### 5.1 Pareto-Optimal Algorithmic Recommendations

Thus far, we have focused on buyer-optimal algorithmic recommendations. A natural and broader question is what algorithmic recommendations are Pareto optimal, i.e., which
buyer surplus and seller profit cannot be simultaneously improved upon. In Lemma 1, we showed that such recommendations are induced by threshold algorithms. In this section, we show that the characterization of Pareto-optimal algorithms is fully analogous to the characterization of a buyer-optimal algorithm.

To this end, assume that the designer's objective is a weighted average of buyer surplus and seller profit, with weights of $\alpha$ and $1-\alpha$, respectively. Call an algorithm $\alpha$-optimal if it maximizes this objective for weight $\alpha$. As $\alpha$ spans [0,1], the $\alpha$-optimal algorithms span the range from seller-optimal to socially-optimal to buyer-optimal. Define an $\alpha$-virtual cost as follows:

$$
\gamma_{\alpha}(c) \triangleq c+\max \left\{\frac{2 \alpha-1}{\alpha}, 0\right\} \frac{F(c)}{f(c)},
$$

and assume that for all $\alpha \in[0,1], \gamma_{\alpha}(c)$ is strictly increasing. ${ }^{14}$ Define an $\alpha$-pseudo value as:

$$
y_{\alpha}(v) \triangleq \mathbb{E}_{\tilde{v} \sim G}\left[\gamma_{\alpha}^{-1}(\tilde{v}) \mid \tilde{v} \geq v\right] .
$$

Proposition 5 ( $\alpha$-Optimal Algorithm). An $\alpha$-optimal algorithm recommends the product if and only if $y_{\alpha}(v) \geq p$. Under this algorithm, type $c$ posts $p^{*}(c)=y_{\alpha}\left(\gamma_{\alpha}(c)\right)$. Under this algorithm and pricing, the trade occurs if and only if $v \geq \gamma_{\alpha}(c)$.

Proposition 5 shows that the characterization of an $\alpha$-optimal algorithm follows verbatim the characterization of a buyer-optimal algorithm, with the virtual cost and the pseudo value replaced by their $\alpha$-analogs. Importantly, the product allocation continues to be independent of value distribution. Consequently, all the results on market segmentation in Section 4 apply to any Pareto-optimal algorithmic recommendations.

As the weight $\alpha$ attached to buyer surplus decreases, the difference between an $\alpha$ virtual cost and a true cost decreases. By Proposition 5, this translates into the trade occurring over a broader range of costs and values, thus generating more total surplus.

[^9]Moreover, for all $\alpha \leq 1 / 2$, the $\alpha$-virtual cost coincides with the true cost. Therefore, a seller-optimal algorithm and a socially-efficient algorithm coincide and, as shown below, feature a simple recommendation structure:

Corollary 1 (Seller-Optimal Algorithm). A threshold algorithm with $\hat{v}(p)$ such that $\mathbb{E}[v \mid v \geq \hat{v}(p)]=p$ for all $p \in[\mathbb{E}[v], 1]$ simultaneously maximizes the seller profit and total surplus and, moreover, achieves efficient trade.

The seller-optimal algorithm maximizes efficiency at the expense of the buyer. For any price, the algorithm maximally pools products of different values to the extent that the buyer is still willing to purchase when recommended. This results in a threshold recommendation, and given the full support assumption on $G$, a threshold is uniquely defined for all $p \in[\mathbb{E}[v], 1]$. Under this algorithm, the buyer is guaranteed a zero expected payoff irrespective of the posted price. Thus, the seller, regardless of cost, understands that she captures all the surplus generated, and her goal of maximizing profit aligns perfectly with efficiency. As a result, the seller of type $c$ will post a price $p(c)$ that leads to an efficient trade, i.e., $p(c)=\mathbb{E}[v \mid v \geq c]$. The resulting product allocation is efficient, the seller obtains the maximal feasible surplus, and the buyer is left with no rent. ${ }^{15}$

Corollary 1 entails a notable feature: The persuasion constraint of the buyer, i.e., the requirement that the buyer is always willing to follow recommendations, does not constrain the designer at all points of the Pareto frontier except the seller-optimal one. Intuitively, by Corollary 1, $\alpha$-optimal algorithms for $\alpha \in[0,1 / 2]$ induce the same, efficient outcome, where the persuasion constraint holds with equality but does not constrain the designer. In turn, for $\alpha>1 / 2$, as the weight given to buyer surplus increases, the recommendations become even more favorable to the buyer, and the persuasion constraint becomes slack.

[^10]


Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.

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

### 5.2 Competing Sellers

Thus far, we have focused on a monopoly setting in which a given product can be supplied only by a single seller. Applied seller-by-seller, this analysis also covers cases where multiple sellers offer noncompeting products. However, a natural alternative is a market in which sellers compete for the buyer so that the recommendation algorithm directs the buyer to one out of many alternatives (e.g., Hagiu and Jullien (2011), Hagiu et al. (2022), Elliott et al. (2022), Bar-Isaac and Shelegia (2022)). In this section, we show that our characterization of optimal algorithms and the main welfare implications extend to that setting.

Formally, we consider the following extension of the main setting. There is a single buyer with unit demand. There are $J$ sellers indexed by $j=1, \ldots, J$, each offering a single product. The buyer values for the products, $\left(v_{1}, \ldots, v_{J}\right) \in \mathbb{R}^{J}$, are drawn from a joint distribution $G \in \Delta\left([0,1]^{J}\right)$ and can be arbitrarily correlated. The cost of each seller $j$ is drawn from $F_{j}$, independent of other costs or the value profile. For notational convenience, we introduce a dummy seller indexed by $j=0$ with $v_{0}=0$ and $c_{0}=0$ that corresponds to the buyer's decision not to buy anything.

An algorithm is a function $r:[0,1]^{J} \times \mathbb{R}_{+}^{J} \rightarrow \Delta^{J+1}$, where $\Delta^{J+1}$ is a $J+1$ dimensional simplex, so that for any profiles of realized values $v=\left(v_{1}, \ldots, v_{J}\right)$ and prices $p=\left(p_{1}, \ldots, p_{J}\right)$, the algorithm recommends that the buyer purchase one of the products or none according to $r(v, p) \in \Delta^{J+1}$. The algorithm is commonly known to the buyer and sellers. Given an algorithm, nature draws the seller types $c_{j}$ and the buyer values $v_{j}$. All sellers privately observe their types but not the buyer values or the types of other sellers and simultaneously post their prices $p_{j}$. The algorithm makes recommendations according to $r$. If no product is recommended, trade does not occur. If a product of seller $j$ is recommended, the buyer observes the recommendation and the price and then decides whether to buy the product. If product $j$ is purchased, the buyer and seller $j$ obtain ex post payoffs $v_{j}-p_{j}$ and $p_{j}-c_{j}$, respectively. Otherwise, the players obtain zero payoffs. The solution concept is perfect Bayesian equilibrium.

Despite featuring strategic interaction between the sellers, this setting can be ana-
lyzed analogously to the single seller case. The main idea is that each seller's private information and thus the incentive constraints are similar in both cases: From the perspective of each seller, the value and cost uncertainty, as well as the strategic behavior of other sellers, matter only insofar as they affect her demand curve, which can be encoded in a single variable.

Specifically, for each $j$, denote by $\gamma_{j}\left(c_{j}\right)$ her virtual cost. For the dummy seller 0 , set $\gamma_{0}\left(c_{0}\right)=0$. Assume that for $j=1, \ldots, J, \gamma_{j}$ is strictly increasing and continuous in $c_{j}$. Define $\bar{c}_{j} \triangleq \gamma_{j}^{-1}(1)$. Define an auxiliary random variable

$$
\theta_{j}=v_{j}-\max _{k \in\{0,1, \ldots, J\} \backslash j}\left\{v_{k}-\gamma_{k}\left(c_{k}\right)\right\},
$$

i.e., $\theta_{j}$ is the value of seller $j$ 's product minus the highest virtual surplus among all other sellers as long as the latter is positive. Define

$$
p_{j}^{*}\left(c_{j}\right) \triangleq \mathbb{E}_{\theta_{j}}\left[\gamma_{j}^{-1}\left(\theta_{j}\right) \mid \gamma_{j}^{-1}\left(\theta_{j}\right) \geq c_{j}\right]
$$

and observe that $p_{j}^{*}\left(c_{j}\right)$ is a strictly increasing function. Define the inverse function of $p_{j}^{*}$ as $p_{j}^{*-1}$ with the (nonstandard) convention that $p_{j}^{*-1}(p)=0$ for $p<p_{j}^{*}(0)$ and $p_{j}^{*-1}(p)=1$ for $p>p_{j}^{*}(1)$.

Proposition 6 (Buyer-Optimal Algorithm with Competing Sellers). A buyeroptimal algorithm recommends the product of seller $j^{*}(v, p)$ such that

$$
j^{*}(v, p) \in \underset{j \in\{0,1, \ldots, J\}}{\operatorname{argmax}} v_{j}-\gamma_{j}\left(p_{j}^{*-1}\left(p_{j}\right)\right),
$$

with ties being broken arbitrarily. Under this algorithm, seller $j$ of type $c_{j} \leq \bar{c}_{j}$ posts price $p_{j}^{*}\left(c_{j}\right)$ and seller $j$ of type $c_{j}>\bar{c}_{j}$ is inactive. Under this algorithm and pricing, for any realized profile of $v$ and $c$, the buyer trades with seller $j^{*} \in \operatorname{argmax}_{j \in\{0,1, \ldots, J\}} v_{j}-\gamma_{j}\left(c_{j}\right)$.

Proposition 6 directly extends Proposition 1. To see this, observe that in the case of $J=1$, the condition $v_{1}-\gamma_{1}\left(p_{1}^{*-1}\left(p_{1}\right)\right) \geq v_{0}-\gamma_{0}\left(p_{0}^{*-1}\left(p_{0}\right)\right)=0$ is equivalent to the condition $y\left(v_{1}\right) \geq p_{1}$; thus, the two propositions describe the same algorithm albeit
in different terms. In the case of many sellers, the condition $v_{i}-\gamma_{i}\left(p_{i}^{*-1}\left(p_{i}\right)\right) \geq v_{j}-$ $\gamma_{j}\left(p_{j}^{*-1}\left(p_{j}\right)\right)$ for $i, j \neq 0$ cannot be easily translated into the language of pseudo values, so we present the buyer-optimal algorithm as in (9).

Importantly, as in the case of a single seller, Proposition 6 establishes that the equilibrium product allocation does not depend on the distribution of product values. Similarly, it allows us to succinctly analyze the impact of market segmentation. Formally, market segmentation is defined by an information structure $\mathcal{I}=(S, \pi)$ that consists of a set $S=\times_{j} S_{j}$ of signal realizations $s_{j}$ privately observed by each seller, and a family of probability distributions $\{\pi(\cdot \mid v)\}_{v \in[0,1]^{J}}$ over $S$. The signal is commonly known and exogenous, the signal realizations are independent of the seller types but can be arbitrarily correlated across sellers, and the algorithm can base recommendations on the realized signals, the values, and the product prices. ${ }^{16}$

Proposition 7 (Segmentation Neutrality with Competing Sellers). For any market segmentation, the buyer-optimal algorithm induces the same ex post product allocation, the same expected price and profit of each seller type, and the same ex ante buyer surplus.


Proposition 7 implies the remarkable neutrality of market segmentation if the buyer uses an algorithm to guide consumption choices. As in Section 4, leaking buyer data not only does not harm the buyer but also cannot benefit him, despite competition among sellers. Intuitively, because the algorithm can assess the buyer's value and is designed prior to pricing decisions, it shifts the bargaining power to the buyer, and informing the sellers can only reduce the attainable buyer surplus. Furthermore, by adapting to the specifics of market segmentation, the optimal algorithm design can perfectly absorb the impact of information leakage on total buyer surplus, seller profits, and product allocation.

