[Paper Review] Anonymous Pricing in Large Markets

Yaonan Jin, Yingkai Li|arXiv (Cornell University)|Jan 23, 2026

Auction Theory and Applications0 citations

TL;DR

The paper proves that in large markets with quasi-regular value distributions and selling k identical units, anonymous pricing achieves a constant-factor approximation to the revenue-optimal mechanism, specifically 2 + O(1/√k), with a worst-case of about 2.47 at k=1 and convergence to 2 as k grows.

ABSTRACT

We study revenue maximization when a seller offers $k$ identical units to ex ante heterogeneous, unit-demand buyers. While anonymous pricing can be $Θ(\log k)$ worse than optimal in general multi-unit environments, we show that this pessimism disappears in large markets, where no single buyer accounts for a non-negligible share of optimal revenue. Under (quasi-)regularity, anonymous pricing achieves a $2+O(1/\sqrt{k})$ approximation to the optimal mechanism; the worst-case ratio is maximized at about $2.47$ when $k=1$ and converges to $2$ as $k$ grows. This indicates that the gains from third-degree price discrimination are mild in large markets.

Motivation & Objective

• Motivate the study of revenue maximization when selling k identical units to ex ante heterogeneous, unit-demand buyers.
• Show that the known worst-case Θ(log k) loss of anonymous pricing in multi-unit settings collapses under a large-market assumption.
• Characterize the worst-case instances and provide a tight constant-factor approximation to the optimal revenue.
• Extend results to quasi-regular distributions and explain why third-degree price discrimination offers mild gains in large markets.

Proposed method

• Model the auction with k identical units and n unit-demand buyers with independent, private values drawn from regularly or quasi-regular distributions.
• Define anonymous pricing and compare its revenue to the revenue-optimal mechanism under IC and IR constraints.
• Reduce worst-case instances to triangular (triangular) distributions and use order-statistics analysis to relate higher-order statistics to the first-order statistic.
• Use revenue curves in quantile space and monopolistic quantiles, along with a decomposition of revenue curves into triangular components (Proposition 2).
• Derive the main bound by showing the worst-case gap is 2 + O(1/√k) and that D_j(p) can be approximated by functions of D_1(p) (Proposition 3).

Experimental results

Research questions

• RQ1 What is the worst-case approximation ratio of anonymous pricing relative to the optimal revenue in k-unit auctions under large-market and (quasi-)regularity assumptions?
• RQ2 Do large markets mitigate the known Θ(log k) revenue loss of anonymous pricing in multi-unit settings, and if so, by how much?
• RQ3 Can worst-case distributions be reduced to triangular forms, and can higher-order statistics be effectively characterized by first-order statistics in large markets?
• RQ4 How does the approximation ratio behave as k grows, and where is the maximum gap attained?
• RQ5 Do results extend to quasi-regular distributions beyond standard regularity?

Key findings

• Under large-market and quasi-regular assumptions, anonymous pricing achieves a (2 + O(1/√k))-approximation to the optimal revenue.
• The worst-case approximation ratio is maximized at about 2.47 when k = 1 and converges to 2 for large k.
• Worst-case instances can be reduced to triangular (triangular) distributions without loss of the approximation, enabling tractable analysis.
• Higher-order statistics in large markets are effectively determined by the first-order statistic, enabling closed-form bounding of the revenue gap.
• The reduction to triangular distributions and order-statistics analysis yield a tight bound on the anonymity price versus optimal revenue in multi-unit auctions.