not require any assumptions about the attribute or type distributions. To appreciate this result, it is instructive to compare the allocation distortions driven by monopoly power in cases of complete and incomplete information about the object.

First, consider the situation in which the object's attributes are commonly known to be $x_{0}$ so that there is no scope for information control. If the seller could observe the type, she would allocate the object efficiently, selling it if and only if $v(\theta) \geq 0$, and would extract full surplus. If the seller could not observe the type, she could attempt to screen by designing a menu of items varying in sale probabilities and prices. This screening is not beneficial as famously resolved by Myerson (1981). Each type $\theta$ is assigned a virtual valuation $\hat{v}(\theta)$ and, under standard regularity conditions, an object is sold if and only if the virtual valuation is positive:

$$
\hat{v}(\theta) \geq 0 .
$$

This allocation is typically inefficient since the virtual valuation differs from the true valuation.

Compare this scenario to the current situation in which the object's attributes are uncertain. If the seller could observe the type $\theta$, then, according to Bergemann and Pesendorfer (2007), she would inform type $\theta$ whether his valuation is positive, charge a maximal acceptable price, and extract the full surplus. If the seller could not observe the type, she could design a menu varying in information content and prices. By Theorem 1, the optimal allocation distortion would be remarkably similar to the case of complete information about the object. Each type $\theta$ is assigned a virtual type $\hat{\theta}(\theta)$. An object is sold when the buyer's virtual valuation is above a specified threshold, possibly with randomization on the boundary:

$$
\hat{v}(\theta)=\hat{\theta}(\theta) \cdot x \geq \alpha_{0}(\theta) .
$$

This allocation is also typically inefficient but is now with an additional distortion since the threshold may differ from 0.

Uniqueness Theorem 1 establishes the existence of an optimal responsive menu with each experiment being a linear disclosure. One might wonder whether there exist optimal responsive menus with non-linear disclosure rules. The proof argument does not preclude this possibility: although the adjustment to linear disclosure strictly relaxes constraints (17) and (18), it preserves the constraint (16); therefore, hypothetically, the relaxed constraints may be not exploited for additional revenue. It can be shown that with only two types, this relaxation can be exploited; thus, linear disclosure is uniquely optimal. With many types, the answer is less clear as the structure of incentive constraints is more complex.

At the same time, observe that the adjustment to linear disclosure strictly decreases the trade probability. Consequently, when the seller faces trading costs or, equivalently, attaches some value to the object, however small, linear disclosures are uniquely optimal.

General Payoffs The arguments behind Theorem 1 might seem to heavily rely on the linearity of the buyer's valuation function. However, Theorem 1 places no structural assumptions on the attribute distribution. This crucial feature allows for extending the optimal disclosure characterization beyond linear environments by carefully defining the relevant attributes. In particular, consider a general valuation function $v(\theta, x)$ and define auxiliary attributes to coincide with the valuations of different buyer types. In this auxiliary formulation, each type's valuation is linear in the relevant attribute, and Theorem 1 applies.

Corollary 1. (General Payoffs)
Let $X$ be an arbitrary attribute set, $v(\theta, x)$ be a general valuation function, and $|\Theta|<\infty$.

Then, there exists an optimal menu in which every experiment has a linear form, i.e., for any experiment, there exist $\alpha: \Theta \rightarrow \mathbb{R}$ and $\alpha_{0} \in \mathbb{R}$, not all zeros, such that:

$$
q(x)= \begin{cases}1, & \text { if } \sum_{\theta \in \Theta} \alpha(\theta) v(\theta, x)>\alpha_{0}, \\ 0, & \text { if } \sum_{\theta \in \Theta} \alpha(\theta) v(\theta, x)<\alpha_{0} .\end{cases}
$$

Note the difference between the definitions of a linear form (25) and a linear disclosure (22). A linear disclosure operates in a space of attributes and can be specified independently of a buyer. A linear form, in contrast, operates in the space of valuations of different buyer types. The richer the buyer heterogeneity is, the more complex the linear form can be. However, in the case of linear payoffs the linear form always reduces to a linear disclosure.

Corollary 1 allows for a characterization of the classes of optimal disclosure rules in general environments with preferences that allow for bliss points or risk aversion. To illustrate, consider the case of location payoffs with the buyer's type capturing his bliss point in the attribute space $X \subseteq \mathbb{R}^{J}, \Theta \subseteq \mathbb{R}^{J}, v_{0}>0$, and:

$$
v(\theta, x)=v_{0}-(x-\theta)^{2} .
$$


Disclosure and Pricing of Attributes

Smolin, Alex. “Disclosure and Pricing of Attributes.” RAND Journal of Economics 54, no. 4 (2023): 570–597.

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

Let there be two types $\theta_{1}, \theta_{2} \in \mathbb{R}^{J}$. Assume that $X$ is bounded and for all $x \in X$, the types' valuations are positive. Optimal disclosure rules can be identified as follows. First, Proposition 2 can be applied to establish that one type is offered no disclosure and always buys. Second, by Corollary 1, the other type is offered a linear form (25) that informs whether a linear combination of valuations $v\left(\theta_{1}, x\right)$ and $v\left(\theta_{2}, x\right)$ is above or below a specified threshold. Generically, this experiment is a neighborhood disclosure: it informs whether the attribute vector is sufficiently close to a virtual type $\hat{\theta}$ located on the line that connects $\theta_{1}$ and $\theta_{2}$.


Disclosure and Pricing of Attributes

## 5 Design of Pricing Mechanisms

I proceed by studying pricing in the revenue-maximizing mechanisms. I identify a general class of optimal pricing mechanisms in the case of a single attribute. With many attributes, I am able to identify key trade-offs and characterize optimal mechanisms for specific classes of buyer types.

From now on, I assume that all types and attributes are positive, $X \subseteq \mathbb{R}_{+}^{J}, \Theta \subseteq \mathbb{R}_{+}^{J}$. In this scenario, it is commonly known that there are positive gains from trade. It makes it possible to ignore the efficiency role of disclosure and to focus solely on its screening effects.


Disclosure and Pricing of Attributes

### 5.1 Single Attribute

I begin with the basic case of a single attribute, $J=1, X \subseteq \mathbb{R}_{+}$. The buyer's type is one dimensional, $\Theta \subseteq \mathbb{R}_{+}$, and the buyer's ex post valuation is

$$
v(\theta, x)=\theta x .
$$

This setting features only vertical type heterogeneity. I establish that providing no attribute information is optimal in this case. The argument starts by considering a more beneficial setting for the seller, in which she can condition payment and allocation directly on the attribute realization, as do Eső and Szentes (2007). In this case, the revelation principle applies, and I can focus on the direct mechanisms in which all payments are front loaded: the buyer reports his type $\theta$, pays the upfront payment $r(\theta)$, and the trade occurs with probability $q(x, \theta)$. The relevant variable is the single-dimensional attribute surplus:

$$
\mathcal{X}(\theta)=\int_{x \in X} x q(x, \theta) \mathrm{d} G(x)
$$

which can be anywhere between 0 and $\mathbb{E}[x]$. I can then rewrite the seller's problem as:

$$
\begin{aligned}
& \max _{r(\theta), 0 \leq \mathcal{X}(\theta) \leq \mathbb{E}[x]} \int_{\theta \in \Theta} r(\theta) \mathrm{d} F(\theta), \\
& \text { s.t. } \theta \mathcal{X}(\theta)-r(\theta) \geq \theta \mathcal{X}\left(\theta^{\prime}\right)-r\left(\theta^{\prime}\right), \quad \forall \theta, \theta^{\prime} \in \Theta, \\
& \theta \mathcal{X}(\theta)-r(\theta) \geq 0, \quad \forall \theta \in \Theta .
\end{aligned}
$$

This problem is analogous to those of Myerson (1981) and Riley and Zeckhauser (1983), with the attribute surplus replacing the allocation probability. The optimal allocation $\mathcal{X}(\theta)$ is a step function equal to 0 for $\theta<\theta^{*}$ and to $\mathbb{E}[x]$ for $\theta \geq \theta^{*}$. The optimal upfront payment $r(\theta)$ is equal to 0 for $\theta<\theta^{*}$ and to $r^{*}=\theta^{*} \mathbb{E}[x]$ for $\theta \geq \theta^{*}$.

The argument concludes by noting that the optimal mechanism can be implemented by providing no disclosure and charging a price of $r^{*}$ for the object. This posted price mechanism is feasible in the original problem with private disclosure and is therefore also optimal there.

Theorem 2. (Single Attribute)
If $J=1, X \subseteq \mathbb{R}_{+}$, and $\Theta \subseteq \mathbb{R}_{+}$, then an optimal menu is a posted price mechanism with no disclosure, i.e., $r(\theta) \equiv 0, E(\theta) \equiv \underline{E}$.

Simple intuition underlies the optimality of no disclosure if the seller can only use a nondiscriminatory mechanism that consists of a single experiment followed by a posted price. Consider an arbitrary disclosure rule. Any signal realization $s$ scales the demand propor-
tionally to the attribute expectation $\mathbb{E}[x \mid s]$. If the seller could observe this realization, she would optimally charge a scaled price and obtain scaled revenue. Importantly, the induced allocation would not depend on the realization $s$. Since any expectation is a martingale, the seller would serve the same population at, on average, the same price. The seller can do equally well using a posted price with no disclosure.

Although intuitive, this argument does not consider discriminatory schemes with upfront payments. Theorem 2 confirms that no disclosure is optimal, even if the seller can use those schemes. Notably, this result requires no assumptions on the type or attribute distributions beyond the common knowledge of positive trade gains.

Remark 1. (Attribute Index) The same argument can be applied to the case of many attributes, $J>1$, if the attributes and the types enter the valuation function through onedimensional indices:

$$
v(\theta, x)=\psi(\theta) \phi(x),
$$

for $\psi, \phi: \mathbb{R}^{J} \rightarrow \mathbb{R}_{+}$. For example, the result applies if all types belong to a ray $\Theta=\left\{\beta \theta_{0}\right\}_{\beta \in \mathbb{R}_{+}}$ for some direction vector $\theta_{0} \in \mathbb{R}_{+}^{J}$. In this case, the indices can be defined as $\psi(\theta)=\beta(\theta)$ and $\phi(x)=\theta_{0} \cdot x$. $\square$

Remark 2. (Uniqueness) The no-disclosure mechanism might not be uniquely optimal. In fact, the analysis of Eső and Szentes (2007) can be applied to show that, if the type distribution $F$ has a monotone hazard rate property, then full disclosure is also optimal. ${ }^{16}$ However, it must be accompanied by a complex structure of upfront payments and object prices. $\square$


Disclosure and Pricing of Attributes

### 5.2 Single-Minded Buyer

I proceed with the case of multiple product attributes, $J \geq 2$. This case is qualitatively different because it may feature horizontal heterogeneity across buyer types: type $\theta$ may have a higher value for the object than type $\theta^{\prime}$ for some attribute realization, yet a lower value than type $\theta^{\prime}$ for another attribute realization. Since the seller should aim to allocate to object to the buyers who are willing to pay the most, she may need to provide attribute information through disclosure. To address such settings, I introduce and study a tractable type structure which allows to capture both vertical and horizontal heterogeneity of tastes.

