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

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.

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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[^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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**Authors:** Alex Smolin

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

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