Bayes-Nash Price of Anarchy for GSP

Generalized Second Price Auction, also known as Ad Word auction, and its variants have been the main mechanism used by search companies to auction positions for sponsored search links. In this paper we study the social welfare of the Nash equilibria of this game under the Bayes-Nash solution concept (i.e., in a partial information setting). In this model, the value of each player for one click is drawn independently from a distribution. Each player knows his own value but he knows only the probability distribution of the other players values. We compare the expected social welfare in a BayesNash equilibrium with the expected value of the optimal social welfare. We obtain a bound of 8 for the Bayes-Nash Price of Anarchy of GSP. Our proof exhibits a combinatorial structure of Nash equilibria and use this structure to bound the price of anarchy. Our proof of this structural property uses novel combinatorial techniques that can be of independent interest.