I call a type single minded if he values only one attribute. For a generic single-minded type, the vector $\theta$ places a positive weight on only one dimension:

$$
\theta=\left(0, \ldots, 0, \theta_{j}, 0, \ldots, 0\right) .
$$

[^10]Thus, single-minded types allow for a simpler notation. I can represent the types by $J$ attribute cohorts $\Theta_{j}$ such that all types within the same cohort value the same attribute. I slightly abuse the notation and let the type subscript identify the attribute cohort and the type value identify the valuation intensity, so that $\Theta_{j} \subseteq \mathbb{R}_{+}$and

$$
v_{j}\left(\theta_{j}, x\right)=\theta_{j} x_{j} \quad \forall j, \theta_{j} \in \Theta_{j} .
$$

I denote the frequency of a cohort $\Theta_{j}$ by $f\left(\Theta_{j}\right)$ and the cumulative type distribution within the cohort by $F_{j}\left(\theta_{j}\right)$.

A buyer is single minded if all types $\theta \in \Theta$ are single minded and the attribute values are independently distributed such that $x_{j} \sim G_{j}$ and $G(x)=\times_{j} G_{j}\left(x_{j}\right) .{ }^{17}$ The independence requirement is substantive. Without it, any buyer can be viewed as being single minded by redefining the attributes as in the proof of Corollary 1.

If the buyer is single minded, then the seller knows that the buyer values only one of many independent attributes but does not know which one or the strength of the preference. Valuations of any two types are either perfectly correlated or independent.

Remark 3. (Customizable Good) The setting of a single-minded buyer resembles a unitdemand multi-good monopolist problem in which each buyer type values only one of the goods. ${ }^{18}$ One difference between these problems is that I allow the seller to provide additional information to the buyer. Another difference is that in the multi-good monopolist problem a type who deviates across cohorts is guaranteed to obtain no value; therefore, the problem can effectively be separated into several single-good problems. In contrast, in my setting, when deviating across cohorts, the buyer can at least obtain the ex ante value of the relevant attribute, which links the problems together. At the same time, the setting of a single-minded buyer admits an interpretation of a sale of a customizable good. In this interpretation, the good that admits several possible configurations and the seller can inform the buyer about them. The buyer values only a single configuration and can set the configuration freely but only one time after the purchase. The question of optimal design can then be translated into what configuration information the seller provides and how this choice interacts with pricing. $\square$

If the buyer is single minded, then the class of optimal experiments can be narrowed. Because attributes are independently distributed, a type $\theta_{j} \in \Theta_{j}$ values only information about attribute $j$. This observation suggests an optimal way to screen single-minded types: if the buyer reports type $\theta_{j} \in \Theta_{j}$, then the seller should provide information only about

[^11]attribute $j$. Providing any other information would make misreporting more appealing without adding value for truth telling. At the same time, a linear disclosure informative only about attribute $j$ is a binary monotone partition defined on this attribute.

Proposition 3. (Directional Disclosure)
If the buyer is single minded, then there exists an optimal menu such that an experiment $E_{j}\left(\theta_{j}\right)$ is a binary monotone partition of attribute $j$.

It follows that an optimal experiment $E_{j}\left(\theta_{j}\right)$ can be characterized by its threshold $\alpha_{0 j}\left(\theta_{j}\right)$ so that it informs the buyer whether attribute $j$ is above or below this threshold. Incentive compatibility requires the buyer to purchase at higher attributes; thus, the attribute surplus can be written as:

$$
\mathcal{X}_{j}\left(\theta_{j}\right)=\int_{\alpha_{0 j}\left(\theta_{j}\right)}^{\infty} x_{j} \mathrm{~d} G_{j}\left(x_{j}\right)
$$

The attribute surplus can take any value between 0 and $\mathbb{E}\left[x_{j}\right]$. The corresponding total trade probability can be written as an increasing and convex function $\mathcal{Q}_{j}\left(\mathcal{X}_{j}\right)$.

In what follows, I assume that that all attribute cohorts admit an upper bound, $\Theta_{j}=$ $\left[0, \bar{\theta}_{j}\right]$, and $\theta_{j}$ are continuously distributed over $\Theta_{j}$ according to distribution function $F_{j}$ with the monotone hazard rate property. The seller's problem is to design the tariff and the attribute surplus functions, $r_{j}\left(\theta_{j}\right), p_{j}\left(\theta_{j}\right)$, and $\mathcal{X}_{j}\left(\theta_{j}\right)$ to maximize her objectives and can be written as follows (cf. Kolotilin et al. (2017)):


Disclosure and Pricing of Attributes

$$
\begin{gathered}
\max _{\left\{r_{j}\left(\theta_{j}\right), \mathcal{X}_{j}\left(\theta_{j}\right), p_{j}\left(\theta_{j}\right)\right\}} \sum_{j=1}^{J} f\left(\Theta_{j}\right) \int_{\theta_{j} \in \Theta_{j}}\left(r_{j}\left(\theta_{j}\right)+\mathcal{Q}_{j}\left(\theta_{j}\right) p_{j}\left(\theta_{j}\right)\right) \mathrm{d} F_{j}\left(\theta_{j}\right) \\
\text { s.t. } \theta_{j} \mathcal{X}_{j}\left(\theta_{j}\right)-p_{j}\left(\theta_{j}\right) \mathcal{Q}_{j}\left(\theta_{j}\right)-r_{j}\left(\theta_{j}\right) \geq\left(\theta_{j} \mathcal{X}_{j}\left(\theta_{j}^{\prime}\right)-p_{j}\left(\theta_{j}^{\prime}\right)\right) \mathcal{Q}_{j}\left(\theta_{j}^{\prime}\right)-r_{j}\left(\theta_{j}^{\prime}\right), \forall j, \theta_{j}, \theta_{j}^{\prime} \in \Theta_{j}, \\
\theta_{j} \mathcal{X}_{j}\left(\theta_{j}\right)-p_{j}\left(\theta_{j}\right) \mathcal{Q}_{j}\left(\theta_{j}\right)-r_{j}\left(\theta_{j}\right) \geq \theta_{j} \mathbb{E}\left[x_{j}\right]-p_{k}\left(\theta_{k}\right)-r_{k}\left(\theta_{k}\right), \forall j, k, \theta_{j} \in \Theta_{j}, \theta_{k} \in \Theta_{k}, \\
\theta_{j} \mathcal{X}_{j}\left(\theta_{j}\right)-p_{j}\left(\theta_{j}\right) \mathcal{Q}_{j}\left(\theta_{j}\right)-r_{j}\left(\theta_{j}\right) \geq 0, \mathcal{X}_{j}\left(\theta_{j}\right) \geq \mathcal{Q}_{j}\left(\theta_{j}\right) \mathbb{E}\left[x_{j}\right], \forall j, \theta_{j} \in \Theta_{j},
\end{gathered}
$$

where $\mathcal{Q}_{j}\left(\theta_{j}\right) \equiv \mathcal{Q}_{j}\left(\mathcal{X}_{j}\left(\theta_{j}\right)\right) \forall j, \theta_{j} \in \Theta_{j}$. This problem resembles a collection of onedimensional mechanism design problems, one per attribute, with the following important differences. First, each item in this problem features both horizontal and vertical components. The upfront payments $r_{j}\left(\theta_{j}\right)$ are purely vertical-all types value them the same. In contrast, the attribute surpluses $\mathcal{X}_{j}\left(\theta_{j}\right)$ are only valuable to types from cohort $\Theta_{j}$. The object prices $p_{j}$ are mixed since they are paid only if the type decides to trade. Second, the problem features nonlinear terms $p_{j}\left(\theta_{j}\right) \mathcal{Q}_{j}\left(\mathcal{X}_{j}\left(\theta_{j}\right)\right)$ and $\mathcal{Q}_{j}\left(\mathcal{X}_{j}\right)$.

Because of these differences, I cannot apply standard mechanism design techniques. Instead, I solve the problem via a sequence of simplifications. In the first step, I observe that
using upfront payments is detrimental. For any $r_{j}\left(\theta_{j}\right)>0$, the seller can reduce the transfer and increase $p_{j}\left(\theta_{j}\right)$ while keeping the total expected transfer $r_{j}\left(\theta_{j}\right)+\mathcal{Q}_{j}\left(\theta_{j}\right) p_{j}\left(\theta_{j}\right)$ the same. This change does not affect the utilities of truth-telling types or the seller's revenue; however, it renders misreporting less appealing. Intuitively, by shifting the expected transfer toward the object price, the seller better discriminates against the types who would always purchase the object.

In the second step, I use standard mechanism design arguments to show that incentive compatibility within the same cohort implies that $\mathcal{X}_{j}\left(\theta_{j}\right)$ is nondecreasing in $\theta_{j}$. That is, higher types must trade with a higher probability but lower conditional expectations of the relevant attribute. Moreover, the expected transfers can be derived from the attribute surplus functions. This allows to rewrite the seller's problem solely as a choice over attribute surplus functions $\mathcal{X}_{j}\left(\theta_{j}\right)$.

Lemma 3. The seller's problem can be written as

$$
\begin{aligned}
& \quad \max _{\left\{\mathcal{X}_{j}\left(\theta_{j}\right)\right\}_{j=1}^{J}} \sum_{j=1}^{J} f\left(\Theta_{j}\right) \int_{0}^{\bar{\theta}_{j}}\left(\theta_{j}-\frac{1-F_{j}\left(\theta_{j}\right)}{f_{j}\left(\theta_{j}\right)}\right) \mathcal{X}_{j}\left(\theta_{j}\right) \mathrm{d} F_{j}\left(\theta_{j}\right) \\
& \text { s.t. } \quad \mathcal{X}_{j}\left(\theta_{j}\right) \text { is non }- \text { decreasing }, \mathcal{X}_{j}\left(\theta_{j}\right) \in\left[0, \mathbb{E}\left[x_{j}\right]\right], \\
& \quad \int_{0}^{\bar{\theta}_{j}} \mathcal{X}_{j}\left(\theta_{j}\right) d \theta_{j} \geq \bar{\theta}_{j} \mathbb{E}\left[x_{j}\right]-\underline{p}\left(\mathcal{X}_{1}(\cdot), \ldots, \mathcal{X}_{J}(\cdot)\right) \quad \forall j=1, \ldots, J .
\end{aligned}
$$

The objective function and the monotonicity constraints capture the incentive-compatibility constraints within each attribute cohort. The integral constraints capture the incentivecompatibility constraints between different cohorts. In particular, these constraints require that the highest type within each cohort not want to purchase the object at the minimal price present in the menu $\underline{p}$.

In an optimal menu, the minimal price must be offered to the highest types. Toward a contradiction, assume that for some cohort $\Theta_{j}$, a neighborhood of $\bar{\theta}_{j}$ is not offered the minimal price. Then, these types are not imposing externalities on other cohorts through the integral constraint. Moreover, to not go for the lowest price, these types should be offered some disclosure so $\mathcal{X}_{j}\left(\theta_{j}\right)<\mathbb{E}\left[x_{j}\right]$. It leads to a contradiction: the seller could marginally increase $\mathcal{X}_{j}\left(\theta_{j}\right)$ for these types, improving the revenue.