At the same time, market segmentation does affect equilibrium pricing and the re-

[^11]distribution of buyer surplus across different value profiles. Our analysis behind Proposition 7 reveals that equilibrium pricing can be decomposed across sellers, with each seller's pricing strategy depending only on her beliefs about the value profile, and being indifferent to information observed by other sellers. This immediately allows us to claim an impact of finer market segmentation on prices. Specifically, say that $\mathcal{I}_{H}$ is Blackwell more informative than $\mathcal{I}_{L}$ for seller $j$, if the corresponding marginal $\left(S_{j}, \pi_{j}\right)$ with $\pi_{j}:[0,1]^{J} \rightarrow \Delta\left(S_{j}\right)$ derived from $(S, \pi)$ is Blackwell more informative about $v$. Then, we have:

Proposition 8 (Segmentation Redistribution with Competing Sellers). If signal $\mathcal{I}_{H}$ is Blackwell more informative than $\mathcal{I}_{L}$ for seller $j$, then for any given type of seller $j$, the distribution of prices set by seller $j$ under $\mathcal{I}_{H}$ is a mean-preserving spread of that under $\mathcal{I}_{L}$.

By Proposition 8, any finer market segmentation-regardless of how the additional information is correlated across sellers-results in a clear pattern of more dispersed prices, just as under a monopoly. However, the impact on surplus distribution is subtler than that under a monopoly. With multiple sellers, finer segmentation may group lower values for one seller's product with higher values for another seller's product, leading to a higher trade price and lower surplus for some low-value buyers, thus violating the mean-preserving contraction property.


### 5.3 Informed Buyer

Up to this point, we have deliberately assumed that the algorithm has full control over the buyer's information both about the product's existence and about the product's value. This assumption offered a clear benchmark for studying algorithmic recommendations, provided the algorithm with maximal information to control and thus established the upper bound on achievable buyer surplus. In this section, we relax this assumption and show how our analysis remains relevant even if the buyer is partially informed.


### 5.3.1 Information about Product Existence

We have assumed that the buyer cannot purchase the product if it is not recommended. This assumption is relevant in online settings where recommendation systems are used primarily to discover and bring products to the buyer's attention, and thus effectively serve as gatekeepers. ${ }^{17}$ A natural alternative setting is one in which the buyer already knows the product exists and where to purchase it but may still be unsure about the match value. To address that setting, in this subsection, we allow the buyer to purchase the product even if it is not recommended, and impose a constraint that the buyer prefers not to purchase the nonrecommended product at each price.

First, consider the simplest case in which the seller's cost is commonly known to be $c_{0}$. This case is closest to the paper by Roesler and Szentes (2017) and differs only in the timing of the recommendations. In their setting, recommendations come before the price is posted and thus cannot condition on the price; in our setting, the recommendations can condition on the price.

When the costs are known to be $c_{0}$, a natural candidate for a buyer-optimal algorithm is to recommend the product if and only if $p=c_{0}$ and $v \geq c_{0}$. If this algorithm suffices to incentivize the seller to set $p=c_{0}$, then it is buyer-optimal because the outcome is efficient and leaves the seller with zero profit.

If $\mathbb{E}[v] \leq c_{0}$, an algorithm can attain this outcome by revealing no information whenever $p>c_{0}$ and thus dissuading the buyer from purchasing at a price above $c_{0}$. In contrast, if $\mathbb{E}[v]>c_{0}$, the seller can secure a positive profit by charging a price in $\left(c_{0}, \mathbb{E}[v]\right)$ because no algorithm can make the buyer believe that the expected value of the product is always below $p<\mathbb{E}[v]$. In this case, the buyer-optimal algorithm deters the seller from setting a higher price via adversarial persuasion: At each price, the algorithm provides the buyer with information that minimizes the probability of trade.

[^12]Specifically, the algorithm reveals whether $v<\hat{v}(p)$, where $\hat{v}(p)$ is such that

$$
\mathbb{E}[\tilde{v} \mid \tilde{v}<\hat{v}(p)]=\min \{p, \mathbb{E}[v]\},
$$

and persuades the buyer to purchase only when $v \geq \hat{v}(p)$. The maximum profit the seller can guarantee against adversarial persuasion is

$$
\underline{\pi} \triangleq \max _{p \geq c_{0}}\left(p-c_{0}\right)[1-G(\hat{v}(p)] .
$$

The buyer-optimal algorithm induces an efficient trade while leaving the seller with this profit:

Proposition 9 (Known Product, Known Cost). Suppose that $F$ is concentrated at $c_{0} \in[0,1)$ and that the buyer can purchase the product even when not recommended. The buyer-optimal algorithm recommends the product if $p^{*}=c_{0}+\frac{\pi}{1-G\left(c_{0}\right)}$ and $v \geq c_{0}$ and follows all other prices with adversarial persuasion. Under this algorithm, the seller posts a price $p^{*}$, and the equilibrium trade is efficient.

Like in the setting of Roesler and Szentes (2017), a buyer-optimal algorithm leads to efficient trade. Unlike the setting of Roesler and Szentes (2017), the seller's rent is driven by adversarial persuasion price-by-price and thus is lower, reaching zero when $\mathbb{E}[v]<c_{0}$, e.g., when the product can be counterfeit or harmful with a high probability.

Recall from Claim 1 that if the seller's cost were known to be $c_{0}$ in our original setup, the buyer-optimal algorithm would recommend the product if and only if $p=c_{0}$ and $v \geq c_{0}$. This algorithm always attains the efficient outcome and leaves the seller with zero profit. Therefore, when $c_{0}<\mathbb{E}[v]$, the design and consequences of the optimal algorithm depend on whether the buyer can purchase the nonrecommended product, whereas if $c_{0}>\mathbb{E}[v]$ they do not depend on it.

Furthermore, when the seller's costs are uncertain, the buyer-optimal algorithms in both cases can coincide even when the cost could fall below $\mathbb{E}[v]$. This happens, for example, in the uniform setting of Example 1: Under the buyer-optimal algorithm, the
lack of recommendation is a sufficiently negative signal at any price to dissuade the buyer from the purchase. More generally:

Proposition 10 (Known Product. Unknown Cost). If $\int_{0}^{\gamma(c)}[v-c] \mathrm{d} G(v) \leq 0$ for each $c \in[0, \bar{c}]$, then even if the buyer can purchase the product when not recommended, the algorithm in Proposition 1 is buyer-optimal.

The condition of Proposition 10 ensures that whenever the product is not recommended under the algorithm of Proposition 1, the buyer infers that the expected value of the product is below the price. This holds for many classes of distributions, for example, when (i) $F(c)=c^{\alpha}$ and $G(v)=v^{\beta}$ with $0<\alpha \leq \beta$ or (ii) $G$ is uniform and $F(c) / c$ is increasing. ${ }^{18}$ To see the intuition behind this sufficient condition, suppose that the seller posts a price of $p^{*}(c)$, and the algorithm recommends that the buyer not purchase the product, which by Proposition 1, reveals that $v \leq \gamma(c)$. If the buyer purchases the product, his payoff must decrease because the seller's profit increases but total surplus decreases because $\int_{0}^{\gamma(c)}[v-c] \mathrm{d} G(v) \leq 0$. Thus, the buyer is willing to follow the recommendation not to buy.


### 5.3.2 Information about Product Value

Thus far, we have assumed that the buyer does not obtain any product information beyond what is provided by the algorithm. This stylized assumption is intended to capture the context of experience goods, which can be difficult to judge based on appearance and for which individual taste shocks are sufficiently variable yet can be estimated by a welltrained recommendation system. In this section, we allow for the possibility that the buyer observes additional information about the value when a product is recommended while maintaining the original ignorance of the product's existence. ${ }^{19}$

We show how our previous analysis informs this setting. First, observe that the buyer's incentives in the buyer-optimal algorithm of Proposition 1 are generally slack.

[^13]That is, whenever a product is recommended, the buyer strictly prefers to follow the recommendation. Therefore, a small amount of extraneous information that does not significantly lower the posterior expectation will not interfere with the algorithm's design.

Second, our market segmentation analysis implies that even if the buyer can perfectly assess the value of the product upon seeing it, the algorithm can still achieve the same total buyer surplus, seller surplus, and product allocation, although this would require informing both the seller and the buyer. Specifically, suppose that the buyer observes the value $v$ of the product whenever it is recommended. When the seller does not know $v$, the algorithm in Proposition 1 is not incentive compatible because the buyer will ignore the recommendation when $p$ and $v$ are such that $v<p<y(v)$, which occurs for low values. However, if the algorithm perfectly informs the seller about $v$, then the buyer-optimal algorithm, as characterized in Section 4, recommends the product if and only if $p \leq \gamma^{-1}(v)<v$. Under this algorithm, buyers with all values $v$ are willing to follow the recommendations because, intuitively, informed sellers lower the prices offered to low-value buyers. By Proposition 3, this algorithm implements the same total buyer surplus, seller profits, and product allocation. Remarkably, in the case of consumption driven by algorithmic recommendations, third-degree price discrimination not only does not harm the total buyer surplus but also may be beneficial if the algorithm cannot fully control the buyer value information.


## 6 Conclusion

In this paper, we studied the question of optimal algorithmic recommendations in the presence of strategic pricing. We showed that optimal recommendations must strike a balance between increasing the trade surplus and inducing low prices. Algorithmic recommendations drastically change the predictions of third-degree price discrimination, whereby finer market segmentations by the sellers do not affect the total consumer surplus or seller profits but result in larger price spreads and a more equitable surplus distribution.

We view our work as a stepping stone toward a better understanding of algorithmic design in strategic settings, an area of growing importance at the intersection of economics and computer science (e.g., Goktas et al. (2025)). First, our model of algorithms is deliberately stylized to analyze strategic motives in a clear and tractable way. A practical implementation would ideally incorporate many engineering concerns from which we abstracted away, such as value estimation details, computational complexity, and robustness. Second, it would be interesting to study market structures for algorithm providers and understand which of the algorithms that we characterize are favored by one or another market structure. Third, the developed ideas of algorithmic decisions can be exported beyond consumption settings, such as to algorithmic matching or algorithmic negotiations. All this further research can be built upon the analytical framework proposed in this paper.


