The Generalized Second Price (GSP) auction is the primary method by which sponsered search advertisements are sold. We study the performance of this auction in the Bayesian setting for players with correlated types. Correlation arises very naturally in the context of sponsored search auctions, especially as a result of uncertainty inherent in the behaviour of the underlying ad allocation algorithm. We demonstrate that the Bayesian Price of Anarchy of the GSP auction is bounded by 4, even when agents have arbitrarily correlated types. Our proof highlights a connection between the GSP mechanism and the concept of smoothness in games, which may be of independent interest.
For the special case of uncorrelated (i.e. independent) agent types, we improve our bound to 2(1 − 1/e)^−1 ≈ 3.16, significantly improving upon previously known bounds. Using our techniques, we obtain the same bound on the performance of GSP at coarse correlated equilibria, which captures (for example) a repeated-auction setting in which agents apply regret-minimizing bidding strategies. Moreoever, our analysis is robust against the presence of irrational bidders and settings of asymmetric information, and our bounds degrade gracefully when agents apply strategies that form only an approximate equilibrium.