This observation allows for the stating of a relaxed problem in which the monotonicity and integral constraints are dropped but all high types are required to be offered a given price. If the type distributions have the monotone hazard rate property, then the solution to the problem is a collection of single-step functions. The solution corresponds to only one item per attribute cohort. It satisfies the original constraints and therefore solves the original problem.

Theorem 3. (Optimal Menu, Single-Minded Buyer)
If the buyer is single minded and the type distributions have the monotone hazard rate property, then in an optimal responsive menu for all $j=1, \ldots, J$ and $\theta_{j} \in \Theta_{j}: r\left(\theta_{j}\right)=0$, $p\left(\theta_{j}\right)=p$, and $E_{j}\left(\theta_{j}\right)=E_{j}$ where $E_{j}$ is a binary monotone partition of $x_{j}$.


Disclosure and Pricing of Attributes

I emphasize the simplicity of both pricing and disclosure components of the optimal mechanism. With regard to pricing, the price for the object is the same for all types and the price of information is nil. With regard to disclosure, the mechanism admits a nondiscriminatory indirect implementation: the seller can simply inform the buyer whether each attribute is above the corresponding threshold and post a fixed price for the object. Because each type values only one attribute and the attributes are independent, each type will use only information about a relevant attribute when deciding whether to buy the object. This simplicity is not a consequence of some exogenous requirement but rather is a feature of a revenue-maximizing mechanism.

Furthermore, my analysis also provides a partial characterization in the case of general distributions $F_{j}$. The first statement of Theorem 3 remains the same-upfront payments are not used with a single-minded buyer. However, the second and the third statements must be modified since an optimal menu may feature limited price discrimination. In particular, the arguments of Samuelson (1984) can be applied to limit the number of optimal items to two per cohort. That is, the highest types are still offered the unique minimal lowest price but per each attribute cohort, there could be one more item that targets lower types.


Disclosure and Pricing of Attributes

### 5.3 Differentiated Types and Separate Persuasion

A notable limit case of the previous section is the case in which each attribute cohort is a singleton, $\Theta_{j}=\left\{\theta_{j}\right\}$. In this case, there is no vertical within-attribute heterogeneity, and the number of types equals the number of attributes $|\Theta|=J$. Any two different types $\theta, \theta^{\prime} \in \Theta$ are orthogonal to one another as vectors in $\mathbb{R}^{J}$; accordingly, this case can be viewed as the setting of orthogonal types. Without loss of generality, all type intensities can be set equal to one, $\theta_{j} \equiv 1$ for all $j$.

By the arguments analogous to those in the previous section, an optimal mechanism is a free-of-charge disclosure followed by a single posted price. Finding optimal experiments is straightforward. The seller should provide minimal information sufficient to convince the buyer to make a purchase at the price posted. If the type $\theta_{j}$ is ex ante sufficiently optimistic, $\mathbb{E}\left[x_{j}\right] \geq p$, then the seller should provide no attribute information, $E_{j}=\underline{E}$. Otherwise, the seller should increase the type's expectation up to the object price.

The corresponding optimal mechanism is illustrated in Figure 4. The mechanism is simple

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 4: Optimal mechanism in the case of orthogonal types. Green indicates attribute regions in which a purchase recommendation is sent for the types with partial disclosure. Blue indicates the types' rent conditional on a trade, for the types with no disclosure. Attributes are ordered by increasing ex ante expectations.

and features a "no distortions at the top, no rents at the bottom" property: All types with the ex ante valuation above the optimal price always buy the object, whereas all other types are indifferent to participating in the mechanism. In this way, "the top" and "the bottom" are not single types, as is typical in mechanism design problems, but are two type classes that partition the type space.

The case of orthogonal types admits a clean illustration of an optimal mechanism and, importantly, suggests a generalization of the current analysis beyond a single-minded buyer. Toward this generalization, consider arbitrary attribute distribution $G(x)$ and valuation function $v(\theta, x)$. For a given price $p$, define a separate persuasion mechanism $M^{S P}(p)$ as a direct menu in which $r^{S P}(\theta)=0, p^{S P}(\theta)=p$, and $E^{S P}(\theta)$ recommends to trade when $v(\theta, x)$ is above a threshold; the threshold is chosen so that $\mathbb{E}\left[v(\theta, x) \mid E^{S P}(\theta), s^{+}\right]=$ $\max \{p, \mathbb{E}[v(\theta, x)]\}$. That is, the seller fixes an object price and for each type provides minimal valuation information to persuade him to make a purchase.

This mechanism is one with discriminatory disclosure that mimics the optimal menu for orthogonal types. If a type reports truthfully, then he is either left with no rents or is provided with no information; in both cases, the experiment brings no value to him. The mechanism may be incentive compatible or not-it depends on whether some types may benefit from information offered to other types. Denote with $p^{*}$ a price that maximizes the revenue, if the buyer is assumed to report his type truthfully.

Theorem 4. (Separate Persuasion)
If the mechanism $M^{S P}\left(p^{*}\right)$ is incentive compatible, then it is an optimal mechanism.

This proposition is based on the observation that a separate persuasion mechanism solves a relaxation of the original problem, in which misreporting types are required to always purchase the object. If incentive compatible, the mechanism satisfies the relaxed constraints and also solves the original problem. ${ }^{19}$

Let me highlight the significance of Theorem 4. A priori, there is no reason to expect the separate persuasion mechanism to be optimal since it does not use all of the flexibility available to the seller: it does not price information and it does not vary the price of the object. Furthermore, the mechanism does not extract the full surplus because it generally induces inefficient allocation. Rather, the result builds on and generalizes the analysis of orthogonal types.

Theorem 4 can be used to find optimal mechanisms in environments in which the types are sufficiently differentiated. For the following statement, let the valuation function be linear (1) and the attributes be independently and continuously distributed.

Corollary 2. (Differentiated Types)
Fix the number of types $J$ and their respective frequencies. Let $p^{*}$ be a uniquely optimal price in a separate persuasion mechanism for some orthogonal types $\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right)$. If $p^{*} \neq$ $\mathbb{E}\left[v\left(\hat{\theta}_{j}, x\right)\right]$ for all $j$, then there exists $\varepsilon>0$ such that for any type profile $\left(\theta_{1}, \ldots, \theta_{J}\right)$ with $\left\|\theta_{j}-\hat{\theta}_{j}\right\| \leq \varepsilon$ for all $j$ a separate persuasion mechanism is optimal.

According to the corollary, separate persuasion mechanisms are generically optimal even if the types value several attributes as long as they place most of the weight on distinct attributes. In these cases, the seller does not benefit from price discrimination but does benefit from information discrimination.

Note the structure of optimal disclosure when the types place weights on several attributes, however small. Even if the attributes are independent, providing information about each of them separately is not optimal. Instead, the disclosure leans towards the tastes of a reported type. At the same time, the informational content regarding a given attribute decreases as the type attaches less weight to it; thus, if the tastes are sufficiently concentrated, then providing information only about the leading attribute is approximately optimal.

[^12]


Disclosure and Pricing of Attributes

## 6 Discussion


Disclosure and Pricing of Attributes

### 6.1 Pricing of Product Information

In the setting of Section 5, the seller cannot benefit from pricing the information that she provides-upfront payments can always be set to zero. Similarly, in reality, it is uncommon to price product information. This fact raises a question regarding why it might be not beneficial to price product information separately from the product itself.

Proposition 4. (Payment Backloading)
In any given responsive menu, $r(\theta)$ can be decreased, and $p(\theta)$ can be increased without any loss of revenue when (i) $\mathcal{Q}(\theta) \leq 1 / 2$ or (ii) no type is willing to act contrary to both recommendations if being offered the experiment $E(\theta)$.

Proposition 4 holds for any valuation function. To understand the logic behind it, take an item of type $\theta$ in a responsive menu and consider the effect of backloading its payment, i.e., reducing the information price and increasing the object price to preserve the expected payment of type $\theta$. By construction, this backloading preserves the incentives of the types who, if offered the item of type $\theta$, would like to follow the recommendations. At the same time, it changes the incentives of the types who would like to disobey recommendations because the object price is paid only if the object is actually purchased. In particular, the backloading relaxes the incentives of the types who would purchase the object more frequently than $\theta$ when faced with $E(\theta)$. Hence, the incentives are clearly relaxed for the types who always purchase the object, because $1 \geq \mathcal{Q}(\theta)$; but also, whenever $\mathcal{Q}(\theta) \leq 1 / 2$, the incentives are relaxed for the types who would like to act contrary to recommendations because in that case $1-\mathcal{Q}(\theta) \geq \mathcal{Q}(\theta)$. The incentives of the types who never purchase the object are captured by their individual rationality. As such, under the conditions of the proposition, the payments can be backloaded without any loss of revenue. In fact, this argument reveals that the second condition of the proposition need only hold for the types along the binding constraints.

By Proposition 4, a given experiment might need to be priced only if it induces a relatively frequent trade and some types would like to mismatch its recommendations. That is, a priced experiment should bring "bad news" by infrequently informing the buyer that the product is not worth its price. Moreover, this information should lead to a disagreement, with some type using it in the opposite fashion. In some cases, one can exclude the latter possibility.

Corollary 3. (No Information Pricing)
In an optimal menu, the price of an experiment $E$ can be set to zero if (i) $J>1$, attributes
are independent, $v(\theta, x)=\theta \cdot x, \Theta \subseteq \mathbb{R}_{+}^{J}$, and $E$ is a linear disclosure with $\alpha \in \mathbb{R}_{+}^{J}$ or (ii) $J=1, v(\theta, x)$ is increasing in $x$ for all $\theta \in \Theta$ and $E$ is a binary monotone partition.

In the cases of Corollary 3, the types agree on the ranking of their interim valuations after different signals of the experiment $E$. Therefore, no type likes to mismatch decisions with recommendations. The first case indicates that only comparative disclosures may need to be priced. The second case explains why binary monotone partitions, frequent in economic analyses, need not be priced.

At the same time, examples can be constructed in a general setting with sufficiently opposed types such that, in an optimal menu, the swapping constraint binds and the information should be priced to deter mismatching deviations.


Disclosure and Pricing of Attributes

### 6.2 Demand Transformation and Shrouded Attributes

As I discussed in Section 3, one way to think about attribute disclosure is in terms of its impact on the demand curve that the seller faces. This particular effect was studied by Johnson and Myatt (2006). They restrict their attention to disclosures that spread type valuations "uniformly." Such disclosures translate into global rotations of the demand curve. The authors show that in many settings, the optimal global rotations are extreme and correspond to either no disclosure or full disclosure: no disclosure is associated with a mass market characterized by a low price and high demand; full disclosure is associated with a niche market characterized by a high price and low demand.

In contrast, I show that attribute disclosure can rotate the demand curve locally (recall Figure 3). The local rotations correspond to partial disclosures that target specific types and, hence, affect the demand curve over a particular price segment. They can outperform full and no disclosure in both mass and niche markets. Multiple attributes are required for this result-recall that no disclosure remains optimal in a one-dimensional framework.