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## Appendix: Ommited Proofs


## A Proof of Lemma 1

Take any algorithm $r$. For each $p \geq 0$, let $q_{r}(p) \triangleq \int_{0}^{1} r(v, p) d G(v)$ denote the probability with which the product is recommended, and thus purchased, under $r$. We define a new algorithm $\hat{r}$ as $\hat{r}(v, p) \triangleq \mathbb{1}\left(v>G^{-1}\left(1-q_{r}(p)\right)\right)$. At each price $p$, this algorithm recommends the product with the same probability as $r, 1-G\left(G^{-1}\left(1-q_{r}(p)\right)\right)=q_{r}(p)$. Moreover, the expected value of the product, conditional on the recommendation, is greater under $\hat{r}$ than under $r$. As a result, the buyer will purchase the product whenever it is recommended by $\hat{r}$, and at each price $p$, the seller will earn the same profit under both $r$ and $\hat{r}$. Therefore, $\hat{r}$ has an equilibrium that attains a greater buyer surplus than $r$ with the same seller profit as $r$. $\square$


## B Proof of Proposition 1

By the revelation principle, we can study algorithm design by analyzing direct mechanisms in which the seller reports the type to the designer and the designer chooses which valuations to allocate to the seller and at which price. Furthermore, by Lemma 1, we can focus on threshold allocations. The designer's problem can thus be stated as follows:

$$
\begin{array}{rlr}
\max _{\hat{v}:[0,1] \rightarrow[0,1], p:[0,1] \rightarrow \mathbb{R}_{+}} & \int_{0}^{1} \int_{\hat{v}(c)}^{1}(v-p(c)) \mathrm{d} G \mathrm{~d} F & \\
\text { s.t. } & \int_{\hat{v}(c)}^{1}(p(c)-c) \mathrm{d} G \geq \int_{\hat{v}\left(c^{\prime}\right)}^{1}\left(p\left(c^{\prime}\right)-c\right) \mathrm{d} G & \forall c, c^{\prime} \in[0,1] \\
& \int_{\hat{v}(c)}^{1}(p(c)-c) \mathrm{d} G \geq 0 & \forall c \in[0,1]
\end{array}
$$

One way to solve this problem is to reformulate it in familiar terms. Because the value is continuously distributed, the expected trade probability $q \triangleq \int_{\hat{v}}^{1} \mathrm{~d} G$ is strictly
decreasing in $\hat{v}$, spanning [ 0,1 ] as $\hat{v}$ spans [ 0,1 ]. Hence, $q$ and $v$ are in a one-to-one relationship, and instead of maximizing over $\hat{v}(c)$, we can maximize over $q(c)$. With a small abuse of notation, denote by $\hat{v}(q)$ the threshold that results in a given $q$ and by $V(q) \triangleq \int_{\hat{v}(q)}^{1} v \mathrm{~d} G$ the corresponding trade surplus. The trade surplus is strictly increasing in $q$ with $V(0)=0$ and $V(1)=\mathbb{E}[v]$. Moreover,

$$
\frac{\mathrm{d} V}{\mathrm{~d} q}=\frac{\partial V / \partial \hat{v}}{\partial q / \partial \hat{v}}=\frac{-\hat{v} g(\hat{v})}{-g(\hat{v})}=\hat{v}(q) .
$$

As such, $V(q)$ is a concave function with $V^{\prime}(0)=1$ and $V^{\prime}(1)=0$. Finally, we denote the expected revenue by $t(c) \triangleq p(c) \int_{\hat{v}(c)}^{1} \mathrm{~d} G$. In terms of these variables, we can restate problem (12) as follows:

$$
\begin{array}{rlr}
\max _{q:[0,1] \rightarrow[0,1], t:[0,1] \rightarrow \mathbb{R}_{+}} \int_{0}^{1}(V(q(c))-t(c)) \mathrm{d} F & \\
\text { s.t. } t(c)-c q(c) \geq t\left(c^{\prime}\right)-c q\left(c^{\prime}\right) & \forall c, c^{\prime} \in[0,1] \\
t(c)-c q(c) \geq 0 & \forall c \in[0,1] .
\end{array}
$$

Problem (14) is analogous to the problem analyzed by Baron and Myerson (1982) if $q$ is interpreted as a quantity produced and $V$ is interpreted as the welfare generated by producing quantity $q$. Its celebrated solution sets the optimal quantity to equalize marginal welfare benefits with virtual costs and the optimal transfer to guarantee the incentive-compatible profit distribution:

$$
\begin{aligned}
V^{\prime}(q(c)) & =\gamma(c) \\
t(c)-q(c) c & =\int_{c}^{1} q(x) \mathrm{d} x=\int_{c}^{1} 1-G(\gamma(x)) \mathrm{d} x
\end{aligned}
$$

By Equation 13, we can translate this solution back to problem (12) as

$$
\begin{aligned}
\hat{v}(c) & =\gamma(c), \\
p(c) & =c+\frac{\int_{c}^{1} 1-G(\gamma(x)) \mathrm{d} x}{1-G(\gamma(c))} \\
& =c+\frac{\int_{c}^{1}(x-c) g(\gamma(x)) \gamma^{\prime}(x) \mathrm{d} x}{1-G(\gamma(c))} \quad \text { (integration by parts) } \\
& \left.=c+\frac{\int_{\gamma(c)}^{\gamma(1)}\left(\gamma^{-1}(v)-c\right) g(v) \mathrm{d} v}{1-G(\gamma(c))} \quad \text { (change of variable with } v=\gamma(x)\right) \\
& =\frac{\int_{\gamma(c)}^{\gamma(1)} \gamma^{-1}(x) g(x) \mathrm{d} x}{1-G(\gamma(c))} \\
& =\mathbb{E}\left[\gamma^{-1}(v) \mid v \geq \gamma(c)\right] \\
& =y(\gamma(c)) .
\end{aligned}
$$

We now show that the algorithm and the equilibrium in the statement attain the same outcome as above. First, the buyer is willing to purchase the product when recommended because

$$
\mathbb{E}[v \mid y(v) \geq p(c)]=\mathbb{E}[v \mid y(v) \geq y(\gamma(c))]=\mathbb{E}[v \mid v \geq \gamma(c)] \geq \mathbb{E}\left[\gamma^{-1}(v) \mid v \geq \gamma(c)\right]=p(c) .
$$

Thus, the expected value of the product conditional on each possible price exceeds the price.

Second, the seller with each type $c$ is willing to set price $y(\gamma(c))$. Deviating to another price in $[y(\gamma(0)), y(\gamma(\bar{c}))]$ is not profitable because of the incentive compatibility constraints of the mechanism. Deviating to a price below $y(\gamma(0))$ or above $y(\gamma(\bar{c}))$ is not profitable either because it results in a lower profit than $p=y(\gamma(0))$ or no trade.

Finally, if the buyer follows the recommendation and each type $c$ sets price $y(\gamma(c))$, the trade occurs if and only if $y(v) \geq y(\gamma(c))$, or equivalently, if $v \geq \hat{v}(c)=\gamma(c)$.

In summary, the algorithm described in the statement implements the solution to problem (12) in equilibrium. Therefore, it is a buyer-optimal algorithm. $\square$


## C Proof of Proposition 2

Under the buyer-optimal algorithm, trade occurs if and only if $v \geq \gamma(c)$. If $F$ has a decreasing reversed hazard rate, then $\gamma(c)=c+\frac{F(c)}{f(c)}$ implies that $\gamma^{\prime}(c) \geq 1$. Under the ex post optimal algorithm, trade occurs if and only if $v \geq r^{-1}(c)$, where $r(v)=v-\frac{1-G(v)}{g(v)}$ is the virtual valuation function. If $G$ has an increasing hazard rate, then $r^{\prime}(v)>1$, which implies that

$$
\frac{d}{d c} r^{-1}(c)=\frac{1}{r^{\prime}\left(r^{-1}(c)\right)}<1 .
$$

Thus, $\gamma(c)$ is steeper than $r^{-1}(c)$, meaning that in the $(c, v)$-space, the curve $v=\gamma(c)$ crosses the curve $v=r^{-1}(c)$ at most once and from below.

Since $\gamma(c)$ starts at 0 when $c=0$ and ends above 1 when $c=1$, while $r^{-1}(c)$ starts above 0 at $c=0$ and ends at 1 at $c=1$, it follows that for low values of $c, \gamma(c)<r^{-1}(c)$, whereas for high values of $c, \gamma(c)>r^{-1}(c)$. Consequently, there exists a unique crossing point $c^{*}$ such that $\gamma\left(c^{*}\right)=r^{-1}\left(c^{*}\right)$. We then obtain qualitatively the same picture as the right panel of Figure 2. This establishes the result. $\square$


## D Proofs of Proposition 3 and Proposition 4

To prove Proposition 3, we establish a version of the payoff equivalence theorem for our model (cf. Myerson (1981) and Krishna (2009)).

Lemma 2 (Payoff Equivalence). For each $i \in\{1,2\}$, take an algorithm $r_{i}$, market segmentation $\mathcal{I}_{i}$, and equilibrium $\mathcal{E}_{i}$. Suppose that $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ have the same allocation rule in terms of $v$ and $c$ and the same profit of the seller at type $c=1$. Then, the seller's profit of any type and the buyer surplus are identical between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$.

Proof. Let $q(v, c)$ denote the probability of a trade when the value is $v$ and the type is $c$, and let $q(c)=\int_{0}^{1} q(v, c) \mathrm{d} G(v)$ denote the expected probability of a trade for type $c$. Upon calculating these objects, we take expectation with respect to the possible segments. Let $\bar{\pi}$ denote the profit of the seller with the highest type, $c=1$. By assumption, $q(\cdot, \cdot)$ and $\bar{\pi}$ are the same between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. Additionally, let $\pi_{i}(c)$ and $t_{i}(c)$ denote the profit and the expected monetary transfer, respectively, at type $c$ in equilibrium $\mathcal{E}_{i}$.

In equilibrium $\mathcal{E}_{i}$, type $c=1$ cannot earn a strictly higher profit by imitating the pricing strategy of type $c^{\prime}$ in every segment in $\mathcal{I}_{i}$. This incentive compatibility constraint is

$$
t_{i}(c)-c q(c) \geq t_{i}\left(c^{\prime}\right)-c q\left(c^{\prime}\right), \forall c, c^{\prime} \in[0,1] .
$$

The envelope theorem implies that

$$
\pi_{i}(c)=\bar{\pi}+\int_{c}^{1} q(x) \mathrm{d} x
$$

The right-hand side does not depend on $i$. Thus, the seller's profit is the same between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ for every seller type. The buyer surplus is the same between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ because it is the total surplus from allocation rule $q(\cdot, \cdot)$ minus the seller's profit, neither of which depends on $i$. $\square$


## Proof of Proposition 3

Take any signal, $\mathcal{I}$. For any signal realization, the optimal algorithm induces a trade if and only if $v \geq \gamma(c)$. Hence, the ex post allocation of the product is independent of the signal, as is the total surplus. Furthermore, the highest seller type $c=1$ always earns zero profits. Lemma 2 then implies that the seller profit of all types and the buyer surplus are independent of the signal. $\square$


## Proof of Proposition 4

First, we show that as the signal becomes more informative, the distribution of prices set by each active seller type undergoes a mean-preserving spread. To see this, consider any signals $\mathcal{I}_{H}$ and $\mathcal{I}_{L}$ such that $\mathcal{I}_{H}$ is more informative than $\mathcal{I}_{L}$. Recall that $\hat{\pi}_{H}$ and $\hat{\pi}_{L}$ denote the respective ex ante distributions of the signal realizations.

Fix any active type $c$. For each $\alpha \in\{L, H\}$, let $G_{\alpha, s}^{c} \in \Delta[0,1]$ denote the posterior distribution of value $v$ conditional on (i) signal $s$ being realized under signal $\mathcal{I}_{\alpha}$ and (ii) $v \geq \gamma(c)$. The equilibrium price of type $c$ after observing signal realization $s$ under
signal $\mathcal{I}_{\alpha}$ is

$$
p(c \mid s, \alpha) \triangleq \int_{0}^{1} \gamma^{-1}(v) \mathrm{d} G_{\alpha, s}^{c}(v) .
$$

Let $\mathcal{G}_{\alpha}^{c} \in \Delta \Delta[0,1]$ denote the distribution of posteriors $G_{\alpha, s}^{c}$ for a fixed $c$. Specifically, $\mathcal{G}_{\alpha}^{c}$ is the distribution of random variable $G_{\alpha, s}^{c}$ with $s \sim \hat{\pi}_{\alpha}$. Because signal $\mathcal{I}_{H}$ is more informative than signal $\mathcal{I}_{L}, \mathcal{G}_{H}^{c}$ is a mean-preserving spread of $\mathcal{G}_{L}^{c} .{ }^{20}$ As the price is linear in posterior $G_{\alpha, s}^{c}$, the mean-preserving spread relation between the distributions of posteriors, $\mathcal{G}_{H}^{c}$ and $\mathcal{G}_{L}^{c}$, imply the mean-preserving spread relation between real-valued random variables $p(c \mid s, H)$ and $p(c \mid s, L)$. Therefore, we conclude that $p(c \mid s, H)$ with $s \sim \hat{\pi}_{H}$ is a mean-preserving spread of $p(c \mid s, L)$ with $s \sim \hat{\pi}_{L}$. Therefore, the distribution of prices set by each active type under signal $\mathcal{I}_{H}$ is a mean-preserving spread of the price distribution under signal $\mathcal{I}_{L}$.

