Does algorithmic recommendation lose its edge as sellers compete more?

"Proposition 3 (Impact of Market Thickness)" states that in a common-value benchmark — all sellers offering the same value v to the buyer with i.i.d. private costs — the ratio of the designer's objective under the optimal recommendation algorithm, WA_α(J), to its value under ordinary Bertrand competition, WB_α(J), is "strictly decreasing in J for all J ≥2, and converges to 1 as J →∞." In plain words: the welfare benefit an AI recommender adds on top of ordinary seller competition is largest when few sellers compete, and shrinks toward zero as the market fills with more competing sellers.

The paper's own explanation is that "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" — cases where trade would occur under ordinary competition but the algorithm withholds a recommendation because the price needed to extract it would cost the buyer too much in information rents. As the number of sellers J grows, "the lowest and second-lowest costs shift toward the bottom of the support, competitive payments approach costs," so these marginal exclusion cases become rarer and lower-stakes, and "the incremental value of algorithmic recommendations decreases with the number of competitors and vanishes in the competitive limit." The paper notes a parallel result for mergers: a merger "weakly lowers the maximal value of every objective that puts positive weight on buyer surplus," the reverse-direction version of the same competition effect.

This qualifies Can an algorithm recommend products without matching an ideal mechanism?, the equivalence result from the same model: that note establishes what an optimal algorithm can achieve in principle, while this one shows the achievable gain over ordinary competition is not constant but shrinks as seller count rises — the equivalence is a ceiling that matters most in thin markets. It also sits on a different axis from Why do consumers delay adopting agentic search?, which concerns a gap between platform and consumer incentives regardless of competitiveness; this result instead says the size of any AI-intermediary advantage, whichever interest it serves, depends on how concentrated the seller side already is.

The result holds only in a common-value benchmark with identical product values, independent and identically distributed costs, and a single buyer, abstracting away product differentiation, repeated purchases, and any seller-side strategic response to algorithm design, such as collusion or steering, which the batch's other sources treat separately. It is a property of this specific mechanism-design model, not a measurement of any deployed shopping agent. The implication, at the strength a theoretical bound supports, is that if a similar commitment logic applies to real AI shopping agents, their welfare contribution — and correspondingly the regulatory stakes of how they are designed — would be largest in markets with few competing sellers and would taper off as seller competition thickens.