This idea of targeted disclosure provides additional justification for selective advertising and attribute shrouding, ubiquitous in practice (Gabaix and Laibson (2006)). Even if customers are perfectly rational, the seller may have incentives to suppress information about some attributes while providing information about the others. Intuitively, different attributes can appeal to different customer cohorts. At a given price, some cohorts may have to be persuaded to purchase the product while others may not have to be.

For example, the customer base of a smartphone company could consist of two main groups: high-value customers, who are primarily interested in reliability and the quality of customer service; and low-value customers, who are primarily interested in entertainment features such as the screen size and camera performance. A smartphone price will optimally
balance the respective cohort valuations. In the absence of additional information, the price will be acceptable for the high-value customers but not for the low-value customers. The advertising campaign could thus optimally focus on the entertainment features to persuade the low-value cohort, while suppressing the information about reliability and services.


Disclosure and Pricing of Attributes

### 6.3 Alternative Disclosure Settings

The multiattribute disclosure setting complements the existing one-dimensional models of disclosure and pricing by Eső and Szentes (2007) and Li and Shi (2017). The main difference, however, lies not in the multidimensionality per se but rather in what kinds of information the seller can provide.

Eső and Szentes (2007) study discriminatory mechanisms in a valuation-rank framework, in which disclosure corresponds to statements such as "Your valuation is in your y-th percentile," with y being the same for all types. The authors obtain two main qualitative results: first, they show that full information disclosure is generally optimal; and second, they show that the seller cannot benefit from conditioning the price on the disclosure realization. ${ }^{20}$

Formally, their seller informs the buyer about an orthogonal shock $\xi(\theta)$, defined as the type's valuation percentile. By construction, these percentiles are uniformly distributed:

$$
\xi(\theta) \sim U[0,1] \quad \forall \theta \in \Theta .
$$

The implicit assumption of the valuation-rank framework is that these shocks are equal, i.e., $\xi(\theta) \equiv \xi\left(\theta^{\prime}\right)$ for all $\theta, \theta^{\prime} \in \Theta$. However, despite having the same distribution, the shocks $\xi(\theta)$ are generally different random variables. This observation is crucial and is particularly evident in the multiattribute setting.

Consider the following example. Let there be two independently distributed attributes $J=2, x_{1} \sim U[0,1], x_{2} \sim U[0,2]$. Let there be two equally likely types: $\theta_{1}=(1,0)$ and $\theta_{2}=(0,1)$. The corresponding orthogonal shocks are $\xi\left(\theta_{1}\right)=x_{1}$ and $\xi\left(\theta_{2}\right)=x_{2} / 2$. Both $\xi\left(\theta_{1}\right)$ and $\xi\left(\theta_{2}\right)$ are uniformly distributed on $[0,1]$. However, they depend on different attributes and are thus independent from each other.

Consequently, neither of the qualitative results of Eső and Szentes (2007) holds in this example. The optimal full-disclosure mechanism can be calculated to be $r_{1}=r_{2}=1 / 2$, $p_{1}=p_{2}=0$ : anticipating information revelation, the seller would prefer to effectively sell the object in advance. The corresponding revenue is 1/2. The seller can do strictly better by providing partial private disclosure. Building on the results of Section 5.2, it can be shown

[^13]that an optimal mechanism informs the buyer, free of charge, whether the first attribute is above or below 1/2 and then follows with a posted object price of 3/4. This mechanism obtains revenue 9/16 > 1/2.

At the same time, the seller could further improve the revenue if she could condition the price directly on the disclosure realization. Consider the following mechanism. The seller provides full disclosure, observes the attributes, and chooses the price optimally given the realized valuation distribution. The corresponding revenue is:

$$
\Pi=\int_{0}^{1} \int_{0}^{2} \frac{1}{2} \max \left\{\min \left\{x_{1}, x_{2}\right\}, \frac{\max \left\{x_{1}, x_{2}\right\}}{2}\right\} \mathrm{d} x_{1} \mathrm{~d} x_{2}=\frac{29}{48}>\frac{9}{16} .
$$

This example also emphasizes the difference between the multiattribute setting and the setting of Li and Shi (2017). These authors study common value settings in which the types represent private information about the object. In their settings, information disclosure can be seen as a valuation-level disclosure that corresponds to statements such as "Your valuation is above x," with x being the same for all types. However, in general, attribute information affects the valuation of different types differently according to the valuation function. In the example above, any experiment informative only about attribute $x_{i}$ affects the valuation of only type $\theta_{i}$. Consequently, attribute information cannot be modeled as an experiment that informs the buyer directly about his valuation. This modeling would misrepresent the buyer incentives in terms of choice across experiments.

Overall, this discussion emphasizes the importance of explicitly modeling demand microstructure, i.e., how the consumer valuation is formed, in trade settings with information control.


Disclosure and Pricing of Attributes

## 7 Conclusion

I studied a monopolist who sells a multiattribute object to a privately informed buyer and showed that the seller can benefit from the disclosure of attribute information. The benefit comes through two channels. First, disclosure can be used as a screening device, leveraging the preferences of different buyer types for learning about different aspects of the object. Second, disclosure can lift the buyer's expectations and persuade him to buy the object at a higher price. Both channels are important. However, I show that in many settings screening is not beneficial and information should be disclosed partially and free of charge. In those settings, the choice of information content is more important than the choice of its pricing.

In this paper, I deliberately focused on the simplest model of pricing and information control. In practice, additional details might be important and should be considered. The
seller could be restricted in the kinds of information that she may provide. The buyer might feature heterogeneity in his ability to process data. The market could involve imperfect competition. Each of these extensions can be approached within the multiattribute disclosure framework that I have outlined.


Disclosure and Pricing of Attributes

## 8 Appendix

Proof of Proposition 1. Consider any menu $M=(r(i), E(i), p(i))_{i \in \mathcal{I}}$. For any type $\theta, M$ induces the allocation distribution $\mu(\theta): X \rightarrow \Delta(A), A=$ \{buy, not buy $\}$, the expected upfront payment $\hat{r}(\theta)$, and the expected object payment, conditional on a trade, $\hat{p}(\theta)$. Consider a direct responsive menu $M^{\prime}=\left(r^{\prime}(\theta), E^{\prime}(\theta), p^{\prime}(\theta)\right)$ with $r^{\prime}(\theta)=\hat{r}(\theta)$, $p^{\prime}(\theta)=\hat{p}(\theta)$, and $E^{\prime}(\theta)=(A, \mu(\theta))$. If all types are truthful and obedient, then $M^{\prime}$ results in the same allocation distribution and the same expected payments as $M$. At the same time, any deviation under $M^{\prime}$ is available to the buyer under $M$. Therefore, reporting truthfully and following the recommendations is incentive-compatible under $M^{\prime}$.

Proof of Proposition 2. Toward a contradiction, assume that a responsive menu $M=$ $(r(\theta), \mathcal{X}(\theta), \mathcal{Q}(\theta), p(\theta))_{\theta \in \Theta}$ is optimal, yet no type buys the object with probability one. Construct a new menu $M^{\prime}$ as follows. Select a type $\bar{\theta}$ with the highest expected payment $\bar{T}=r(\bar{\theta})+p(\bar{\theta}) \mathcal{Q}(\bar{\theta})$. Since $\Theta$ is finite, this type exists. Change this type's item to no disclosure followed by an object price as follows:

$$
\left(r^{\prime}(\bar{\theta}), \mathcal{X}^{\prime}(\bar{\theta}), \mathcal{Q}^{\prime}(\bar{\theta}), p^{\prime}(\bar{\theta})\right)=(0, \mathbb{E}[x], 1, \bar{T}+\bar{\theta} \cdot(\mathbb{E}[x]-\mathcal{X}(\bar{\theta}))) .
$$

Keep all other items the same. In this menu, type $\bar{\theta}$ chooses the new item and always buys the object. This strategy gives him exactly the same payoff as that of the original menu:

$$
\bar{\theta} \cdot \mathcal{X}^{\prime}(\bar{\theta})-p^{\prime}(\bar{\theta})=\bar{\theta} \cdot \mathcal{X}(\bar{\theta})-p(\bar{\theta}) \mathcal{Q}(\bar{\theta})-r(\bar{\theta}) .
$$

As $X \subseteq \mathbb{R}_{++}^{J}$, the uninformative experiment achieves a maximal attribute surplus, $\mathbb{E}[x]=$ $\int_{x \in X} x d G>\mathcal{X}(\theta)$. As $\Theta \subseteq \mathbb{R}_{++}^{J}$, the new expected payment from type $\bar{\theta}$ is strictly higher than that in the original menu, $p^{\prime}(\bar{\theta})>\bar{T}$.

The new menu $M^{\prime}$ is not necessarily direct. The no disclosure item may be attractive to some types other than $\bar{\theta}$. However, the only profitable strategy under no disclosure is always buying. Such a deviation would only increase the seller's profit, as $\bar{T}$ was chosen to be the highest expected payment. Accordingly, menu $M^{\prime}$ brings strictly greater revenue than menu $M$, which is a contradiction.

Proof of Lemma 1. Set $\mathcal{F}$ is an image of a set of measures dominated by the prior distribution. That set of measures is compact in the weak* topology. Moreover, the corresponding map is continuous by the dominated convergence theorem, since the ex-ante attribute expectations exist. Therefore, $\mathcal{F}$ is compact.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 5: Feasibility set $\mathcal{F}$ of attribute surplus $\mathcal{X}=\left(\mathcal{X}_{1}, \mathcal{X}_{2}\right)$ and trade probability $\mathcal{Q}$ for the case of two attributes distributed uniformly over a unit square $X=[0,1]^{2}$.

Set $\mathcal{F}$ is convex as a linear image of a convex set. Indeed, take any two points $\left(\mathcal{X}_{1}, \mathcal{Q}_{1}\right),\left(\mathcal{X}_{1}, \mathcal{Q}_{2}\right) \in$ $\mathcal{F}$ and $\gamma \in[0,1]$. By construction, there exist trade functions $q_{1}, q_{2}: X \rightarrow[0,1]$ that generate these two points. Then, the function $q_{3} \triangleq \gamma q_{1}+(1-\gamma) q_{2}$ is an admissible trade function that generates the attribute surplus:

$$
\begin{aligned}
\mathcal{X}_{3}(\theta) & =\int_{x \in X} x q_{3}(x) \mathrm{d} G(x)=\int_{x \in X} x\left(\gamma q_{1}(x)+(1-\gamma) q_{2}(x)\right) \mathrm{d} G(x) \\
& =\gamma \int_{x \in X} x q_{1}(x) \mathrm{d} G(x)+(1-\gamma) \int_{x \in X} x q_{2}(x) \mathrm{d} G(x)=\gamma \mathcal{X}_{1}(\theta)+(1-\gamma) \mathcal{X}_{2}(\theta)
\end{aligned}
$$

The same argument can be applied to the trade probability. Thus, $\left(\mathcal{X}_{3}, \mathcal{Q}_{3}\right)$ is a convex combination of $\left(\mathcal{X}_{1}, \mathcal{Q}_{1}\right)$ and $\left(\mathcal{X}_{2}, \mathcal{Q}_{2}\right)$, and it belongs to the feasibility set $\mathcal{F}$. Figure 5 illustrates the feasibility set for the case of two uniformly and independently distributed attributes.

