Bayes-Nash Equilibria of the Generalized Second Price Auction

Abstract

In this paper we develop a Bayes-Nash analysis of the Generalized Second Price (GSP) auction, the mechanism used by search engines to sell sponsored advertising positions. Interestingly, our results are in sharp contrast with the previous literature that adopts a complete information framework. First, we characterize the efficient Bayes-Nash equilibrium of the GSP when such an equilibrium exists. We then obtain sufficient conditions on click-through rates that guarantee existence and show that an efficient equilibrium may fail to exist if click-through rates across slots are sufficiently close. Next, we derive the counter-intuitive result that the seller's revenue may decrease as click-through rates of the second to last slots increase, provided the distribution of advertisers' values per click is concentrated on high values. Fortunately, we show that setting optimal reserve prices reverses this result. Further, we prove that the GSP possesses no mixed strategy equilibrium and that no inefficient equilibrium can be symmetric.