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

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.


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.

F Proofs of Proposition 6 and Proposition 7

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

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