Since $\mathcal{F}$ is a finite-dimensional closed set, the supporting hyperplane theorem (Rockafellar (1970), Theorem 11.6, Corollary 11.6.1, p. 100) can be applied. ${ }^{21}$ A point $(\hat{\mathcal{X}}, \hat{\mathcal{Q}})$ belongs to the boundary of $\mathcal{F}$ if and only if there are coefficients $\left(\lambda, \lambda_{0}\right)$, not all zero, such that:

$$
(\hat{\mathcal{X}}, \hat{\mathcal{Q}}) \in \arg \max _{(\mathcal{X}, \mathcal{Q}) \in \mathcal{F}} \lambda \cdot \mathcal{X}+\lambda_{0} \mathcal{Q} .
$$

[^14]By the definition of $\mathcal{F}$, the trade function $\hat{q}$ that generates the point $(\hat{\mathcal{X}}, \hat{\mathcal{Q}})$ is such that:

$$
\begin{aligned}
\hat{q}(x) & \in \arg \max _{q: X \rightarrow[0,1]} \lambda \cdot \int_{x \in X} x q(x) \mathrm{d} G(x)+\lambda_{0} \int_{x \in X} q(x) \mathrm{d} G(x)= \\
& \in \arg \max _{q: X \rightarrow[0,1]} \int_{x \in X}\left(\lambda \cdot x+\lambda_{0}\right) q(x) \mathrm{d} G(x)
\end{aligned}
$$

The integral is maximized pointwise. Any maximizer of it is a linear disclosure (22) with coefficients $\alpha=\lambda$ and $\alpha_{0}=\lambda_{0}$.

Proof of Lemma 2. The seller's problem can be written in terms of attribute surpluses and trade probabilities (15). Consider any profile $(r(\theta), \mathcal{X}(\theta), \mathcal{Q}(\theta), p(\theta))_{\theta \in \Theta}$ that satisfies constraints (16), (17), (18), (19), (20). Define functions $\mathcal{Q}^{\prime}: \Theta \rightarrow[0,1]$ and $p^{\prime}: \Theta \rightarrow \mathbb{R}_{+}$as:


Disclosure and Pricing of Attributes

$$
\begin{aligned}
\mathcal{Q}^{\prime}(\theta) & =\min _{Q:(\mathcal{X}(\theta), Q) \in \mathcal{F}} Q, \\
p^{\prime}(\theta) & = \begin{cases}\frac{\mathcal{Q}(\theta) p(\theta)}{\mathcal{Q}^{\prime}(\theta)}, & \text { if } \mathcal{Q}^{\prime}(\theta)>0, \\
p(\theta), & \text { if } \mathcal{Q}^{\prime}(\theta)=0 .\end{cases}
\end{aligned}
$$

Intuitively, these functions are perturbations of the original mechanism that minimize trade probability while keeping the expected revenue fixed. By Lemma 1, $\mathcal{F}$ is compact, so for each $\theta \in \Theta, \mathcal{Q}^{\prime}(\theta)$ is well defined and $\left(\mathcal{X}(\theta), \mathcal{Q}^{\prime}(\theta)\right)$ belongs to the boundary of $\mathcal{F}$. At the same time, by the measurable maximum theorem (Aliprantis and Border (2006), Thm. 18.19) $\mathcal{Q}^{\prime}$, and hence $p^{\prime}$, is a measurable function. By construction, for any type $\theta \in \Theta, \mathcal{Q}^{\prime}(\theta) \leq \mathcal{Q}(\theta), p^{\prime}(\theta) \geq p(\theta)$, and $\mathcal{Q}^{\prime}(\theta) p^{\prime}(\theta)=\mathcal{Q}(\theta) p(\theta)$. As such, allocation $\left(\mathcal{X}(\theta), \mathcal{Q}^{\prime}(\theta)\right)_{\theta \in \Theta}$ is implementable via tariff functions $r(\theta), p^{\prime}(\theta)$ and satisfies all conditions of the lemma. The result follows.

Proof of Theorem 1. The seller's problem (15) can be seen as a maximization of a continuous function over a compact set. Therefore, an optimal menu exists. By Lemma 2, there exists an optimal menu with all allocations located on the boundary of the feasibility set $\mathcal{F}$. By Lemma 1, such allocations are achieved by linear disclosures.

Proof of Corollary 1. Define an auxiliary attribute $x_{\theta}^{\prime}$ as the valuation of a type $\theta$, $x_{\theta}^{\prime} \triangleq v(\theta, x)$. By construction, the valuation of each type can be defined as $v^{\prime}\left(\theta, x^{\prime}\right)=x_{\theta}^{\prime}$. This instance is a special case of the formulation (1). Thus, Theorem 1 applies and there exists an optimal menu with every experiment in it being a linear disclosure of auxiliary
attributes $x^{\prime}$ :

$$
q\left(x^{\prime}\right)= \begin{cases}1, & \text { if } \sum_{\theta \in \Theta} \alpha_{\theta} x_{\theta}^{\prime}>\alpha_{0}, \\ 0, & \text { if } \sum_{\theta \in \Theta} \alpha_{\theta} x_{\theta}^{\prime}<\alpha_{0},\end{cases}
$$

for $\alpha \in \mathbb{R}^{|\Theta|}, \alpha_{0} \in \mathbb{R}$, not all zeros. In the original formulation, these are linear forms.

Calculations behind Location Payoffs Example. Consider a linear form. If $\alpha_{1}+\alpha_{2} \neq$ 0, then the sum can be normalized to equal 1. The linear form can be rewritten as:

$$
q(x)= \begin{cases}1, & \text { if }-\left(x-\left(\alpha_{1} \theta_{1}+\alpha_{2} \theta_{2}\right)\right)^{2} \gtrless \alpha_{0}^{\prime}, \\ 0, & \text { if }-\left(x-\left(\alpha_{1} \theta_{1}+\alpha_{2} \theta_{2}\right)\right)^{2} \lessgtr \alpha_{0}^{\prime},\end{cases}
$$

with $\alpha_{0}^{\prime}=-v_{0}+\alpha_{0}+\alpha_{1} \alpha_{2}\left(\theta_{1}-\theta_{2}\right)^{2}$ and the inequality sign depending on the sign of the original $\alpha_{1}+\alpha_{2}$. This is a neighborhood disclosure with $\hat{\theta}=\alpha_{1} \theta_{1}+\alpha_{2} \theta_{2}$ and $\alpha_{1}+\alpha_{2}=1$.

If $\alpha_{1}=\alpha_{2}=0$, then the linear form provides no disclosure and, as $X$ is bounded, is equivalent to a neighborhood disclosure for a sufficiently large $\left|\alpha_{0}\right|$.

If $\alpha_{1}+\alpha_{2}=0$ and $\alpha_{1} \neq 0$, then the linear form is a linear disclosure:

$$
q(x)= \begin{cases}1, & \text { if }\left(\theta_{1}-\theta_{2}\right) \cdot x \gtrless \alpha_{0}^{\prime}, \\ 0, & \text { if }\left(\theta_{1}-\theta_{2}\right) \cdot x \lessgtr \alpha_{0}^{\prime},\end{cases}
$$

with $\alpha_{0}^{\prime}=\alpha_{0} /\left(2 \alpha_{1}\right)+\left(\theta_{1}^{2}-\theta_{2}^{2}\right) / 2$ and the inequality sign depending on the sign of $\alpha_{1}$. However, the proof of Lemma 1 established that the attribute surplus and probability achieved by a linear form with parameters $\left(\alpha_{1}, \alpha_{2}, \alpha_{0}\right)$ correspond to a boundary point of $\mathcal{F}$ in the auxiliary attributes, supported by the hyperplane orthogonal to the vector $\left(\alpha_{1}, \alpha_{2}, \alpha_{0}\right)$. If $\theta_{1} \neq \theta_{2}$, then $\mathcal{F}$ has a strict interior. Thus, the set of boundary points supported by hyperplanes with $\alpha_{1}+\alpha_{2}=0$ has a measure of zero.

Proof of Theorem 2. The argument is given in the text. The only difference from the standard problems is that $\mathcal{X}$ can take values in $[0, \mathbb{E}[x]]$, not in $[0,1]$. However, this difference does not affect the extreme nature of the solution.

Lemma 4. (Directional Decomposition)
Let $\left(x_{1}, \ldots, x_{J}\right)$ be $J$ attributes independently distributed over $X \subseteq \mathbb{R}^{J}$ according to distributions $G_{1}, \ldots, G_{J}$. Let $E=(S, \pi)$ be an arbitrary experiment. Let $(\mu(s, E), \operatorname{Pr}(s, E))$ be the belief distribution induced by $E$ so that $\mu(s, E)$ is a distribution over $X$ conditional on $s$ given $E$. Denote by $\mu_{j}(s, E)$ the $j$ th marginal distribution of $\mu(s, E)$. Then, there exist ex-
periments $\left\{E_{j}\right\}_{j=1}^{J}$ such that: $E_{j}=\left(S, \pi_{j}\right)$ induces a belief distribution $\left(\mu\left(s, E_{j}\right), \operatorname{Pr}\left(s, E_{j}\right)\right)$ with $\mu\left(s, E_{j}\right)=\left(\mu_{j}(s, E), G_{-j}\right)$ and $\operatorname{Pr}\left(s, E_{j}\right)=\operatorname{Pr}(s, E)$ for all $s \in S$.

Proof. The proof is constructive. Introduce dummy variables $\left(x_{1}^{\prime}, \ldots, x_{J}^{\prime}\right)$ which are distributed as $\left(x_{1}, \ldots, x_{J}\right)$ but are drawn independently of them. For a given $j$, construct $E_{j}$ as an experiment that provides information about the vector $\left(x_{j}, x_{-j}^{\prime}\right)$ according to $\pi$. By construction, $E_{j}$ induces the same marginal distribution of beliefs about attribute $j$. However, since $\left(x_{1}, x_{1}^{\prime}, \ldots, x_{J}, x_{J}^{\prime}\right)$ are independent, it provides no information about other attributes. $\square$


Disclosure and Pricing of Attributes

Proof of Proposition 3. Consider an arbitrary responsive experiment $E_{j}\left(\theta_{j}\right)$. By Lemma 4, there exists a linear disclosure $E_{j}^{\prime}\left(\theta_{j}\right)$ such that $\mathcal{X}_{j}^{\prime}\left(\theta_{j}\right)=\mathcal{X}_{j}\left(\theta_{j}\right), \mathcal{Q}\left(E_{j}^{\prime}\left(\theta_{j}\right)\right)=\mathcal{Q}\left(E_{j}\left(\theta_{j}\right)\right)$, and $\mathcal{X}_{k}^{\prime}\left(\theta_{j}\right)=\mathbb{E}\left[x_{k}\right]$ for all $k \neq j$. Replacing $E_{j}\left(\theta_{j}\right)$ with $E_{j}^{\prime}\left(\theta_{j}\right)$ does not change the incentive compatibility within cohort $\Theta_{j}$, but by Blackwell's Theorem, it relaxes the incentive compatibility of other cohorts. By Theorem 1, the result follows.

