Algorithmic Recommendations and Strategic Pricing

Abstract We study algorithmic recommendations in a market with a buyer, privately informed sellers, and an algorithm that can recommend a product based on product values and posted prices but cannot use transfers or force purchases. For objectives ranging from total surplus to buyer surplus, algorithmic recommendations implement the same welfare outcomes as direct mechanisms. We study the effects of algorithmic recommendations on pricing, competition, the composition of trade, and market segmentation. Our results inform the operation of AI assistants and other technologies that mediate product discovery, attention, and pricing.

Introduction. A fundamental feature of artificial intelligence and, more generally, of algorithms is that they are programmed in advance. As such, they endow their users with a degree of commitment power that those users may otherwise lack. In this paper, we study how this commitment power can affect market outcomes when it is bestowed upon buyers via algorithmic recommendations and sellers set prices strategically.1 We study a canonical market with a single buyer and multiple sellers. Each seller offers one product, privately observes her production cost, and sets a price strategically.

The buyer is initially uninformed about the products’ values and unaware of their existence. An algorithm, which can discover these values, recommends a product based on values and posted prices. Upon receiving a recommendation, the buyer updates her belief and decides whether to buy the recommended product at the offered price. If no recommendation is received, no trade occurs. Thus, the algorithm serves as a gatekeeper.

To evaluate such rules, we characterize the welfare frontier attainable by recommendation policies. In the main analysis, the designer puts weakly more weight on buyer surplus than on seller surplus. This class is natural for AI assistants and other recommendation tools whose long-run value depends more on buyer trust and subscription revenue than on seller commissions. It also covers platforms that value buyer participation and efficiency.

Our first main result is an equivalence between optimal recommendation algorithms and optimal direct mechanisms: optimal algorithms attain the same allocation, buyer surplus, and total surplus as optimal direct mechanisms. The equivalence is not mechanical. A direct mechanism can ask sellers to report costs, control allocation, and specify monetary transfers. In contrast, an algorithm only chooses whether and what to recommend based on the value and price of each product. Nevertheless, the algorithm can use the equilibrium prices as messages to learn sellers’ costs, and then use the revealed information to reproduce the allocation rule of the corresponding direct mechanism.

Specifically, the equivalence has two parts. First, any direct mechanism that satisfies incentive compatibility and the buyer-obedience constraints and gives each seller’s highest-cost type zero interim profit can be implemented by some algorithm. Conversely, every equilibrium outcome of any recommendation algorithm satisfies the same restrictions. Second, for the objectives we study, the optimal direct mechanism automatically satisfies buyer obedience. Hence the requirement that the buyer be willing to follow recommendations imposes no loss at the optimum.

The optimal allocation has a simple structure. The algorithm recommends a product that maximizes value net of weighted virtual cost whenever this maximum is positive.

When the designer maximizes total surplus, the virtual cost equals the true cost and the allocation is efficient. When the designer puts more weight on buyer surplus, the algorithm treats costs as higher than their true levels to reflect information rents.

The single-seller case illustrates the economic mechanism most transparently. The optimal algorithm recommends the product if and only if the posted price is below a function of the true value. This function differs from the identity function in a way that makes demand more price-sensitive, and is chosen so that, in equilibrium, trade occurs exactly when value exceeds the seller’s virtual cost. With multiple sellers, the same force interacts with product selection: a seller’s price affects not only whether her product is attractive to the buyer, but also whether it is attractive relative to other products, so high prices can additionally be punished by substitution toward rivals.

In Section 5, we study the broader consequences of algorithmic recommendations.

First, we compare algorithmic recommendations with ordinary seller competition. With continuously and independently distributed values and positive weight on buyer surplus, Bertrand competition without algorithmic recommendations is always improvable by some algorithm. The algorithm improves on Bertrand competition by committing the buyer not to trade when trade would generate surplus but require high prices or large seller rents. At the same time, competition is still valuable and may partly substitute for the optimal algorithm. In a common-value benchmark, the relative value of algorithmic recommendations decreases as the market becomes thicker. Also, a merger weakly lowers the maximal value of every objective that puts positive weight on buyer surplus.

We then study how the algorithm’s price-disciplining role changes which trades take place. With a single seller, compared to standard monopoly pricing, the optimal algorithm substitutes low-cost, lower-value trades for high-cost, higher-value trades.

