Bayes-Nash Equilibrium of the Generalized Second-Price Auction

Renato Gomes, and Kane Sweeney


We develop a Bayes–Nash analysis of the generalized second-price (GSP) auction, the multi-unit auction used by search engines to sell sponsored advertising positions. Our main result characterizes the efficient Bayes–Nash equilibrium of the GSP and provides a necessary and sufficient condition that guarantees existence of such an equilibrium. With only two positions, this condition requires that the click–through rate of the second position is sufficiently smaller than that of the first. When an efficient equilibrium exists, we provide a necessary and sufficient condition for the auction revenue to decrease as click–through rates increase. Interestingly, under optimal reserve prices, revenue increases with the click–through rates of all positions. Further, we prove that no inefficient equilibrium of the GSP can be symmetric. Our results are in sharp contrast with the previous literature that studied the GSP under complete information.


Generalized second-price auction; Bayes–Nash equilibrium; Position auctions; Sponsored search; Click–through rates;

JEL codes

  • C78: Bargaining Theory • Matching Theory
  • D44: Auctions
  • L86: Information and Internet Services • Computer Software
  • M37: Advertising

See also

Published in

Games and Economic Behavior, vol. 86, July 2014, pp. 421–437