Proof of Lemma 3. Define the expected transfer function as $T_{j}\left(\theta_{j}\right) \triangleq \mathcal{Q}_{j}\left(\theta_{j}\right) p_{j}\left(\theta_{j}\right)$. I can use standard one-dimensional arguments within each cohort to establish the connection between the attribute surplus and the expected transfer function:

$$
T_{j}\left(\theta_{j}\right)=\theta_{j} \mathcal{X}_{j}\left(\theta_{j}\right)-\int_{0}^{\bar{\theta}_{j}} \mathcal{X}_{j}(z) \mathrm{d} z .
$$

Individual rationality and incentive compatibility within each cohort are satisfied by construction. However, deviations between different cohorts impose additional constraints,

$$
\int_{0}^{\bar{\theta}_{j}} \mathcal{X}_{j}\left(\theta_{j}\right) d \theta_{j} \geq \bar{\theta}_{j} \mathbb{E}\left[x_{j}\right]-\underline{p},
$$

where $\underline{p}$ is the minimal object price in the menu $\left\{\mathcal{X}_{j}\right\}_{j=1}^{J}$. The deviations from all other types $\theta_{j} \in \Theta$ follow because the indirect utility function is convex and grows slower than $\theta_{j} \mathbb{E}\left[x_{j}\right]$. Applying double integration to the objective function completes the derivation.

Proof of Theorem 3. The argument in the text establishes that all high types are offered the minimal price. The optimal mechanism should then solve the problem (36) with the additional constraints that all high types are offered the same fixed price $\underline{p}^{*}$ and are served
the fixed attribute surplus $\mathcal{X}_{j}^{*}\left(\bar{\theta}_{j}\right)$. These constraints can be written as:

$$
\begin{aligned}
\int_{0}^{\bar{\theta}_{j}} \mathcal{X}_{j}\left(\theta_{j}\right) d \theta_{j} & =\mathcal{X}_{j}\left(\bar{\theta}_{j}\right)-\underline{p}^{*} \mathcal{Q}_{j}\left(\mathcal{X}_{j}\left(\bar{\theta}_{j}\right)\right) \\
\mathcal{X}_{j}\left(\bar{\theta}_{j}\right) & =\mathcal{X}_{j}^{*}\left(\bar{\theta}_{j}\right)
\end{aligned}
$$

Consider a relaxed problem with the original integral constraints and the monotonicity constraints dropped. In this problem, by Luenberger (1969) (Chapter 8, Theorem 1), there exist Lagrange multipliers $\left\{\lambda_{j}\right\}$ such that the optimal $\mathcal{X}_{j}\left(\theta_{j}\right)$ maximize the Lagrange function:

$$
\mathcal{L} \sim \sum_{j=1}^{J} f\left(\Theta_{j}\right) \int_{0}^{\bar{\theta}_{j}}\left(\theta_{j}-\frac{1-F_{j}\left(\theta_{j}\right)}{f_{j}\left(\theta_{j}\right)}-\lambda_{j}\right) \mathcal{X}_{j}\left(\theta_{j}\right) \mathrm{d} F_{j}\left(\theta_{j}\right)
$$

over a domain $\mathcal{X}_{j}\left(\theta_{j}\right) \in\left[0, \mathcal{X}^{*}\left(\bar{\theta}_{j}\right)\right]$. If all type distributions have the monotone hazard rate property, then the integrands increase in $\theta_{j}$. Therefore, the optimal $\mathcal{X}_{j}\left(\theta_{j}\right)$ are bang-bang: $\mathcal{X}_{j}\left(\theta_{j}\right)=0$ for $\theta_{j}<\theta_{j}^{*}, \mathcal{X}_{j}\left(\theta_{j}\right)=\mathcal{X}^{*}\left(\bar{\theta}_{j}\right)$ for $\theta_{j}>\theta_{j}^{*}$. This solution corresponds to a single item per each attribute cohort and satisfies the relaxed constraints.

Proof of Theorem 4. I begin by characterizing the optimal mechanism in the case of orthogonal types, since formally it is not covered by 3. The class of orthogonal types features particularly tractable incentive constraints. If type $\theta_{j}$ misreports, then he is offered an experiment tailored to another orthogonal type that is hence not informative about attribute $j$. Thus, the type has no reason to act on the experiment realization, and the tightest incentive-compatibility constraint is one in which he always buys. All others can be dropped. The seller's problem can be written as:

$$
\begin{array}{ll}
& \max _{\left\{r_{j}, \mathcal{X}_{j}, p_{j}\right\}_{j=1}^{J}} \sum_{j=1}^{J} f\left(\theta_{j}\right)\left(r_{j}+\mathcal{Q}_{j} p_{j}\right) \\
\text { s.t. } & \mathcal{X}_{j}-p_{j} \mathcal{Q}_{j}-r_{j} \geq \mathbb{E}\left[x_{j}\right]-p_{k}-r_{k}, \forall j, k=1, \ldots, J \\
& \mathcal{X}_{j}-p_{j} \mathcal{Q}_{j}-r_{j} \geq 0, \mathcal{X}_{j} \in\left[0, \mathbb{E}\left[x_{j}\right]\right], \mathcal{Q}_{j}=\mathcal{Q}_{j}\left(\mathcal{X}_{j}\right), \forall j=1, \ldots, J .
\end{array}
$$

I show that the solution to this problem does not feature price discrimination. First, by the arguments of Theorem 3, upfront payments can be without loss of revenue set to zero, $r_{j} \equiv 0$. Second, consider an arbitrary solution to the seller's problem such that $r_{j} \equiv 0$ and define $\underline{p}=\min _{j}\left\{p_{j}\right\}$. Toward the contradiction, assume that $p_{j}>\underline{p}$ for some $j$.

If $\mathbb{E}\left[x_{j}\right] \geq \underline{p}$, then the incentive-compatibility constraint is binding. Hence, $\mathcal{Q}_{j} p_{j}=$ $\mathcal{X}_{j}-\mathbb{E}\left[x_{j}\right]+\underline{p}$. For small $\varepsilon>0$, consider a modified mechanism with $\mathcal{X}_{j}^{\prime}=\mathcal{X}_{j}+\varepsilon, \mathcal{Q}_{j}^{\prime} p_{j}^{\prime}=$
$\mathcal{X}_{j}^{\prime}-\mathbb{E}\left[x_{j}\right]+\underline{p}$. Because $\mathcal{Q}_{j}\left(\mathcal{X}_{j}\right)$ is continuous, the mechanism remains incentive compatible yet brings higher revenue, which is a contradiction. Moreover, note that as $\mathcal{X}_{j}^{\prime}>\mathcal{X}_{j}, \alpha_{0 j}^{\prime}<$ $\alpha_{0 j}$; hence, $\mathcal{X}_{j}^{\prime} / \mathcal{Q}_{j}^{\prime}=\mathbb{E}\left[x_{j} \mid x_{j} \geq \alpha_{0 j}^{\prime}\right]<\mathbb{E}\left[x_{j} \mid x_{j} \geq \alpha_{0 j}\right]=\mathcal{X}_{j} / \mathcal{Q}_{j}$ and $\mathcal{Q}_{j}^{\prime}>\mathcal{Q}_{j}$. Thus, $p_{j}^{\prime}<p_{j}$.


Disclosure and Pricing of Attributes

If $\mathbb{E}\left[x_{j}\right]<\underline{p}$, then the individual-rationality constraint is binding. Hence, $\mathcal{Q}_{j} p_{j}=\mathcal{X}_{j}$. For small $\varepsilon>0$, consider the modified mechanism with $p_{j}^{\prime}=p_{j}-\varepsilon, \mathcal{X}_{j}^{\prime} / \mathcal{Q}_{j}^{\prime}=\mathcal{X}_{j} / \mathcal{Q}_{j}-\varepsilon$. The mechanism remains incentive compatible yet brings higher revenue, which is a contradiction.

Now, consider the optimal disclosure for a given object price. According to feasibility and individual rationality, $\mathcal{X}_{j} / \mathcal{Q}_{j} \geq \max \left\{p, \mathbb{E}\left[x_{j}\right]\right\}$. If $\mathcal{X}_{j} / \mathcal{Q}_{j}>\max \left\{p, \mathbb{E}\left[x_{j}\right]\right\}$, then for small $\varepsilon>0$, the mechanism with $\mathcal{X}_{j}^{\prime} / \mathcal{Q}_{j}^{\prime}=\mathcal{X}_{j} / \mathcal{Q}_{j}-\varepsilon$ is incentive compatible and increases trade probability, $\mathcal{Q}_{j}^{\prime}>\mathcal{Q}_{j}$, and consequently, revenue. This is a contradiction.

These arguments establish that in the case of orthogonal types an optimal mechanism sets $r_{j}=0, p_{j}=p$, and $E_{j}$ is a binary monotone partition of $x_{j}$ such that $\mathbb{E}\left[x_{j} \mid E_{j}, s^{+}\right]=$ $\max \left\{p, \mathbb{E}\left[x_{j}\right]\right\}$ for all $j=1, \ldots, J$.

Now, consider a general setting. Introduce auxiliary attributes as in the proof of Corollary 1. By the arguments above, $M^{S P}\left(p^{*}\right)$ solves a relaxed problem in which the constraints (16) and (17) are dropped. If $M^{S P}\left(p^{*}\right)$ is incentive compatible, then these relaxed constraints are satisfied and, thus, $M^{S P}\left(p^{*}\right)$ solves an original problem.

Proof of Corollary 2. For a profile $\left(\theta_{1}, \ldots, \theta_{J}\right)$, denote by $p^{*}\left(\theta_{1}, \ldots, \theta_{J}\right)$ an optimal price in a separate persuasion mechanism, by $\alpha_{0 j}\left(\theta_{1}, \ldots, \theta_{J}\right)$ optimal persuasion thresholds, and by $\Pi^{S P}\left(p, \theta_{1}, \ldots, \theta_{J}\right)$ the revenue function. As attributes are continuously distributed, $\Pi^{S P}(\cdot)$ is continuous. If $p^{*}\left(\theta_{1}, \ldots, \theta_{J}\right)$ is a singleton, then, by the Maximum Theorem, $p^{*}(\cdot)$ and $\alpha_{0}(\cdot)$ are continuous functions in a neighborhood of $\left(\theta_{1}, \ldots, \theta_{J}\right)$.

