Anonymous Pricing in Large Markets

TL;DR

This paper demonstrates that in large markets with many buyers, anonymous pricing nearly matches the optimal revenue, with only a small loss, especially as the number of units increases.

Contribution

It shows that in large markets, anonymous pricing achieves near-optimal revenue, reducing the advantage of complex price discrimination.

Findings

Anonymous pricing achieves a 2+O(1/√k) approximation to optimal revenue.

Worst-case ratio is about 2.47 for k=1 and approaches 2 as k increases.

Gains from third-degree price discrimination are limited in large markets.

Abstract

We study revenue maximization when a seller offers $k$ identical units to ex ante heterogeneous, unit-demand buyers. While anonymous pricing can be $\Theta(\log k)$ 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.