On the efficiency of equilibria in generalized second price auctions.

In sponsored search auctions, advertisers compete for a number of available advertisement slots of different quality. The auctioneer decides the allocation of advertisers to slots using bids provided by them. Since the advertisers may act strategically and submit their bids in order to maximize their individual objectives, such an auction naturally defines a strategic game among the advertisers. In order to quantify the efficiency of outcomes in generalized second price auctions, we study the corresponding games and present new bounds on their price of anarchy, improving the recent results of Paes Leme and Tardos [16] and Lucier and Paes Leme [13]. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria. Given the existing lower bounds, this bound denotes that the number of advertisers has almost no impact on the price of anarchy. The proof exploits the equilibrium conditions developed in [16] and follows by a detailed reasoning about the structure of equilibria and a novel relation of the price of anarchy to the objective value of a compact mathematical program. For more general equilibrium classes (i.e., mixed Nash, correlated, and coarse correlated equilibria), we present an upper bound of 2.310 on the price of anarchy. We also consider the setting where advertisers have incomplete information about their competitors and prove a price of anarchy upper bound of 3.037 over Bayes-Nash equilibria. In order to obtain the last two bounds, we adapt techniques of Lucier and Paes Leme [13] and significantly extend them with new arguments.