Fix an orthogonal type profile $\hat{\Theta}=\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right)$. If $\mathbb{E}[\hat{\theta} \cdot x] \neq p\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right)$ for all $\hat{\theta} \in \hat{\Theta}$, then the types, when deviating, strictly prefer to not act contrary to their no-information action. As $\hat{\Theta} \geq 0$, the mismatching strategies are irrelevant; moreover, for all $\hat{\theta}_{j}, \hat{\theta}_{k} \in \hat{\Theta}$, $k \neq j$ :

$$
\begin{aligned}
& \mathbb{E}\left[\hat{\theta}_{j} \cdot x \mid \hat{\theta}_{k} \cdot x>\alpha_{0 k}\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right)\right]<p^{*}\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right), \text { if } \mathbb{E}\left[\hat{\theta}_{j} \cdot x\right]<p, \\
& \mathbb{E}\left[\hat{\theta}_{j} \cdot x \mid \hat{\theta}_{k} \cdot x<\alpha_{0 k}\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right)\right]>p^{*}\left(\hat{\theta}_{1}, \ldots, \hat{\theta}_{J}\right), \text { if } \mathbb{E}\left[\hat{\theta}_{j} \cdot x\right]>p .
\end{aligned}
$$

Now, replace orthogonal $\hat{\Theta}$ by a generic $\Theta$. As long as all types in $\Theta$ are positive, the mismatching strategies remain irrelevant. As attributes are continuously distributed, in some neighborhood of $\hat{\Theta}$ both sides of the inequalities are continuous and the constraints remain satisfied. Hence, the buyer cannot benefit from misreporting, and the result follows
from Theorem 4.

Proof of Proposition 4. Consider any optimal menu and a type $\theta$ with $r(\theta)>0$. Let $r^{\prime}(\theta)=r(\theta)-\varepsilon, p^{\prime}(\theta)=p(\theta)+\varepsilon / \mathcal{Q}(\theta)$ for a small $\varepsilon>0$. If incentive compatible, this modification preserves the seller's revenue and the buyer's payoff. The constraint (16) remains the same. The constraint (18) is relaxed and is strictly so when $\mathcal{Q}(\theta)<1$. The constraint (19) is satisfied by the individual rationality. For the constraint (17):

$$
r^{\prime}(\theta)+(1-\mathcal{Q}(\theta)) p^{\prime}(\theta)=r(\theta)+(1-\mathcal{Q}(\theta)) p(\theta)+\varepsilon \frac{1-2 \mathcal{Q}(\theta)}{\mathcal{Q}(\theta)} .
$$

For the modification to violate incentive constraints, it must be that $1-2 \mathcal{Q}(\theta)<0$ and the constraint (17) binds for some type $\theta^{\prime}$.

Proof of Corollary 3. I present the arguments for the case when the higher signal is sent with probability one at the threshold. The cases with randomization are analogous.

i. ) Whenever $\alpha_{j} \geq 0$ :
$$
\mathbb{E}\left[x_{j} \mid \alpha x \geq \alpha_{0}\right]=\mathbb{E}\left[x_{j} \mid \alpha_{j} x_{j} \geq \alpha_{0}-\sum_{k \neq j} \alpha_{k} x_{k}\right] \geq \mathbb{E}\left[x_{j} \mid \alpha_{j} x_{j}<\alpha_{0}-\sum_{k \neq j} \alpha_{k} x_{k}\right] .
$$
$\mathbb{E}\left[v(\theta, x) \mid \alpha x \geq \alpha_{0}\right]=\sum_{j=1}^{J} \theta_{j} \mathbb{E}\left[x_{j} \mid \alpha x \geq \alpha_{0}\right] \geq \sum_{j=1}^{J} \theta_{j} \mathbb{E}\left[x_{j} \mid \alpha x<\alpha_{0}\right]=\mathbb{E}\left[v(\theta, x) \mid \alpha x<\alpha_{0}\right]$.
ii. ) For any increasing function $v(\theta, \cdot)$, and $x_{0} \in \mathbb{R}, \mathbb{E}\left[v(\theta, x) \mid x \geq x_{0}\right] \geq \mathbb{E}\left[v(\theta, x) \mid x<x_{0}\right]$.


Disclosure and Pricing of Attributes

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Krähmer, D. (2020): "Information Disclosure and Full Surplus Extraction in Mechanism Design," Journal of Economic Theory, 105020.

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[^0]:    *Toulouse School of Economics, alexey.v.smolin@gmail.com. I thank Dirk Bergemann and Daniel Krähmer for helpful conversations as well as Heski Bar-Isaac and Levent Celik for excellent discussions. I am grateful to the participants in research seminars at numerous institutions. Finally, I thank the coeditor, Nicola Persico, and the anonymous referees for their many productive suggestions. I acknowledge funding from ANR under grant ANR-17-EURE-0010 (Investissements d'Avenir program).

[^1]:    ${ }^{1}$ "ZipRecruiter Is Valued at \$1.5 Billion in a Bet on AI Hiring," (Carville (2018)).

[^2]:    ${ }^{2}$ See, for example, Anderson and Renault (2006), Eső and Szentes (2007), and Li and Shi (2017).
    ${ }^{3}$ This terminology follows Rayo and Segal (2010) and is not to be confused with the voluntary disclosure of Dye (1985), studied for example by Koessler and Renault (2012).

[^3]:    ${ }^{4}$ Chakraborty and Harbaugh (2010) use linear disclosure rules to construct informative equilibria in a multidimensional cheap talk game.
    ${ }^{5}$ This intuition leads in the right direction but does not consider discriminatory menus and information pricing. I formally complete the argument and confirm the result by building on single-dimensional mechanism-design machinery.


Disclosure and Pricing of Attributes

[^4]:    ${ }^{6}$ See, however, Anderson and Renault (2006), who show that optimal disclosure rule is partial if the purchase is associated with search costs and the seller cannot commit to prices. Similarly, Dworczak (2020) shows that optimal disclosure rule can be partial in the presence of aftermarkets. See also Bar-Isaac, Caruana, and Cuñat (2010).
    ${ }^{7}$ Wei and Green (2019) show that withholding information may also be optimal if the information must be provided free of charge.

[^5]:    ${ }^{8}$ Almost all of this literature studies one-dimensional settings. The works of Rayo and Segal (2010) and Dworczak and Martini (2019) provide elegant exceptions.
    ${ }^{9}$ As I discuss in Sections 4.4 and 5.3, this formulation can be generalized.

[^6]:    ${ }^{10} S$ can be any Polish space. Throughout the paper, all introduced functions are (Borel) measurable.
    ${ }^{11}$ This setting is equivalent to one in which the seller provides information for free but can charge the buyer for opting out from the consecutive sale.
    ${ }^{12}$ For instance, the buyer cannot claim a refund ex post. See Krähmer and Strausz (2015b), Heumann (2020) and Bergemann et al. (2020) for recent studies of ex post incentive constraints.

[^7]:    ${ }^{13}$ Krähmer (2020) and Doval and Ely (2020) emphasize the usefulness of such schemes in screening problems and in general games of incomplete information, respectively.

[^8]:    ${ }^{14}$ Rochet and Choné (1998), Bergemann et al. (2012) and Daskalakis et al. (2017) highlight the difficulties associated with the multidimensional screening problems.

[^9]:    ${ }^{15}$ The attribute surplus should not be confused with the trade surplus which depends on the match between the attribute surplus and the buyer type.

[^10]:    ${ }^{16} F$ has the monotone hazard rate property if $f(\theta) /(1-F(\theta))$ is monotonically increasing.

[^11]:    ${ }^{17}$ The name is inspired by "single-minded" bidders in combinatorial auctions who value specific bundles. See, for example, Lehmann et al. (2002).
    ${ }^{18}$ For example, see Section 8.4 of Hartline (2020).

[^12]:    ${ }^{19}$ In any given setting, the incentive compatibility of this mechanism can be straightforwardly checked.

[^13]:    ${ }^{20}$ Eső and Szentes (2017) generalize the latter finding to dynamic environments. Krähmer and Strausz (2015a) discuss settings in which the full-disclosure distributional assumptions are violated.

[^14]:    ${ }^{21}$ This step might fail if there are infinitely many attributes, $|J|=\infty$. If $\mathcal{F}$ has infinite dimensions, then it might have some boundary points that cannot be supported by a hyperplane. In such cases, a sufficient condition for the existence of a supporting hyperplane is that $\mathcal{F}$ has a nonempty interior.


Disclosure and Pricing of Attributes

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Dynamic Evaluation Design

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Dynamic Evaluation Design

# Dynamic Evaluation Design

**Authors:** Alex Smolin

**Manuscript date:** 2020-10-28

#### Abstract

A principal owns a firm, hires an agent of uncertain productivity, and designs a dynamic policy for evaluating his performance. The agent observes ongoing evaluations and decides when to quit. When not quitting, the agent is paid a wage that is linear in his perceived productivity; the principal claims the residual performance. After quitting, the players secure fixed outside options. I show that equilibrium evaluation policies are Pareto efficient. In a minimally informative equilibrium, for a broad class of performance technologies, the agent's wage deterministically grows with tenure. My analysis suggests that endogenous performance evaluation plays an important role in shaping careers in organizations.

Keywords: evaluation, information design, career concerns, bandit experimentation, downward wage rigidity, up-or-out, internal labor markets

JEL Codes: C72, D82, D83, M52

[^0]


Dynamic Evaluation Design

## 1 Introduction

Performance evaluation is an important part of organizational life. Although much evaluation is informal, most organizations have formal evaluation policies designed to collect and distribute performance information to employees. ${ }^{1}$ As communication and information technologies advance, many companies find it easier to provide more evaluation. As a recent example, in August 2015 General Electric (GE) announced an ongoing shift from its legacy system of annual performance reviews to more frequent conversations between managers and employees via an online application. ${ }^{2}$ In this way, GE joined other high-profile companies such as Microsoft, Accenture, and Adobe in a move towards more frequent, exhaustive, and real-time evaluation. However, whenever adopting new evaluation policies, companies should ask: What is their effect on overall performance? Would other evaluation policies perform better? Ultimately, which evaluation policy is the best for the company?

In this paper, I develop a framework to analyze the design of evaluation policies. I consider a principal who owns a firm and hires an agent to work over time. The agent's productivity, his type, is initially uncertain to both players but affects the agent's ongoing performance via a general production technology. The agent's performance is not directly observed but can be revealed through evaluations. While at the firm, the agent's wage is linear in his expected productivity; the principal claims the residual performance. In every period, the agent evaluates his career prospects and decides whether to quit. When the agent quits, the players secure exogenous outside options. Both players are risk neutral and discount the future at the same rate.

The principal designs and adopts a dynamic evaluation policy. The policy is a sequence of statistical experiments that are informative of past performance. The experiments can vary in what and when performance is assessed. The evaluation is costless but its design should take into account the agent's incentives. On the one hand, the promise of future evaluations motivates the agent to stay at the firm and learn whether he is able to perform well. On the other hand, an evaluation may turn out negative and persuade the agent to quit.

In Section 4, I investigate equilibrium evaluation policies by developing a novel efficiency argument. First, I show that the design problem can be viewed as a dynamic persuasion problem in a bandit experimentation setting. It allows me to apply the revelation principle and characterize the set of feasible payoffs that can be possibly achieved in the relationship.