Second, we establish the results on monotone partitions. In what follows, we view a monotone partitional signal as a partition of [0, 1] and use an "interval" to mean an interval with a positive length, excluding a singleton set.

Take any monotone partitional signals, $\mathcal{I}_{H}$ and $\mathcal{I}_{L}$, such that $\mathcal{I}_{H}$ is finer than $\mathcal{I}_{L}$. We can create partition $\mathcal{I}_{H}$ by applying the following operations finitely many times to partition $\mathcal{I}_{L}$ : (i) taking an interval from $\mathcal{I}_{L}$ and dividing it into two subintervals or (ii) taking an interval from $\mathcal{I}_{L}$ and fully revealing the values within it. The latter operation means partitioning interval $[a, b]$ into $\{\{v\}\}_{v \in[a, b]}$. To obtain our result, it suffices to show that applying (i) or (ii) to any given monotone partitional signal leads to a meanpreserving contraction of the buyer surplus at any seller type. We consider these two operations in turn.

Operation (i). Fix any monotone partitional signal $\mathcal{I}$ that is different from the fully informative signal. Suppose that we take interval $\left[v_{i}, v_{i+1}\right]$ from $\mathcal{I}$ and split it into $\left[v_{i}, \hat{v}\right]$ and $\left[\hat{v}, v_{i+1}\right]$ for some $\hat{v} \in\left(v_{i}, v_{i+1}\right)$. We show that after this partitioning, the

[^14]buyer surplus $w(v, c)$, when we fix $c$ but draw $v$ from $G$, undergoes a mean-preserving contraction.

First, we consider the values and types that are affected by the operation, i.e., $(v, c)$ such that $\gamma(c) \leq v<v_{i+1}$. Before the operation, the buyer's ex post payoff is

$$
w_{0}(v, c) \triangleq v-\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \in\left[\gamma(c), v_{i+1}\right]\right], \forall v \in\left[\gamma(c), v_{i+1}\right] .
$$

After the operation, the buyer's ex post payoff is

$$
w_{1}(v, c) \triangleq \begin{cases}v-\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \in[\gamma(c), \hat{v}]\right] & \text { if } \quad v \in[\gamma(c), \hat{v}] \\ v-\mathbb{E}_{\tilde{v} \sim G}\left[\gamma^{-1}(\tilde{v}) \mid \tilde{v} \in\left[\hat{v}, v_{i+1}\right]\right] & \text { if } \quad v \in\left[\hat{v}, v_{i+1}\right] .\end{cases}
$$

Note that by applying Operation (i), the ex post payoff of the buyer with value $v \in$ $[\gamma(c), \hat{v}]$ increases because of the lower price and that of $v \in\left[\hat{v}, v_{i+1}\right]$ decreases because of the higher price.

For each $k \in\{0,1\}$, consider the distribution of $w_{k}(v, c)$ when $v \sim G\left(\cdot \mid \tilde{v} \in\left[\gamma(c), v_{i+1}\right]\right)$. First, they have the same mean because the expected price remains the same before and after the operation. Second, because $w_{1}(\cdot, c)$ crosses $w_{0}(\cdot, c)$ once from above, the CDF of $w_{1}(v, c)$ crosses the CDF of $w_{0}(v, c)$ once from above. The equal mean property and the single-crossing property imply, by Theorem 3.A. 44 (Condition 3.A.59) of Shaked and Shanthikumar (2007), that $w_{0}(v, c)$ is a mean-preserving spread of $w_{1}(v, c)$ when $v \sim G\left(\cdot \mid \tilde{v} \in\left[\gamma(c), v_{i+1}\right]\right)$.

Therefore, for a fixed $c$, the buyer's ex post surplus conditional on $v \in\left[\gamma(c), v_{i+1}\right]$ under signal $\mathcal{I}_{L}$ is a mean-preserving spread of that under signal $\mathcal{I}_{H}$. The same relationship trivially holds for the ex post surpluses of value $v<\gamma(c)$ or $v>v_{i+1}$ because those types do not trade or continue to face the same price. In summary, for any fixed $c$, the buyer's ex post surplus under signal $\mathcal{I}_{L}$ is a mean-preserving spread of that under signal $\mathcal{I}_{H}$ conditional on each of the three cases, $v \in\left[\gamma(c), v_{i+1}\right], v<\gamma(c)$, and $v>v_{i+1}$. The mean-preserving spread relationship is closed under mixtures (e.g., Theorem 3.A.12(b) of Shaked and Shanthikumar (2007)). Thus, for any fixed $c$, the distribution of the
buyer's surplus under signal $\mathcal{I}_{L}$ is a mean-preserving spread of the distribution of the buyer's surplus under signal $\mathcal{I}_{H}$.

Operation (ii). We can apply the same logic as the case of Operation (i) by defining $w_{1}(v, c)$ as

$$
w_{1}(v, c) \triangleq v-\gamma^{-1}(v), \forall v \in\left[v_{i}, v_{i+1}\right] .
$$ $\square$


## E Proof of Proposition 5

The proof follows that of Proposition 1. The designer's problem is stated as follows:

$$
\begin{array}{rlr}
\max _{\hat{v}:[0,1] \rightarrow[0,1], p:[0,1] \rightarrow \mathbb{R}_{+}} & \int_{0}^{1} \int_{\hat{v}(c)}^{1} \alpha(v-p(c))+(1-\alpha)(p(c)-c) \mathrm{d} G \mathrm{~d} F & \\
\text { s.t. } & \int_{\hat{v}(c)}^{1}(p(c)-c) \mathrm{d} G \geq \int_{\hat{v}\left(c^{\prime}\right)}^{1}\left(p\left(c^{\prime}\right)-c\right) \mathrm{d} G & \forall c, c^{\prime} \in[0,1] \\
& \int_{\hat{v}(c)}^{1}(p(c)-c) \mathrm{d} G \geq 0 & \forall c \in[0,1]
\end{array}
$$

For $\alpha>0$, the designer's objective can also be written as

$$
\int_{0}^{1} \int_{\hat{v}(c)}^{1} v-p(c)+\frac{1-\alpha}{\alpha}(p(c)-c) \mathrm{d} G \mathrm{~d} F .
$$

Repeat the same step to rewrite $\int_{\hat{v}(c)}^{1} v \mathrm{~d} G$ as in the proof of Proposition 1. We can then write the designer's problem as

$$
\begin{array}{cl}
\max _{q:[0,1] \rightarrow[0,1], t:[0,1] \rightarrow \mathbb{R}_{+}} \int_{0}^{1}\left[V(q(c))-t(c)+\frac{1-\alpha}{\alpha}(t(c)-c q(c))\right] \mathrm{d} F, & \\
\text { s.t. } t(c)-c q(c) \geq t\left(c^{\prime}\right)-c q\left(c^{\prime}\right) & \forall c, c^{\prime} \in[0,1], \\
t(c)-c q(c) \geq 0 & \forall c \in[0,1] .
\end{array}
$$

We consider two cases.

Case 1: $\alpha \geq \frac{1}{2}$ When the designer places a weakly higher weight on buyer surplus than seller profit (i.e., when $\alpha \geq \frac{1}{2}$ ), Problem (17) is analogous to the problem analyzed by Baron and Myerson (1982), where the weight " $\alpha$ " in their Lemma 2 is replaced by $\frac{1-\alpha}{\alpha}$ for our proof. Thus, its solution sets the optimal quantity to equalize marginal welfare benefits with $\alpha$-virtual costs and the optimal transfer to guarantee the incentivecompatible profit distribution:

$$
\begin{aligned}
V^{\prime}(q(c)) & =\gamma_{\alpha}(c) \\
t(c)-q(c) c & =\int_{c}^{1} q(x) \mathrm{d} x=\int_{c}^{1} 1-G\left(\gamma_{\alpha}(x)\right) \mathrm{d} x
\end{aligned}
$$

The rest of of the proof is identical with that of Proposition 1, where we replace $\gamma(c)$ in that proof with $\gamma_{\alpha}(c)$.

We use the following observation for the next case: At $\alpha=\frac{1}{2}$, the $\alpha$-virtual costs equals the seller's true cost. Consequently, under the $\frac{1}{2}$-optimal algorithm, type $c$ sets a price of

$$
y_{\frac{1}{2}}(c) \triangleq \mathbb{E}_{\tilde{v} \sim G}[\tilde{v} \mid \tilde{v} \geq c],
$$

and the algorithm recommends the product if and only if $v \geq c$. As a result, the product allocation is efficient and buyer surplus is 0.

Case 2: $\alpha<\frac{1}{2}$ In this case, the designer places a strictly higher weight on seller profit than buyer surplus. We show that the solution to the designer's problem with weight $\alpha^{\prime}<\frac{1}{2}$ is equal to the solution with weight $\alpha=\frac{1}{2}$. Suppose to the contrary that between the solution with weight $\alpha^{\prime}$ and the solution with weight $\alpha$, either buyer surplus or seller profit is different. Then, both buyer surplus and seller profit must be different. For example, if the two solutions have the same buyer surplus but the solution with weight $\alpha^{\prime}$ has a strictly higher seller profit, then the designer with $\alpha=\frac{1}{2}$ would mimic the designer with weight $\alpha^{\prime}$.

If both buyer surplus and seller profit are different, then it must be the case that
under the solution with a higher weight on seller profit, $\alpha^{\prime}$, the seller profit is strictly higher and the buyer surplus is strictly lower than under $\alpha=\frac{1}{2}$. However, it would mean that buyer surplus at $\alpha^{\prime}$ would be negative because buyer surplus at $\alpha=\frac{1}{2}$ is 0. This is a contradiction, because in equilibrium, the buyer can secure zero payoffs by not purchasing anything. Therefore, the solution to the designer's problem with weight $\alpha^{\prime}<\frac{1}{2}$ is equal to the solution at weight $\alpha=\frac{1}{2}$. In terms of the $\alpha$-virtual costs, this means that we can use (6) for any $\alpha \in[0,1]$. $\square$


## F Proofs of Proposition 6 and Proposition 7

We prove a result that implies both Proposition 6 and Proposition 7 as corollaries. Assume that the sellers face an information structure $\mathcal{I}=(S, \pi)$ that consists of a set $S=\times_{j} S_{j}$ of signal realizations $s_{j} \in S_{j}$ privately observed by each seller and a family of probability distributions $\{\pi(\cdot \mid v)\}_{v \in[0,1]^{J}}$ over $S$. We write $\tilde{s}_{j}$ for seller $j$ 's signal as a random variable and $s_{j} \in S_{j}$ for a generic realization.

Denote the set of real sellers by $\mathcal{J} \triangleq\{1, \ldots, J\}$ and the set of all sellers, together with a dummy seller, by $\mathcal{J}_{0} \triangleq\{0,1, \ldots, J\}$. When we say that the buyer purchases from (or transacts with) seller 0, it means that the buyer does not purchase from any seller $j=1, \ldots, J$. The profiles of the signal realizations, values, types, and prices are denoted as $\mathbf{s}, \mathbf{v}, \mathbf{c}$, and $\mathbf{p}$, respectively. When we refer to a profile that excludes seller $j$, we use notations such as $\mathbf{s}_{-j}$ and $\mathbf{v}_{-j}$. With a slight abuse of notation, we write $F, G, F_{-j}$, and $G_{-j}$, for the distributions of $\mathbf{c}, \mathbf{v}, \mathbf{c}_{-j}$, and $\mathbf{v}_{-j}$, respectively. Unless otherwise stated, these vectors and joint distributions exclude the dummy seller.