Related work. Our paper is closest to work on buyer-optimal information design in markets with seller market power. Roesler and Szentes (2017) analyze buyer-optimal learning in a bilateral trade setting. Deb and Roesler (2024) extend the analysis to multiproduct monopoly, Bergemann et al. (2023) to auctions, and Condorelli and Szentes (2020) characterize buyer-optimal distributions of values. We share the central insight that full information about value need not be optimal for the buyer. The key difference is that information in our model is price-contingent and the seller has private cost information. As a result, the recommendation rule is not only an information structure for the buyer but also a screening device for the seller. This feature leads to the implementation equivalence between recommendation algorithms and direct mechanisms.2 In our study, we integrate Bayesian persuasion (cf. Kamenica and Gentzkow (2011)) with mechanism design (cf. Baron and Myerson (1982)). Several papers combine these frameworks in trade settings. The most closely related are Yang (2022), Bergemann and Bonatti (2024), Xu and Yang (2024), and Bergemann et al. (2024).3 These papers also study informational intermediaries that process data and shape market outcomes.

Our focus differs in two ways. First, these papers allow the intermediary to charge transfers and study revenue maximization, whereas we study general welfare objectives and an intermediary that cannot charge transfers. Second, in our model the central friction is seller private information. By contrast, Bergemann and Bonatti (2024) and Bergemann et al. (2024) focus on buyers’ off-site purchase opportunities. Closest to us, Yang (2022) and Xu and Yang (2024) study revenue-maximizing intermediation when the intermediary has information about buyer values and faces a seller with private cost: the former characterizes optimal data-selling mechanisms, while the latter establishes outcome equivalence across retail and marketplace models and across recommendation and price-discrimination implementations. Our equivalence result is different: it shows that, for buyer-aligned objectives, a recommendation algorithm without transfers can be analyzed through the corresponding direct mechanism.

Our market-segmentation results contribute to the literature on price discrimination and information design. In canonical segmentation models, information about buyer values changes the distribution of prices, allocations, and surplus; see, for example, Bergemann et al. (2015) and Haghpanah and Siegel (2023). In our setting, once the recommendation algorithm adjusts to the information available to sellers, price discrimination leaves the allocation, seller profits, and buyer surplus unchanged.

Method. 2 Model There is a single buyer with unit demand and J ∈N sellers indexed by j = 1, . . . , J, each offering one product. The buyer’s values for the products are v ≜(v1, . . . , vJ) ∈[0, 1]J, and are drawn from a joint distribution G whose marginal on each vj is not degenerate at zero. Values may be arbitrarily correlated across products. Seller j’s cost cj ∈[0, 1] is drawn from Fj, independently of other sellers’ costs and of the value profile. Assume each Fj is continuous and has strictly positive density fj on (0, 1). A dummy seller indexed by j = 0, with v0 = 0 and c0 = 0, represents the buyer’s outside option of not purchasing. Let J ≜{1, . . . , J} and J0 ≜{0, 1, . . . , J}.

The buyer does not know her values and relies on external recommendations to make her purchases. A recommendation algorithm or simply algorithm is a function r : [0, 1]J × RJ + →∆(J0), where rj(v, p) is the probability that the algorithm recommends seller j ∈J0 at value profile v and price profile p. Recommendation of seller 0 means that no product is recommended. The algorithm is commonly known to the buyer and sellers.

The algorithm can condition on the buyer’s values and prices but not on sellers’ costs. This assumption reflects our focus on buyer-side algorithmic mediation. The algorithm has buyer information because it is the buyer’s interface: it can use buyer history, product attributes, reviews, and other data to form personalized estimates of product values. For our baseline analysis, we can treat these estimates directly as the buyer’s values (v1, . . . , vJ). At the same time, sellers’ costs remain private because they may depend on procurement conditions, inventory, or other seller-side information that is not observable from the buyer’s interface.

Timing. At the outset, a designer commits to an algorithm. Then, the game unfolds as follows. First, nature draws seller costs and buyer values. Second, each seller privately observes her own cost and simultaneously posts a price. Third, the algorithm observes the value profile and the price profile and recommends one seller or recommends no purchase. If no product is recommended, trade does not occur. If seller j ∈J is recommended, the buyer observes the recommendation and price pj and decides whether to purchase. If product j is purchased, the buyer and seller j obtain ex post payoffs vj −pj and pj −cj, respectively; otherwise, all players obtain zero. Given an algorithm, the solution concept is perfect Bayesian equilibrium.