[^1]Second, I study the set of implementable payoffs-the payoffs that can be achieved by some evaluation policy and the agent's best response to it. I show that this set includes all Pareto-efficient payoffs that deliver the agent at least his safe option, and I conclude that any equilibrium evaluation policy is efficient. Under a minimally informative equilibrium policy, the agent is informed about whether he would quit if he could fully observe past performance but had a lower outside option.

In Section 5, I study the effects of optimal evaluation on the agent's career and wage dynamics. I observe several qualitative properties that hold in the minimally informative equilibrium. First, the agent's wage is a deterministic function of tenure. Second, the agent's continuation value grows with tenure so that the agent becomes more optimistic about his prospects the longer he remains at the firm. Third, for a broad class of performance technologies, the agent's wage also grows with tenure. The increase reflects the ongoing positive selection and the corresponding growth of expected productivity. The shape of the wage profile depends on the performance technology. If the technology is coarse, such that performance comes as a stream of infrequent successes, then the wage increases at the revision dates, which are spaced sparsely over the agent's career. In contrast, if the technology is detailed, such that performance can always reveal the agent's incompetence, then performance is constantly monitored, and the wage gradually grows in time.

In Section 6, I study the joint design of wage contract parameters and an evaluation policy. I show that once the principal can control both the information flow and monetary incentives, she is able to extract the full surplus from the relationship: in equilibrium, the joint surplus is maximized, and the agent is left with no rents. The principal can achieve this by offering a fixed-wage contract and, in many cases, by offering a pure-bonus contract.

I discuss the findings in Section 7. First, my analysis suggests that endogenous evaluation policies may be important in explaining economic dynamics commonly observed within firms. These include a lack of wage variation within same-tenure cohorts, downward wage rigidity, and up-or-out contracts. My results further speak in favor of retrospective evaluation and provide a rationale for rating compression and leniency bias as techniques to maintain workforce morale. Second, I highlight the important commitment role that human resource (HR) departments may play in implementing optimal evaluation policies. Third, I show that the equilibrium evaluation policy is robust to a multitude of performance leaks. Finally, I discuss how the framework can be extended to incorporate advisory feedback.

Related Literature My paper contributes to the literature on dynamic persuasion and information design built from the static models of Rayo and Segal (2010) and Kamenica and Gentzkow (2011). Orlov (2016) studies the joint design of performance evaluations and
monetary contracts when the agent exerts private effort. Renault, Solan, and Vieille (2017) and Ely (2017) study dynamic persuasion with an exogenous information flow and a myopic agent. Orlov, Skrzypacz, and Zryumov (2020) investigate a setting in which the principal lacks commitment across different periods. Importantly, my paper introduces the efficiency argument that permits the characterization of the optimal information policies by studying Pareto-efficient allocations. ${ }^{3}$


Dynamic Evaluation Design

My framework highlights the interplay between career concerns and performance evaluations. It complements the turnover theory of Jovanovic (1979) by endogenizing the information flow. Relatedly, several papers investigate evaluation effects on private efforts in the framework of Holmström (1999). Hansen (2013) studies static incentives and focuses on partition evaluations. Hörner and Lambert (2020) study dynamic incentives with a focus on Gaussian policies.

Similar incentive effects are present in multistage contests and tournaments. Ederer (2010) compares the effectiveness of complete- and no-evaluation policies in two-stage tournaments. Halac, Kartik, and Liu (2016) study the optimal design of general multistage contests and similarly focus on the extreme evaluation policies within each period. Nevertheless, Goltsman and Mukherjee (2011) highlight that the optimal evaluation policies in tournaments are generally partially informative.

Finally, my paper contributes to the literature on dynamic contracts without transfers. Guo (2016) studies dynamic delegation when the agent is privately informed. Hörner and Guo (2015) study dynamic resource allocation when the agent's private information evolves over time. My paper highlights that information control may complement delegation and action control as a powerful management tool.


Dynamic Evaluation Design

## 2 Model

A principal owns a firm and hires an agent. The relationship takes place in consecutive periods $t=0,1,2, \ldots$ At time 0, the agent's productivity $\theta \in \Theta \subseteq \mathbb{R}$ is drawn according to a cumulative distribution $G_{0}$. The productivity is fixed throughout the relationship and is not directly observed by either principal or agent. The players are symmetrically informed about productivity with the prior expectation of productivity, $\mathbb{E}[\theta]$, being equal to $\theta_{0}$.

Performance Productivity affects the agent's performance at firm $y_{t} \in Y \subseteq \mathbb{R}$. Conditional on productivity, performance is independently and identically distributed across

[^2]periods according to a cumulative distribution $F_{\theta}$. The collection of distributions $\left\{F_{\theta}\right\}_{\theta \in \Theta}$ defines production capabilities of the firm and is called the (performance) technology. I associate productivity with its expected performance: $\mathbb{E}\left[y_{t} \mid \theta\right]=\theta$.

It follows that performance is informative of the agent's productivity: consistently higher performance suggests higher productivity. The overall informativeness and details of the learning process are determined by technology in place. I impose no assumptions on technology in the characterization of equilibrium payoffs in Section 4. I will impose a regularity assumption in Section 5 to establish downward wage rigidity. Performance is not directly observed by either party but can be revealed through evaluations as discussed below.

Strategies The principal can publicly reveal past performance through an evaluation policy that she designs. The policy is costless and governs when and what performance information is available. The evaluations are objective; their outcomes cannot be manipulated by the principal. At the same time, I place no restrictions on which evaluations the principal can conduct. That is, she can conduct a complete evaluation, no evaluation, periodic reviews, grade evaluations, and so forth.

Formally, the principal chooses an evaluation policy $m$ among all stochastic processes measurable with respect to past performance and evaluations. ${ }^{4}$ The policy can be represented by a sequence of random messages $\left\{m_{t}\right\}_{t=0}^{\infty}$ that are sent to the agent: ${ }^{5}$

$$
m_{t}: Y^{t-1} \times M^{t-1} \rightarrow \Delta(M) .
$$

The message space $M$ is the same in all periods and can be freely chosen by the principal. The exact message labels are irrelevant because their meaning is determined solely by the law of $m$. Associate a complete-evaluation policy $\bar{m}$ with $m_{t} \equiv y_{t-1}$ and a no-evaluation policy $\underline{m}$ with $m_{t} \equiv \emptyset$. Denote the set of all possible evaluation policies of the form (1) by $\mathcal{M}$.

The concept of an evaluation policy is an extension of Kamenica and Gentzkow (2011)'s static persuasion policy and admits two possible interpretations. First, it can be viewed as a disclosure policy. In this interpretation, the principal constantly monitors performance but is bound to communicate according to the chosen policy. Second, it can be viewed as a sequence of public experiments. In this interpretation, the principal does not directly observe performance but commits to a sequential policy of public tests to inform both players of past

[^3]performance.
The evaluations may be understood as being conducted by the HR department of a firm. In this case, a realization $m_{t}$ corresponds to an outcome of a particular evaluation. The evaluation policy $m$ corresponds to the operating rules of the department and specifies which and how past performance is evaluated in any given period. I discuss the implementation details in Section 7.2.

Faced with the evaluation policy, the agent chooses whether and when to quit the firm. He decides based on past evaluations, which he correctly interprets according to Bayes' rule. Quitting is irreversible and ends the game.

Formally, the agent chooses a quitting time $\tau$, which is a stopping time measurable with respect to the evaluation policy $m$ :

$$
\tau \text { is a stopping time w.r.t. } m_{0}, m_{1}, \ldots
$$

The quitting time is a random variable. If $\tau \equiv 0$, then the agent quits at time 0 and does not generate any performance. If $\tau \equiv \infty$, then the agent stays at the firm forever, irrespective of past evaluations. Denote the set of all possible quitting times by $\mathcal{T}$.

Payoffs As long as the agent stays at the firm, the principal appropriates the performance outcomes and pays the agent a wage $w_{t}$. I assume that the wage is set according to a linear contract:

$$
w_{t}\left(m^{t}\right)=w^{F}+\alpha \mathbb{E}\left[y_{t} \mid m^{t}\right],
$$

with $w^{F}>0$ being a fixed base wage and $\alpha \in(0,1)$ being a bonus rate. Linear contracts are widely used in practice and capture in the simplest form the reputation effects of performance evaluations (see Carroll (2015) and the references therein). The agent wants to receive positive evaluations to be perceived as more productive because, in this case, he will be paid a higher bonus. For now, I will treat $w^{F}$ and $\alpha$ as exogenously fixed. I study endogenous contracts in Section 6.

The timing within each period $t$ is as follows. First, the worker receives an evaluation $m_{t}$. Then, he decides whether to stay at the firm. If he stays, then the output $y_{t}$ is produced and the agent is paid wage $w_{t}$ according to (3).

As soon as the agent quits the firm, he secures a total payoff of $V^{A} \in \mathbb{R}$, and the principal secures $V^{P} \in \mathbb{R}$. These payoffs are exogenous, commonly known, and fixed throughout the relationship. They can be viewed as the opportunity costs of the players.

Both players are risk neutral and discount the future with a common discount factor $\delta$. For given evaluation policy $m \in \mathcal{M}$ and quitting strategy $\tau \in \mathcal{T}$, the normalized expected
payoffs of the players are:

$$
\begin{aligned}
& U^{P}(m, \tau)=\mathbb{E}_{m, \tau}\left[(1-\delta) \sum_{t=0}^{\tau-1} \delta^{t}\left(y_{t}-w_{t}\right)+\delta^{\tau} V^{P}\right] \\
& U^{A}(m, \tau)=\mathbb{E}_{m, \tau}\left[(1-\delta) \sum_{t=0}^{\tau-1} \delta^{t} w_{t}+\delta^{\tau} V^{A}\right]
\end{aligned}
$$


Dynamic Evaluation Design

Note that evaluation policy plays two roles in the payoffs. First, it shapes the agent's wage. Second, it provides the agent with information that guides his quitting decision.

Equilibrium I study perfect Bayesian equilibria of this game. For a given evaluation policy, the agent chooses a quitting time to maximize his total expected payoff. The principal anticipates the agent's best response and designs the evaluation policy to maximize her expected payoffs.

Definition 1. An evaluation policy $m^{*}$ and a quitting time $\tau^{*}$ constitute an equilibrium if they solve the problem:

$$
\begin{array}{ll}
& \max _{m \in \mathcal{M}, \tau \in \mathcal{T}} U^{P}(m, \tau) \\
\text { s.t. } & \tau \in \arg \max _{\tau \in \mathcal{T}} U^{A}(m, \tau) .
\end{array}
$$

My goal is to characterize an equilibrium evaluation policy and payoffs in this game. This characterization further allows me to study equilibrium wage dynamics. To this end, for given strategies $m$ and $\tau$, let observed wage $W_{t}$ equal $w_{t}$ if the agent remains at the firm, $\tau>t$, and, to complete the definition, equal to 0 otherwise. The observed wage at time $t$ is a random variable that may take many values because the agent can possibly stay at the firm under a wide range of past evaluations. Define a wage profile as a collection of observed wages at different times $W=\left\{W_{t}\right\}_{t=0}^{\infty}$. The wage profile captures the dynamics of the agent's wage throughout his career at the firm.