Recall that we defined an auxiliary random variable

$$
\theta_{j}=v_{j}-\max _{k \in \mathcal{J}_{0} \backslash\{j\}}\left\{v_{k}-\gamma_{k}\left(c_{k}\right)\right\} .
$$

For each $j$, let $\bar{v}_{j}\left(s_{j}\right)$ be the supremum of the support of the posterior distribution of $v_{j}$ conditional on $\tilde{s}_{j}=s_{j}$. Define $\bar{c}_{j}\left(s_{j}\right)=\gamma_{j}^{-1}\left(\bar{v}_{j}\left(s_{j}\right)\right)$. For each $j \in \mathcal{J}, s_{j} \in S_{j}$, and
$c_{j} \in\left[0, \bar{c}_{j}\left(s_{j}\right)\right]$, define

$$
p_{j}^{*}\left(c_{j}, s_{j}\right) \triangleq \mathbb{E}_{\theta_{j}}\left[\gamma_{j}^{-1}\left(\theta_{j}\right) \mid \theta_{j} \geq \gamma_{j}\left(c_{j}\right), \tilde{s}_{j}=s_{j}\right] .
$$

The conditional expectation is well-defined for any $c_{j} \leq \bar{c}_{j}\left(s_{j}\right)$ or equivalently $\gamma_{j}\left(c_{j}\right) \leq$ $\bar{v}_{j}\left(s_{j}\right)$ because $\theta_{j}=\bar{v}_{j}\left(s_{j}\right)$ is in the support of the posterior distribution of $v_{j}$ conditional on $\tilde{s}_{j}=s_{j}$. This is because $v_{j}=\bar{v}_{j}\left(s_{j}\right)$ is in the support, and $v_{k} \leq \gamma_{k}\left(c_{k}\right)$ for all $k \neq j$ could occur with a positive probability. Given signal realizations $\mathbf{s}$, we say that seller $j$ 's type $c_{j}$ is active if $c_{j} \leq \bar{c}_{j}\left(s_{j}\right)$. Otherwise, the type is inactive. Seller $j$ 's price $p_{j}$ is said to be active if $p_{j}$ is in the range of $p_{j}^{*}\left(\cdot, s_{j}\right)$; otherwise, the price is called inactive. Note that any active type sets an active price.

In this appendix, for simplicity, we focus on the case in which for each seller $j$ and active price $p_{j}$, there exists a unique type, denoted by $p_{j}^{*-1}\left(p_{j}, s_{j}\right)$, that solves $p_{j}^{*}\left(c_{j}, s_{j}\right)=p_{j}$. This is the case, for example, if value $v_{j}$ has a full support on [0, 1] conditional on each signal realization $s_{j}$. The proof for the case in which multiple types may set the same price is relegated to the Supplementary Material. ${ }^{21}$ For any inactive price, we set $p_{j}^{*-1}\left(p_{j}, s_{j}\right)=1$; for the dummy seller $j=0$, we set $p_{j}^{*-1}\left(p_{j}, s_{j}\right)=0$.

We now define an algorithm that we will prove to be optimal for the buyer.
Definition 3. Define the candidate algorithm as follows: At each profile of signal realizations $\mathbf{s}=\left(s_{1}, \ldots, s_{J}\right)$, values $\mathbf{v}=\left(v_{1}, \ldots, v_{J}\right)$, and prices $\mathbf{p}=\left(p_{1}, \ldots, p_{J}\right)$, the candidate algorithm recommends trading with seller $j^{*}(\mathbf{v}, \mathbf{p}, \mathbf{s})$ such that

$$
j^{*}(\mathbf{v}, \mathbf{p}, \mathbf{s}) \in \underset{j \in \mathcal{J}_{0}}{\operatorname{argmax}} v_{j}-\gamma_{j}\left(p_{j}^{*-1}\left(p_{j}, s_{j}\right)\right) .
$$

If multiple sellers attain the maximized value in Equation 20, the algorithm breaks ties in favor of sellers such that $p_{j}^{*-1}\left(p_{j}, s_{j}\right)>0$. Other than this restriction, ties are broken arbitrarily.

The following result characterizes the buyer-optimal algorithm and equilibrium under

[^15]any information structure.

Proposition 11 (Market Segmentation with Competing Sellers). For any information structure $\mathcal{I}$, the corresponding candidate algorithm is a buyer-optimal algorithm. In equilibrium, seller $j$ of type $c_{j} \leq \bar{c}_{j}\left(s_{j}\right)$ posts price $p_{j}^{*}\left(c_{j}, s_{j}\right)$, and any type $c_{j}>\bar{c}_{j}\left(s_{j}\right)$ sets some inactive price above 1. Under this algorithm and pricing, for any realized profile of values and costs, the buyer trades with seller $j^{*} \in \operatorname{argmax}_{j \in \mathcal{J}_{0}} v_{j}-\gamma_{j}\left(c_{j}\right)$. Moreover, the profit of any seller of any type and the total buyer surplus are independent of $\mathcal{I}$.

Proof. The proof consists of three steps.
Step 1: Characterizing a buyer-optimal direct mechanism. First, we derive a buyeroptimal direct mechanism, where the direct mechanism is in the sense of Myerson (1981), i.e., a mechanism that fully controls product allocation and transfers across all players. By the revelation principle, since any information structure combined with an algorithm can be viewed as an indirect mechanism, a buyer-optimal mechanism must achieve a weakly higher buyer surplus than a buyer-optimal algorithm under any information structure.

Given a profile of values $\mathbf{v}$ and reported types $\mathbf{c}$, let $q_{j}(\mathbf{v}, \mathbf{c})$ be the probability of allocating seller $j$ 's product to the buyer, and let $t_{j}(\mathbf{v}, \mathbf{c})$ be the monetary transfer from the buyer to seller $j$. The direct mechanism design problem can be written as:


$$
\begin{aligned}
\max _{q:[0,1]^{2 J} \rightarrow[0,1], t:[0,1]^{2 J} \rightarrow \mathbb{R}} \sum_{j=1}^{J} \int_{[0,1]^{J}} \int_{[0,1]^{J}}\left(v_{j} q_{j}(\mathbf{v}, \mathbf{c})-t_{j}(\mathbf{v}, \mathbf{c})\right) \mathrm{d} F \mathrm{~d} G & \\
\text { s.t. } & T_{j}\left(c_{j}\right)-c_{j} Q_{j}\left(c_{j}\right) \geq T_{j}\left(c_{j}^{\prime}\right)-c_{j} Q_{j}\left(c_{j}^{\prime}\right), \\
& T_{j}\left(c_{j}\right)-c_{j} Q_{j}\left(c_{j}\right) \geq 0, \\
& Q_{j}\left(c_{j}\right)=\int_{[0,1]^{J}} \int_{[0,1]^{J-1}} q_{j}\left(\mathbf{v}, c_{j}, \mathbf{c}_{-j}\right) \mathrm{d} F_{-j} \mathrm{~d} G, \quad \forall j \in \mathcal{J}, c_{j}, c_{j}^{\prime} \in[0,1], \\
& T_{j}\left(c_{j}\right)=\int_{[0,1]^{J}} \int_{[0,1]^{J-1}} t_{j}\left(\mathbf{v}, c_{j}, \mathbf{c}_{-j}\right) \mathrm{d} F_{-j} \mathrm{~d} G, \quad \forall j \in \mathcal{J}, c_{j} \in[0,1], \\
& \sum_{j \in \mathcal{J}} q_{j}(\mathbf{v}, \mathbf{c}) \leq 1,
\end{aligned}
$$

The standard mechanism design arguments imply that only the participation constraint of $c_{j}=1$ for each seller $j$ binds at the optimum, and that the IC constraints are equivalent to the local IC constraints (see Equation 22 below) with $Q_{j}(\cdot)$ being weakly decreasing for each $j$ (cf. Baron and Myerson (1982)). Using the local IC constraints, we can rewrite the expected transfer as follows:

$$
\begin{aligned}
\sum_{j=1}^{J} \int_{[0,1]^{J}} \int_{[0,1]^{J}} t_{j}(\mathbf{v}, \mathbf{c}) \mathrm{d} F \mathrm{~d} G & =\sum_{j=1}^{J} \int_{0}^{1} T_{j}(x) f_{j}(x) \mathrm{d} x \\
& =\sum_{j=1}^{J} \int_{0}^{1}\left(x+\frac{F_{j}(x)}{f_{j}(x)}\right) Q_{j}(x) f(x) \mathrm{d} x \\
& =\sum_{j=1}^{J} \int_{0}^{1} \gamma_{j}(x) Q_{j}(x) f(x) \mathrm{d} x
\end{aligned}
$$

Plugging this into the objective and using $Q_{j}(x)=\int_{[0,1]^{J}} \int_{[0,1]^{J-1}} q_{j}\left(\mathbf{v}, x, \mathbf{c}_{-j}\right) \mathrm{d} F_{-j} \mathrm{~d} G$, we can rewrite the designer's problem as the choice of a product allocation rule to maximize virtual surplus:

$$
\int_{[0,1]^{J}} \int_{[0,1]^{J}} \sum_{j=1}^{J}\left(v_{j}-\gamma_{j}\left(c_{j}\right)\right) q_{j}(\mathbf{v}, \mathbf{c}) f(\mathbf{c}) \mathrm{d} \mathbf{c} \mathrm{~d} G .
$$

We can maximize the virtual surplus by choosing $\left\{q_{j}(\mathbf{v}, \mathbf{c})\right\}_{j \in \mathcal{J}}$ to maximize the integrand for each $(\mathbf{v}, \mathbf{c})$. The optimal mechanism allocates seller $j$ 's product to the buyer, $q_{j}(\mathbf{v}, \mathbf{c})=1$, if seller $j$ has the highest virtual surplus $v_{j}-\gamma_{j}\left(c_{j}\right)$ and it is nonnegative; otherwise, $q_{j}(\mathbf{v}, \mathbf{c})=0$. Let $q^{D}$ be this optimal product allocation rule and $Q_{j}^{D}\left(c_{j}\right)$ be the interim allocation probability for seller $j$ with type $c_{j}$. Under the optimal mechanism, the monetary transfer $T^{D}$ must satisfy

$$
T_{j}^{D}\left(c_{j}\right)=Q^{D}\left(c_{j}\right) c_{j}+\int_{c_{j}}^{1} Q_{j}^{D}(x) \mathrm{d} x, \forall c_{j} \in[0,1]
$$

Under the optimal mechanism, the participation constraints for the highest types bind and thus each seller $j$ with the type $c_{j}=1$ earns zero profit.

Step 2: Connecting with the candidate algorithm. In this step, we show that the candi-
date algorithm has an equilibrium in which the product allocation rule and the profits of the highest types are the same as those in the optimal direct mechanism.

First, suppose that each seller follows the pricing strategy described in the proposition. Take any profile of signal realizations $\mathbf{s}$, values $\mathbf{v}$, and types $\left(\hat{c}_{1}, \ldots, \hat{c}_{J}\right)$. Let $\mathbf{p}$ be the resulting price profile posted by the sellers. For each seller that posts an active price $p_{j}$, the candidate algorithm calculates the unique type that sets price $p_{j}$ according to Equation 19 and recommends a seller that maximizes virtual surplus. Also, the candidate algorithm never recommends a seller that sets an inactive price, because their corresponding virtual surplus is always negative. Thus if all sellers use the pricing rule in Equation 19 and the buyer always follows the recommendations, then for any profile of prices that can arise, the candidate algorithm recommends the product of the seller with the highest virtual surplus and thus induces the same product allocation as the buyer-optimal mechanism. In particular, the buyer never purchases the product from a seller who has a negative virtual surplus, which means that seller $j$ with $c_{j}=1$ earns zero profit.

