We study revenue optimization learning algorithms for repeated posted-price auctions where a seller interacts with a single strategic buyer that holds a fixed private valuation for a good and seeks to maximize his cumulative discounted surplus. For this setting, first, we propose a novel algorithm that never decreases offered prices and has a tight strategic regret bound in @math under some mild assumptions on the buyer surplus discounting. This result closes the open research question on the existence of a no-regret horizon-independent weakly consistent pricing. The proposed algorithm is inspired by our observation that a double decrease of offered prices in a weakly consistent algorithm is enough to cause a linear regret. This motivates us to construct a novel transformation that maps a right-consistent algorithm to a weakly consistent one that never decreases offered prices. Second, we outperform the previously known strategic regret upper bound of the algorithm PRRFES, where the improvement is achieved by means of a finer constant factor @math of the principal term @math in this upper bound. Finally, we generalize results on strategic regret previously known for geometric discounting of the buyer's surplus to discounting of other types, namely: the optimality of the pricing PRRFES to the case of geometrically concave decreasing discounting; and linear lower bound on the strategic regret of a wide range of horizon-independent weakly consistent algorithms to the case of arbitrary discounts.
Most of studies on online advertising auctions lies in the field of game theory @cite_13 @cite_22 : a large part of them focused on characterizing different aspects of equilibria, and recent ones was devoted (but not limited) to: position auctions @cite_18 @cite_12 @cite_37 @cite_27 , different generalizations of second-price auctions @cite_44 @cite_25 , efficiency @cite_17 , mechanism expressiveness @cite_15 , competition across auction platforms @cite_3 , buyer budget @cite_40 , experimental analysis @cite_31 @cite_23 @cite_53 , etc.
@cite_37 @cite_18 @cite_31 @cite_22 @cite_53 @cite_3 @cite_44 @cite_27 @cite_40 @cite_23 @cite_15 @cite_13 @cite_25 @cite_12 @cite_17
1967821428 2022890659 2564127794 1990656299
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. The Generalized Second Price Auction has been the main mechanism used by search companies to auction positions for advertisements on search pages. In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full information setting, socially optimal Nash equilibria are known to exist (i.e., the Price of Stability is 1). This paper is the first to prove bounds on the price of anarchy, and to give any bounds in the Bayesian setting. Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated strategies. In the full information setting we prove a bound of 1.618 for the price of anarchy for pure Nash equilibria, and a bound of 4 for mixed Nash equilibria. We also prove a bound of 8 for the price of anarchy in the Bayesian setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the distributions used are common knowledge. Our proof exhibits a combinatorial structure of Nash equilibria and uses this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest. We consider the problem of online keyword advertising auctions among multiple bidders with limited budgets, and propose a bidding heuristic to optimize the utility for bidders by equalizing the return-on-investment for each bidder across all keywords. We show that natural auction mechanisms combined with this heuristic can experience chaotic cycling (as is the case with many current advertisement auction systems), and therefore propose a modified class of mechanisms with small random perturbations. This perturbation is reminiscent of the small time-dependent perturbations employed in the dynamical systems literature to convert many types of chaos into attracting motions. We show that our perturbed mechanism provably converges in the case of first-price auctions and experimentally converges in the case of second-price auctions. Moreover, we show that our bidder-optimal system does not decrease the revenue of the auctioneer in the sense that it converges to the unique market equilibrium in the case of first-price auctions. In the case of second-price auctions, we conjecture that it converges to the non-unique “supplyaware” market equilibrium. We also observe that our perturbed auction scheme is useful in a broader context: In general, it can allow bidders to “share” a particular item, leading to stable allocations and pricing for the bidders, and improved revenue for the auctioneer. 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 @math , 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.