Posted Pricing and Competition in Large Markets

TL;DR

This paper analyzes the welfare and revenue guarantees of fixed-price postings in large, iid markets. By leveraging extreme value theory, it proves a tight fixed-price welfare guarantee of $0.712$ for single-item sales and extends to $k$-unit cases with a bound of $1-1/\sqrt{2k\pi}$, showing the large-market advantage diminishes as $k$ grows. It also characterizes the constant competition complexity in large markets, linking it to the EVT parameter $\gamma$ and giving explicit formulas for Fréchet, Gumbel, and Reverse Weibull families. A case study using eBay auction data demonstrates practical robustness, with a fixed-threshold policy achieving roughly $73\%$ of the optimal welfare, validating the theoretical results in real-world settings. Overall, the work highlights the practicality of fixed-price policies in large-scale marketplaces, offering provable performance guarantees and insights into design trade-offs versus dynamic optimal mechanisms.

Abstract

Posted price mechanisms are prevalent in allocating goods within online marketplaces due to their simplicity and practical efficiency. We explore a fundamental scenario where buyers' valuations are independent and identically distributed, focusing specifically on the allocation of a single unit. Inspired by the rapid growth and scalability of modern online marketplaces, we investigate optimal performance guarantees under the assumption of a significantly large market. We show a large market benefit when using fixed prices, improving the known guarantee of $1-1/e\approx 0.632$ to $0.712$. We then study the case of selling $k$ identical units, and we prove that the optimal fixed price guarantee approaches $1-1/\sqrt{2k π}$, which implies that the large market advantage vanishes as $k$ grows. We use real-world auction data to test our fixed price policies in the large market regime. Next, under the large market assumption, we show that the competition complexity for the optimal posted price mechanism is constant, and we identify precise scaling factors for the number of bidders that enable it to match benchmark performance. Remarkably, our findings break previously established worst-case impossibility results, underscoring the practical robustness and efficiency of posted pricing in large-scale marketplaces.

Figures (2)

• Figure 1: Our optimal welfare guarantee over $k$ (continuous line) vs. $1-1/\sqrt{2k\pi}$ (dashed line).
• Figure 2: Frequency histograms of real (\ref{['fig:density_bids']}) and simulated data (\ref{['fig:density_samples']}) representing the valuations of $509$ bidders for 7-day auctions of Cartier watches on eBay, organized into bins of width 200. The density of the Frechet distribution $Fr(m = 0, s = 289, \alpha = 2.24)$, the best fit for the real data, is overlaid on top of both. The simulated data is drawn randomly from the same distribution.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Corollary 2
• Proposition 1
• proof : Proof of Proposition \ref{['prop:numerator-multi']}
• Theorem 5: see, e.g., libroextremo
• Theorem 6: see, e.g., LLR12