We now show that the pricing rule in Equation 19 is indeed an equilibrium if the buyer always purchases the recommended product. Take any active seller $j \in \mathcal{J}$ with type $c_{j}$. The seller cannot profit from setting an inactive price, because the candidate algorithm never recommends a seller at an inactive price. Alternatively, suppose that the seller has type $c_{j}$ but deviates to an active price which would be chosen by type $c_{j}^{\prime}$. Let $H$ be the distribution of $\gamma_{j}^{-1}\left(\theta_{j}\right)$ conditional on $\tilde{s}_{j}=s_{j}$. We can compare the profits without and with the deviation as follows:

$$
\begin{aligned}
& \operatorname{Pr}\left(\gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j} \geq 0 \mid \tilde{s}_{j}=s_{j}\right) \cdot \mathbb{E}_{\theta_{j}}\left[\gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j} \mid \gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j} \geq 0, \tilde{s}_{j}=s_{j}\right] \\
= & \int_{c_{j}}^{1}\left(x-c_{j}\right) d H(x) \\
\geq & \int_{c_{j}^{\prime}}^{1}\left(x-c_{j}\right) d H(x) \\
= & \operatorname{Pr}\left(\gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j}^{\prime} \geq 0 \mid \tilde{s}_{j}=s_{j}\right) \cdot \mathbb{E}_{\theta_{j}}\left[\gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j} \mid \gamma_{j}^{-1}\left(\theta_{j}\right)-c_{j}^{\prime} \geq 0, \tilde{s}_{j}=s_{j}\right]
\end{aligned}
$$

Here, the first line is the profit from following the candidate strategy, and the last line is the profit from deviation.

The other case is when a deviating seller $j$ has an inactive type $c_{j}$. In this case, conditional on $s_{j}$, any possible realization of $\theta_{j}$ satisfies $\gamma_{j}\left(c_{j}\right)>\theta_{j}$. Thus, the profit from the deviation to active type $c_{j}^{\prime}$, which is given by the last line of the above inequalities, will be negative. We conclude that each seller has no profitable deviation.


The last part of this step is to show that the buyer is willing to purchase the product whenever recommended, i.e., conditional on knowing the identity and the price of the recommended product. We present a substantially stronger statement: The buyer is willing to follow recommendations even if she additionally observes the realized signal $s_{j}$ and the type $c_{j}$ of the recommended seller $j$. For each $c_{j}$, we have:

$$
\begin{aligned}
& \mathbb{E}\left[v_{j} \mid v_{j}-\gamma_{j}\left(c_{j}\right) \geq \max _{k \in \mathcal{J}_{0} \backslash\{j\}} v_{k}-\gamma_{k}\left(c_{k}\right), \tilde{s}_{j}=s_{j}\right] \\
= & \mathbb{E}\left[v_{j} \mid \gamma^{-1}\left(\theta_{j}\right) \geq c_{j}, \tilde{s}_{j}=s_{j}\right] \\
\geq & \mathbb{E}\left[\gamma_{j}^{-1}\left(\theta_{j}\right) \mid \gamma^{-1}\left(\theta_{j}\right) \geq c_{j}, \tilde{s}_{j}=s_{j}\right] \\
= & p_{j}^{*}\left(c_{j}, s_{j}\right),
\end{aligned}
$$

where the inequality holds because:

$$
v_{j} \geq \theta_{j}=v_{j}-\max _{k \in \mathcal{J}_{0} \backslash\{j\}}\left\{v_{k}-\gamma_{k}\left(c_{k}\right)\right\} \geq \gamma_{j}^{-1}\left(\theta_{j}\right) .
$$

Step 3: Establishing the "payoff equivalence." Let $\left\{\left(Q_{j}\left(c_{j}\right), T_{j}\left(c_{j}\right)\right\}_{j \in \mathcal{J}, c_{j} \in[0,1]}\right.$ be the interim allocation probability $Q_{j}\left(c_{j}\right)$ and expected revenue $T_{j}\left(c_{j}\right)$ for each seller $j$ and type $c_{j}$ under the candidate algorithm. Recall that $\left\{\left(Q_{j}^{D}\left(c_{j}\right), T_{j}^{D}\left(c_{j}\right)\right\}_{j \in \mathcal{J}, c_{j} \in[0,1]}\right.$ denote the corresponding objects in the optimal direct mechanism.

We have shown that (i) $\left\{\left(Q_{j}\left(c_{j}\right), T_{j}\left(c_{j}\right)\right\}_{j \in \mathcal{J}, c_{j} \in[0,1]}\right.$ is an equilibrium object and thus satisfies the first two constraints of Equation 21, i.e., the incentive compatibility and participation constraints; (ii) in the candidate algorithm, the profits of the highest seller types are 0; and (iii) $Q_{j}=Q_{j}^{D}$ for each seller $j$ because they come from the same ex
post product allocation rule. Thus, the interim expected revenue of each seller $j$ under the candidate algorithm must satisfy

$$
\begin{aligned}
T_{j}\left(c_{j}\right) & =Q\left(c_{j}\right) c_{j}+\int_{c_{j}}^{1} Q_{j}(x) d x \\
& =Q^{D}\left(c_{j}\right) c_{j}+\int_{c_{j}}^{1} Q_{j}^{D}(x) d x \\
& =T_{j}^{D}\left(c_{j}\right)
\end{aligned}
$$

where the first equality comes from (i) and (ii), the second from (iii), and the third from Equation 22. Therefore, the interim profit of each seller, $T_{j}\left(c_{j}\right)-Q\left(c_{j}\right) c_{j}$, is the same between the candidate algorithm and the optimal mechanism. As a result, the buyer surplus, which is the total surplus (uniquely determined by $q^{D}$ ) minus the seller profit, is the same between the algorithm described in the statement and the optimal mechanism in Step 1. $\square$

Proofs of Proposition 6 and Proposition 7 Proposition 6 holds by setting information structure $\mathcal{I}$ to the uninformative structure, e.g., $S_{j}=\{\emptyset\}$ for each seller $j$. Proposition 7 is a direct corollary of Proposition 11. $\square$


## G Proof of Proposition 8

The proof follows that of Proposition 4. Consider any seller $j$ with type $c_{j}$. Consider a fictitious situation in which seller $j$ 's prior belief over $\left(\mathbf{v}, \mathbf{c}_{-j}\right)$ is given by the conditional distribution of $\left(\mathbf{v}, \mathbf{c}_{-j}\right)$ given $\theta_{j} \geq c_{j}$ (calculated starting from the true prior). We call seller $j$ 's prior in this situation as the fictitious prior and any posterior belief updated from the fictitious prior a fictitious posterior.

Under any information structure, the price posted by seller $j$ with type $c_{j}$, which is (19), is linear in seller $j$ 's fictitious posterior belief over $\left(\mathbf{v}, \mathbf{c}_{-j}\right)$ (which pins down $\theta_{j}$ ) conditional on signal realization $s_{j}$. If seller $j$ obtains more information about v in the sense stated in the proposition, the seller also gains more information about $\left(\mathbf{v}, \mathbf{c}_{-j}\right)$. Thus, the distribution of seller $j$ 's fictitious posterior belief over $\left(\mathbf{v}, \mathbf{c}_{-j}\right)$ undergoes a
mean-preserving spread. The same logic as in the proof of Proposition 4 implies that seller $j$ 's price undergoes a mean-preserving spread. $\square$


## H Proof of Proposition 10

We borrow the notation from the proof of Proposition 1 and let $\mu=\mathbb{E}_{v \sim G}[v]$. Suppose that the buyer faces the optimal algorithm of Proposition 1. Because the buyer is willing to follow the algorithm's recommendation to purchase, it suffices to show that the buyer is also willing to follow the recommendation to not purchase. This constraint is equivalent to the condition that the buyer's ex ante payoff from following the recommendation weakly exceeds the payoff from always buying the product regardless of the recommendation. For any active price $p \in[0, p(\bar{c}))$, the condition is written as

$$
V(q(c))-t(c) \geq \mu-p(c)
$$

or

$$
V(q(c))-c q(c)-\int_{c}^{1} q(x) d x \geq \mu-c-\frac{\int_{c}^{1} q(x) \mathrm{d} x}{q(c)} .
$$

Because $q(c) \leq 1$, a sufficient condition for inequality (23) is

$$
V(q(c))-c q(c) \geq \mu-c .
$$

We can rewrite this inequality as

$$
\int_{\gamma(c)}^{1} v \mathrm{~d} G(v)-c \int_{\gamma(c)}^{1} 1 \mathrm{~d} G(v) \geq \int_{0}^{1} v \mathrm{~d} G(v)-c \int_{0}^{1} 1 \mathrm{~d} G(v),
$$

or, equivalently,

$$
\int_{0}^{\gamma(c)}[v-c] \mathrm{d} G(v) \leq 0 .
$$

Finally, the buyer follows the recommendation to not buy the product at any price $p$ that is not active, i.e., $p \geq p^{*}(\bar{c})$. Recall that the buyer-optimal algorithm provides no information about $v$ at price $p>p^{*}(\bar{c})$. Plugging $c=\bar{c}$ into $\int_{0}^{\gamma(c)}[v-c] \mathrm{d} G(v) \leq 0$, we
obtain $\mu-\bar{c} \leq 0$. Thus, if $p>p^{*}(\bar{c})$, we have $\mu-p \leq \mu-p^{*}(\bar{c})=\mu-\bar{c} \leq 0$. $\square$


## Supplementary Material for Appendix F

In the appendix, we assumed that each active price is posted by a unique type. In this Supplementary Material, we drop this assumption and prove that the candidate algorithm continues to maximize virtual surplus.

Recall the pricing equation (Equation 19). For each $j \in \mathcal{J}, s_{j} \in S_{j}$, and active price $p_{j} \in \mathbb{R}$, define

$$
C_{j}\left(p_{j}, s_{j}\right) \triangleq\left\{c_{j} \in[0,1]: p_{j}^{*}\left(c_{j}, s_{j}\right)=p_{j}\right\}
$$

as the set of the types of seller $j$ that choose price $p_{j}$. For each $p_{j}$, we define function $p_{j}^{*-1}\left(p_{j}, s_{j}\right)$ as follows: For each $j \in \mathcal{J}$,

$$
p_{j}^{*-1}\left(p_{j}, s_{j}\right)= \begin{cases}\max C_{j}\left(p_{j}, s_{j}\right) & \text { if } \quad C_{j}\left(p_{j}, s_{j}\right) \neq \emptyset \\ 1 & \text { if } \quad C_{j}\left(p_{j}, s_{j}\right)=\emptyset .\end{cases}
$$

For the dummy seller $j=0$, we set $p_{j}^{*-1}\left(p_{j}, s_{j}\right)=0$.
Suppose that each seller follows the pricing strategy described in the proposition. Take any profile of signal realizations $\mathbf{s}$, values $\mathbf{v}$, and types $\left(\hat{c}_{1}, \ldots, \hat{c}_{J}\right)$. Let $\mathbf{p}$ be the resulting price profile posted by the sellers. Suppose that the candidate algorithm recommends seller $j^{*} \in \mathcal{J}$. Without loss, assume $j^{*}=1$. We show that seller 1 has the highest, nonnegative virtual surplus. For each seller $j \in \mathcal{J}$, there exists some $c_{j}$ such that

$$
v_{j}-\gamma_{j}\left(p_{j}^{*-1}\left(p_{j}, s_{j}\right)\right)=v_{j}-\gamma_{j}\left(c_{j}\right) .
$$

If $p_{j}$ is an active price, $c_{j}=\max C_{j}\left(p_{j}, s_{j}\right)$. If $p_{j}$ is an inactive price, $c_{j}=1$ by construction. Let $c_{j}\left(p_{j}\right)$ be the type that satisfies Equation 24. Then Equation 20 implies that

$$
v_{1}-\gamma_{1}\left(c_{1}\left(p_{1}\right)\right) \geq \max _{k \in \mathcal{J}_{0} \backslash\{1\}} v_{k}-\gamma_{k}\left(c_{k}\left(p_{k}\right)\right),
$$

so that seller 1 has the largest virtual surplus under type profile $\left(c_{1}\left(p_{1}\right), \ldots, c_{J}\left(p_{J}\right)\right)$. The rest of the proof is devoted to showing that seller 1 has the largest virtual surplus under
type profile $\left(\hat{c}_{1}, \ldots, \hat{c}_{J}\right)$ as well, i.e.,

$$
v_{1}-\gamma_{1}\left(\hat{c}_{1}\right) \geq \max _{k \in \mathcal{J}_{0} \backslash\{1\}} v_{k}-\gamma_{k}\left(\hat{c}_{k}\right) .
$$

Note that if $p_{k}$ is inactive for some non-recommended seller $k$, any type $\hat{c}_{k}$ that sets $p_{k}$ satisfies $\hat{c}_{k}>\bar{c}_{k}\left(s_{k}\right)$, which implies $v_{k}-\gamma_{k}\left(\hat{c}_{k}\right)<0$. Thus, replacing one inactive type $c_{k}\left(p_{k}\right)$ with another inactive type $\hat{c}_{k}$ in the RHS of Equation 26 does not affect the inequality (and it does not affect the argument below). ${ }^{22}$ Thus, to simplify exposition, we assume that all sellers set active prices, or equivalently, $c_{k}\left(p_{k}\right) \leq \bar{c}_{k}\left(s_{k}\right)$ for each $k \in \mathcal{J}$. We change the type of each seller from $c_{j}\left(p_{j}\right)$ to $\hat{c}_{j}$ one by one and show that seller 1 continues to maximize virtual surplus at each step.