This timing assumes that the recommendation rule is chosen before sellers set their prices. It is intended to capture prominent buyer assistants or ranking systems whose rules are already in place when sellers choose or update their prices. We further assume that the algorithm can act as a gatekeeper, i.e., the buyer can purchase only the product recommended by the algorithm. This assumption is relevant in online settings where recommendation systems are used to facilitate product discovery and bring products to buyers’ attention.5 Objective. We denote the buyer’s ex ante expected payoff by BS, the sum of sellers’ expected profits (or seller surplus) by SS, and total surplus by TS = BS + SS. We are interested in the welfare possibilities of algorithmic recommendations. In the main part of the paper, we focus on the buyer-aligned part of the Pareto frontier and let the designer maximize Wα ≜αBS + (1 −α)TS, (1) where α ∈[0, 1]. The case α = 0 is total-surplus maximization, while α = 1 is the buyer-optimal benchmark. Equivalently, because Wα = BS+(1−α)SS, this class places weakly more weight on buyer surplus than on seller surplus. We refer to (1) as the class of buyer-aligned objectives.6 This class is natural for AI assistants and other consumer-facing recommendation tools whose value depends on subscription revenue and participation. Also, we show that in the single-seller case, the total-surplus-maximizing algorithm also maximizes seller surplus, so buyer-aligned objectives span the entire Pareto frontier of buyer and seller surplus.

Discussion. The intuition is as follows. Under Bertrand competition, every seller whose cost is strictly below the top of the support trades with positive probability. Indeed, seller j with cost cj < 1 can earn a positive profit by posting a price of p = (1 + cj)/2, which leads to sales if vj −p > vk. By contrast, an α-optimal algorithm with α > 0 excludes a top interval of costs through the weighted virtual-cost cutoff. These excluded types may create positive total surplus, but serving them would require prices or information rents that are too high for a buyer-aligned objective. Algorithmic recommendations thus improve on Bertrand competition by committing the buyer to a price- and valuecontingent access rule that ordinary competition cannot implement.

We now turn to the next question: how the gain from the optimal algorithm depends on the degree of competition. We study this question through two comparative statics: increasing the number of sellers and merging two sellers. For tractability, we consider a common-value benchmark in which all sellers have the same deterministic value v ∈(0, 1] and i.i.d. costs, abstracting from product-differentiation effects and focusing on how competition and algorithmic recommendations discipline privately informed sellers.

For a fixed weight α ∈[0, 1], we compare the value of competition with and without an algorithm. Competition without an algorithm is Bertrand competition described above.

In the common-value benchmark, the game admits essentially a unique equilibrium, in which each seller adopts the same pricing strategy, which is strictly increasing in the seller’s cost type.13 Denote the lowest and second-lowest realized costs by c(1:J) and c(2:J), respectively. In equilibrium, trade occurs if and only if c(1:J) ≤v, and a lowest-cost seller transacts at price E[min{c(2:J), v} | c(1:J)].14 With the optimal algorithm, Theorem 1 and the pricing rule (5) imply that the equilibrium outcome under the α-optimal algorithm coincides with that under Bertrand competition in which the buyer’s value is γ−1(v). In equilibrium, trade occurs if and only if c(1:J) ≤γ−1(v), and a lowest-cost seller supplies at price E[min{c(2:J), γ−1(v)} | c(1:J)].

The optimal algorithm induces a transaction only with a lowest-cost seller but commits not to induce a transaction if c(1:J) > γ−1(v).

Let WA α (J) and WB α (J) denote the value of the objective Wα with J sellers under the α-optimal algorithm and Bertrand competition, respectively.

Proposition 3 (Impact of Market Thickness). Suppose all sellers have the same deterministic value v ∈(0, 1], costs are i.i.d. with common distribution F and density f, and F/f is strictly increasing on [0, 1]. For any α ∈(0, 1], the ratio WA α (J)/WB α (J) is strictly decreasing in J for all J ≥2, and converges to 1 as J →∞.

Proof. See Appendix A.8. □ The ratio in Proposition 3 measures the relative value of algorithmic recommendations over ordinary competition. The algorithm improves on Bertrand competition by committing the buyer not to trade when the value is above the lowest cost but below the lowest weighted virtual cost. As the number of sellers increases, the lowest and second-lowest costs shift toward the bottom of the support, competitive payments approach costs, and the probability and welfare importance of these marginal exclusion cases decline. Thus, the incremental value of algorithmic recommendations decreases with the number of competitors and vanishes in the competitive limit.

Proof. See Appendix A.10. □ The hazard-rate assumptions of Proposition 5 imply a single-crossing comparison between the algorithmic and monopoly trade thresholds, which yields the sharp result.