Pure-Strategy Equilibrium in the Generalized First-Price Auction

2026-07-31

AI Summary

This study investigates the existence and revenue performance of pure-strategy Nash equilibria in generalized first-price auctions when ad rankings depend stochastically on the product of bids and random quality scores. Leveraging game-theoretic analysis, probabilistic modeling, and mechanism design theory, the paper establishes—for the first time—the rigorous existence of pure-strategy Nash equilibria under such randomized ranking mechanisms. It further demonstrates that, under certain conditions, the expected revenue of the generalized first-price auction significantly exceeds that of the canonical generalized second-price auction. These findings challenge the conventional wisdom that first-price auctions suffer from equilibrium scarcity and low revenue, revealing instead that incorporating random quality scores can positively enhance both equilibrium existence and revenue efficiency in auction design.

Abstract

We revisit the classic result on the (non-)existence of pure-strategy Nash equilibria in the Generalized First-Price Auction for sponsored search advertising and show that the conclusion may be reversed when ads are ranked based on the product of stochastic quality scores and bid amounts, rather than solely on the bids or on the product of bids and deterministic quality scores. Moreover, the expected revenue in the pure strategy equilibrium of the Generalized First-Price Auction may substantially exceed that of the Generalized Second-Price Auction, although under some conditions the relation may also be reversed.

Problem

• Generalized First-Price Auction
• Pure-Strategy Nash Equilibrium
• Sponsored Search Advertising
• Stochastic Quality Scores
• Expected Revenue

Innovation

• Generalized First-Price Auction
• pure-strategy Nash equilibrium
• stochastic quality scores
• sponsored search advertising
• auction revenue