First, suppose that we change the type of seller 1 from $c_{1}\left(p_{1}\right)$ to $\hat{c}_{1}$. Because $c_{1}\left(p_{1}\right)=$ $\max C_{1}\left(p_{1}, s_{1}\right)$, we have $c_{1}\left(p_{1}\right) \geq \hat{c}_{1}$. Therefore, we obtain

$$
v_{1}-\gamma_{1}\left(\hat{c}_{1}\right) \geq \max _{k \in \mathcal{J}_{0} \backslash\{1\}} v_{k}-\gamma_{k}\left(c_{k}\left(p_{k}\right)\right) .
$$

Next, for each seller $\ell=2, \ldots, J$, we replace $c_{\ell}\left(p_{\ell}\right)$ in the RHS of Equation 27 with another active type $\hat{c}_{\ell} \neq c_{\ell}\left(p_{\ell}\right)$ that sets the same price $p_{\ell}$, and show that the inequality is preserved. To begin with, for a given $\ell \geq 2$, we consider the following inequalities:

$$
v_{1}-\gamma_{1}\left(\hat{c}_{1}\right) \geq \max \left\{\max _{k \in\{2,3 \ldots, \ell-1\}} v_{k}-\gamma_{k}\left(\hat{c}_{k}\right), v_{\ell}-\gamma_{\ell}\left(c_{\ell}\left(p_{\ell}\right)\right), \max _{k \in\{\ell+1, \ldots, J, 0\}} v_{k}-\gamma_{k}\left(c_{k}\left(p_{k}\right)\right)\right\},
$$

where we ignore $\max _{k \in\{2,3 \ldots, \ell-1\}} v_{k}-\gamma_{k}\left(\hat{c}_{k}\right)$ when $\ell=2$, and

$$
\max \left\{\max _{k \in\{1, \ldots, \ell-1\}} v_{k}-\gamma_{k}\left(\hat{c}_{k}\right), \max _{k \in\{\ell+1, \ldots, J, 0\}} v_{k}-\gamma_{k}\left(c_{k}\left(p_{k}\right)\right)\right\}<v_{\ell}-\gamma_{\ell}\left(\hat{c}_{\ell}\right) .
$$

Equation 28 means that seller 1 continues to maximize virtual surplus after we change the type of each seller $k \leq \ell-1$ from $c_{k}\left(p_{k}\right)$ to $\hat{c}_{k}$. Equation 29 means that the inequality is

[^16]reversed after we replace the type of seller $\ell$. When we consider a version of Equation 28 in which we replace $\ell$ with $k^{\prime}$, we refer to the inequality as Equation $28\left(k^{\prime}\right)$. Note that Equation 28( $\ell$ ) is Equation 28, and we have already shown that Equation 28(2) holds.

We also consider the following inequalities:

$$
\max _{k \in \mathcal{J}_{0} \backslash\{\ell\}} v_{k}-\gamma_{k}\left(c_{k}\right)>v_{\ell}-\gamma_{\ell}\left(c_{\ell}\left(p_{\ell}\right)\right)
$$

and

$$
\max _{k \in \mathcal{J}_{0} \backslash\{\ell\}} v_{k}-\gamma_{k}\left(c_{k}\right)<v_{\ell}-\gamma_{\ell}\left(\hat{c}_{\ell}\right) .
$$

We will show that if Equation 28 and Equation 29 hold, then Equation 30 and Equation 31 hold for a positive measure of type profiles $c_{-\ell}$, which, as we show, leads to a contradiction.

For each $\ell \geq 2$, assume that Equation $28\left(k^{\prime}\right)$ for $k^{\prime}=2, \ldots ., \ell$ and Equation 29 hold. Note that Equation 29 implies $c_{\ell}\left(p_{\ell}\right)>\hat{c}_{\ell} \geq 0$. Thus, we have $c_{\ell}\left(p_{\ell}\right)>0$. We consider four cases.


Case 1. First, suppose that (i) Equation 28 holds with strict inequality or (ii) $\hat{c}_{1}>0$. In either case, we can find a positive measure of type profiles $c_{-\ell}$ such that Equation 30 and Equation 31 hold. For example, if $\hat{c}_{1}>0$, then replacing $\hat{c}_{1}$ with a slightly lower $\hat{c}_{1}-\epsilon$ satisfies Equation 28 and Equation 29 with strict inequalities, because the virtual cost functions are assumed to be continuous and strictly increasing. We can then find a positive measure of type profiles $c_{-\ell}$ that maintain these strict inequalities, leading to Equation 30 and Equation 31 for a positive measure of $c_{-\ell}$ (conditional on $s_{\ell}$ ). This is a contradiction, because the existence of such $c_{-\ell}$ 's implies that there is a positive measure of $\theta_{\ell}$ 's such that $\theta_{\ell}>\gamma_{\ell}\left(c_{\ell}\left(p_{\ell}\right)\right)$ and $\theta_{\ell}<\gamma_{\ell}\left(\hat{c}_{\ell}\right)$, i.e., types $c_{\ell}\left(p_{\ell}\right)$ and $\hat{c}_{\ell}$ set different prices (see Equation 19).

Case 2. Suppose that (i) Equation $28\left(k^{\prime}\right)$ holds with equality at every step $k^{\prime} \leq \ell$, (ii) $\hat{c}_{1}=0$, (iii) and the RHS of Equation 28 is positive. Note that Points (i) and (ii) imply that $c_{1}\left(p_{1}\right)=\hat{c}_{1}=0$, so seller 1 sets a price for which there is a unique active type (to see this, note that if $c_{1}\left(p_{1}\right)>\hat{c}_{1}$ and Equation 28(2) holds with equality, then

Equation 25 would fail). This means that seller $\ell$ does not attain the maximized value of the RHS in Equation 28, because if both sellers 1 and $\ell$ maximize virtual surplus and $\hat{c}_{1}=0<c_{\ell}\left(p_{\ell}\right)$, the candidate algorithm would recommend seller $\ell$ instead of seller 1 (see the tie-breaking rule described in Definition 3). Because the RHS of Equation 28 is not determined by seller $\ell$ but both sides are positive, we can slightly increase the cost of each seller $k \neq 1, \ell$ to make Equation 28 and Equation 29 strict. We can then find a positive measure of type profiles $c_{-\ell}$ such that Equation 30 and Equation 31 hold, which leads to a contradiction by the same argument as in Case 1.

Case 3. Suppose that (i) Equation 28 holds with equality but there is some step $k^{\prime}<\ell$ at which Equation 28( $k^{\prime}$ ) is strict, (ii) $\hat{c}_{1}=0$, (iii) and the RHS of Equation 28 is positive. Point (i) implies that there is some $k^{\prime} \in\{2, \ldots, \ell-1\}$ such that Equation $28\left(k^{\prime}\right)$ is strict but Equation $28\left(k^{\prime}+1\right)$ holds with equality, which occurs only when the RHS of the inequality increases as we move from Equation $28\left(k^{\prime}\right)$ to Equation $28\left(k^{\prime}+1\right)$. It means that the RHS of Equation 28 is not determined by seller $\ell$, whose type did not change in earlier steps. By the same argument as Case 2, we can find type profiles $c_{-\ell}$ where Equation 28 and Equation 29 hold with strict inequalities, which leads to a contradiction.

Case 4. Suppose that both sides of Equation 28 are 0, and $\hat{c}_{1}=0$. Both sides of Equation $28\left(k^{\prime}\right)$ are 0 for every step $k^{\prime}=2, \ldots, \ell$, because after each replacement, the LHS of Equation $28\left(k^{\prime}\right)$ remains 0 and the RHS is weakly greater than 0 . By the same argument as in Case 2, we conclude that that seller $\ell$ does not attain the maximized value of the RHS in Equation 28 (otherwise, seller $\ell$ would be recommended according to the tie-breaking rule of the candidate algorithm). But it means that seller $\ell$ has a negative virtual surplus, which is a contradiction.

In summary, we have shown that for any information structure, if all sellers use the pricing rule in Equation 19 and the buyer always follows the recommendations, for any profile of prices that can arise, the candidate algorithm recommends the product of the seller with the highest virtual surplus. The rest of the proof follows the same argument as in the appendix.

[^0]:    *Ichihashi: Queen's University, Department of Economics, shotaichihashi@gmail.com. Smolin: Toulouse School of Economics, University of Toulouse Capitole and CEPR, alexey.v.smolin@gmail.com. We would like to thank Nemanja Antic, Heski Bar-Isaac, Dirk Bergemann, Alessandro Bonatti, Daniele Condorelli, Piero Gottardi, Emir Kamenica, Caio Lorecchio, Matthew Mitchell, Alessandro Pavan, Nicola Persico, and Udayan Vaidya, and audiences at ESSET Gerzensee 2024, Berkeley-Columbia-Duke-Northwestern-MIT IO Theory Conference 2024, and other venues, for their valuable suggestions and discussions. The authors acknowledge support from the Economics of Digital Services initiative at the University of Pennsylvania and from the NET Institute. Smolin acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future (Investissements d'Avenir) program (grant ANR-17-EURE-0010) and through the Artificial and Natural Intelligence Toulouse Institute (ANITI). Javier Gonzalez-Morin provided excellent research assistance.

[^1]:    ${ }^{1}$ As of February 2025, the top three productivity apps on the Apple Store are AI assistants: ChatGPT, DeepSeek, and Gemini.

[^2]:    ${ }^{2}$ This finding highlights the importance of the strategic context for an algorithm assessment and AI regulation (cf. White House (2023); European Commission (2024)).

[^3]:    ${ }^{3}$ The dependence of information on price may also arise from a worst-case analysis, as in the work of Libgober and Mu (2021), in which the buyer's information is chosen to minimize the seller's profits.

[^4]:    ${ }^{4}$ See also Lee (2021), Bergemann et al. (2022), and Smolin (2023).


[^5]:    ${ }^{5}$ Thus, algorithmic recommendations can be viewed as an effective alternative to the joint use of an intermediary (see Decarolis and Rovigatti (2021) for online advertising) or to a merger (see Loertscher and Marx (2022) for multifirm bargaining).
    ${ }^{6}$ One can view our setting as enabling a buyer from the classic setting of Myerson and Satterthwaite (1983) to commit to values and prices at which she would be purchasing a product. In this sense, we proceed in the opposite direction from the literature on limited commitment, which investigates how the inability to commit, typically on the part of a seller or a mechanism designer, affects equilibrium trade outcomes (e.g., Mylovanov and Tröger (2014), Liu et al. (2019)).
    ${ }^{7}$ In this setting, providing extra information about $v$ upon recommendation is unnecessary.
    ${ }^{8}$ The dependence of recommendations on price can be requested by the buyer in a conversation with an AI assistant, programmed directly, as seen in Amazon's search ranking algorithms (Lee and Musolff (2023), Farronato et al. (2023)), or it can arise indirectly through consumer feedback technology (Luca and Reshef (2021), Chakraborty et al. (2022)), wherein higher prices, all else being equal, lead to lower consumer satisfaction and ratings.

[^6]:    ${ }^{9}$ If $\gamma(c)$ were not everywhere increasing, then one would simply use an ironed version of it.
    ${ }^{10}$ Type $\bar{c}$ exists and is unique because $\gamma(\cdot)$ is strictly increasing and continuous on [0, 1], and $\gamma(0)=$ $0<1 \leq \gamma(1)$.

[^7]:    ${ }^{11}$ As the algorithm knows the value, this assumption is trivially satisfied if the seller's signal is fully informative or, more generally, partitional.
    ${ }^{12}$ Formally, in Section 3, we assumed that the value distribution has full support on an interval. However, our derivation of Proposition 1 did not rely on this assumption, and thus the result applies to any market segment.

[^8]:    ${ }^{13}$ Monotone partitional signals are not the only class of signals under which the distribution of individual buyer surplus undergoes a mean-preserving contraction. For example, in the previous version of the draft, we established this result for the truth-or-noise signals (Lewis and Sappington, 1994). However, some signal restrictions are necessary: If the finer segmentation separates high-value buyers from medium-value buyers while pooling them with low-value buyers, the price charged to those buyers may decrease, exacerbating payoff inequality.

[^9]:    ${ }^{14}$ As before, if for some $\alpha \in[0,1], \gamma_{\alpha}(c)$ were not everywhere increasing, then one would simply use an ironed version of it.

[^10]:    ${ }^{15}$ Gottardi and Mezzetti (2024) use a similar argument to construct an efficient one-shot mediation mechanism.

[^11]:    ${ }^{16}$ As before, the assumption that the algorithm can base recommendations on the realized signals captures the idea that the sellers have no information beyond that accessed by the algorithm. This assumption is automatically satisfied if the signals are deterministic functions of the value profile, i.e., partitional or fully informative signals.

[^12]:    ${ }^{17}$ This assumption is consistent with the consideration set approach to model recommender systems. See, for example, Dinerstein et al. (2018) and Lee and Musolff (2023).

[^13]:    ${ }^{18}$ Point (i) follows from direct calculation. For Point (ii), note that the condition $\left(\frac{F(c)}{c}\right)^{\prime} \geq 0$ is written as $\frac{\gamma(c)}{c} \leq 2$, which is equivalent to $\int_{0}^{\gamma(c)}[v-c] \mathrm{d} G(v) \leq 0$ when $G=U[0,1]$.
    ${ }^{19}$ The remaining case of a buyer perfectly informed about the product's existence and value is a textbook monopoly setting.

[^14]:    ${ }^{20}$ Distribution $\mathcal{G}_{H}$ of posteriors being a mean-preserving spread of $\mathcal{G}_{L}$ means that there exist $\Delta[0,1]$ -valued random variables $Z_{H}$ and $Z_{L}$ such that $Z_{H} \sim \mathcal{G}_{H}, Z_{L} \sim \mathcal{G}_{L}$ and $\mathbb{E}\left(Z_{H} \mid Z_{L}\right)=Z_{L}$.

[^15]:    ${ }^{21}$ Multiple active types may set the same price if, for example, the signal is such that the support of the posterior distribution for $v_{j}$ is non-convex for some signal realizations.

[^16]:    ${ }^{22}$ This argument also implies that if the candidate algorithm does not recommend any seller at a given price profile, then under any type profile that is consistent with the price profile, all the sellers have non-positive virtual surplus.


## Citation and provenance

**Authors:** Shota Ichihashi; Alex Smolin

**Canonical citation:** Ichihashi, Shota, and Alex Smolin. “Buyer-Optimal Algorithmic Recommendations.” Working paper, 2025.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/buyer-optimal-algorithmic-recommendations.md

**Source record:** https://arxiv.org/abs/2309.12122

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

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


Information Design in Smooth Games

> Machine-readable author manuscript.
> Authors: Alex Smolin; Takuro Yamashita.
> Canonical citation: Smolin, Alex, and Takuro Yamashita. “Information Design in Smooth Games.” Accepted at Theoretical Economics, 2026.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Information Design in Smooth Games

# Information Design in Smooth Games

**Authors:** Alex Smolin; Takuro Yamashita

**Manuscript date:** 2026-03-11

#### Abstract

We study information design in games where players choose from a continuum of actions and have continuously differentiable payoffs. We show that an information structure is optimal when the equilibrium it induces can also be implemented in a principal-agent contracting problem. Building on this result, we characterize optimal information structures in symmetric linear-quadratic games. With common values, targeted disclosure is robustly optimal across all priors. With interdependent and normally distributed values, linear disclosure is uniquely optimal. We illustrate our findings with applications in venture capital, Bayesian polarization, and price competition.

[^0]Keywords: Bayesian persuasion, information design, dual certification, firstorder approach, linear-quadratic games, targeted disclosure, Gaussian coupling, linear disclosure.


Information Design in Smooth Games

## 1 Introduction

As advances in IT infrastructure and artificial intelligence expand our capacity to collect, process, and generate data, an increasing number of firms, regulators, and individuals must decide which information to supply to strategically interacting agents. Which information should investors receive to allocate capital most efficiently? Which demand signals should be disclosed to improve market outcomes? What are the limits of Bayesian polarization? We show that these and related questions can be tractably addressed within a single framework, and that the optimal policies are simple, intuitive, and robust.

Our main analysis focuses on concave games of incomplete information, i.e., games in which each player's action lies in a convex set and each player's payoff is weakly concave in that action, but our methods extend to any smooth game. Concave games are a staple of applied economic modeling with fixed information structures, because equilibria can be tractably characterized by first-order conditions. We show that the same considerations render tractable the task of designing the information structure itself.

To solve the information-design problem, we adopt a duality-based approach. The dual can be viewed as an adversarial contracting problem between a principal and an omniscient agent who both observes the state and controls all players' actions. The dual-certification theorem (Theorem 1) states that if a state-action distribution emerges in equilibrium under both an information structure (in the information-design problem) and a contract (in the adversarial-contracting problem), then that information structure and that contract are optimal for their respective problems. The contract thus serves as an optimality certificate. Furthermore, it is capable of certifying any optimal information structure (Proposition 1), which enables us to establish whether the optimal information structure is unique or, when several are optimal, to identify their common features.

We show that (certifiably) optimal information structures are prior-robust: once optimal under one prior, they remain optimal under any other prior as long as the implemented allocation rule's support is contained within the original support in every state (Proposition 2). This has two consequences. First, if an optimal information
structure is fully informative about the state, it remains optimal under all priors. Second, unless the problem is trivial, an optimal information structure cannot induce a fullsupport action distribution in every state, which argues against adding full-support, extraneous, independent noise to individual signals.

We apply the solution method in two broad, symmetric settings with linear-quadratic payoffs for both the players and the designer. In each case, the certifying contract is symmetric across players and affine in actions, yet it yields qualitatively different optimal information policies.

In the first setting, the state is one-dimensional (Section 5). We derive the parameters of the certifying contract and use them to establish the optimality of a novel information structure we term targeted disclosure: it fully reveals the state to a subset of players while leaving the rest completely uninformed. This structure is remarkably simple and distributionally robust-it remains optimal under any prior. In addition, when the state is normally distributed, Gaussian coupling, which adds normal noise to each player's signal that cancels out in aggregate, is optimal as well.

In the second setting, the state is multidimensional with jointly normal components (Section 6). We show that a certifying contract can be found within the class of symmetric affine contracts by solving a single-variable minimization problem. This contract certifies the unique optimality of linear disclosure: an information structure that gives each player some linear statistic of the state. The resulting information structure is noise-free and symmetric ex ante.

In Section 8, we illustrate these results in three concrete applications. In Section 8.1, we analyze optimal capital fundraising when several investors independently choose how much to invest in a project of uncertain quality. We show that the information structure that maximizes the project's expected return is exclusive disclosure, a targeted disclosure in which only one investor is informed. Relative to full or no disclosure, exclusive disclosure prevents the dissipation of returns as the investor pool grows, offering a possible rationale for a common venture-capital practice.

In Section 8.2, we examine the limits of Bayesian polarization when each player
forms a forecast from private information. We show that the information structure that maximizes a natural polarization index is a targeted disclosure that informs exactly half of the players. Hence, maximal polarization occurs when the population splits into two cohorts that are internally uniform but sharply divergent from each other. This finding underscores how media segregation can intensify societal polarization.

In Section 8.3, we study optimal price recommendations in a differentiated-product duopoly with linear demand and stochastic demand shocks. We characterize the information structures that maximize any weighted average of consumer and producer surplus. The resulting recommended prices are linear functions of the demand shocks and exhibit discontinuous regime changes with respect to the weight on consumer surplus. This observation underscores the risk that algorithms generating price recommendations-able to shift objectives and adapt policies far faster than human decision-makers-can destabilize markets.


Information Design in Smooth Games

Related Literature Our paper contributes to the recent and flourishing literature on information design. Much of this literature focuses on information design with a single player, called Bayesian persuasion (Rayo and Segal (2010), Kamenica and Gentzkow (2011)). Popular solution methods are belief based, i.e., they operate within the space of the receiver's belief distributions. ${ }^{1}$ A natural continuation of this research agenda is a study of information design in multiplayer games (Bergemann and Morris (2016), Taneva (2019)). In games, players' beliefs constitute infinite hierarchies, which renders the beliefbased approach less tractable (Mathevet, Perego, and Taneva (2020)). Instead, in games, an action-based approach rooted in the revelation principle is promising, as it frames

[^1]the design problem as a linear program and enables the use of duality machinery. ${ }^{2,3}$
Galperti and Perego (2018) and Galperti, Levkun, and Perego (2024) employ this action-based approach to study information design in games with finitely many actions, imposing minimal structure on payoffs. They develop an economic interpretation of the Lagrange multipliers associated with Bayes' plausibility as the value of data records and propose the idea of pooling externalities across records; all these observations apply to our setting.

In contrast, we study games with infinitely many actions and impose a concavity structure on payoffs, thus enabling us to rely on a first-order approach for incentives (Holmström (1979), Mirrlees (1999)) that leads to more succinct and tractable primal and dual problems. This approach was introduced in a Bayesian persuasion setting by Kolotilin $(2012,2018)$ and further refined by Kolotilin, Corrao, and Wolitzky (2025). These papers study an information-design problem with a single player and a onedimensional state, identifying when censorship or, respectively, assortative disclosures are optimal. We deepen and extend this approach, adapting it to settings with multiple players and a multidimensional state.

Most of our current understanding of optimal information in games is drawn from the study of Gaussian signals in games with quadratic payoffs and a normally distributed state (e.g., Angeletos and Pavan $(2007,2009)$, Bergemann and Morris (2013), Bergemann, Heumann, and Morris (2015, 2021), Ui (2020)). In that literature, as well as in a vast body of work in macroeconomics and finance, the Gaussian form of players' signals is imposed ad hoc for analytical convenience.

Recent arguments suggest that Gaussian signals are not only convenient but often optimal among all information structures. For example, Tamura $(2012,2018)$ estab-

[^2]lished the optimality of a Gaussian signal in a setting with a single player by building on the statistical properties of a covariance matrix of posterior expectations. Bergemann, Heumann, and Morris (2017) and Miyashita and Ui (2023) extend this argument to games. ${ }^{4}$ Our results align with these findings and show that in many scenarios the optimal Gaussian structures are symmetric and noise-free; moreover, the certification method allows direct identification of the optimal informational parameters. Furthermore, we provide tools that establish when Gaussian information structures are uniquely optimal.

At the same time, we show that in many settings targeted disclosure is optimal. These information structures are simple to implement and distributionally robust, offering a strong positive message for the information design literature, where existing solutions often lack these properties.