Polymatroid Prophet Inequalities

Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen. The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint. In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems

Revenue Guarantees in the Generalized Second Price Auction

Sponsored search auctions are the main source of revenue for search engines. In such an auction, a set of utility maximizing advertisers competes for a set of ad slots. The assignment of advertisers to slots depends on the bids they submit; these bids may be different than the true valuations of the advertisers for the slots. Variants of the celebrated VCG auction mechanism guarantee that advertisers act truthfully and, under some assumptions, lead to revenue or social welfare maximization. Still, the sponsored search industry mostly uses generalized second price (GSP) auctions; these auctions are known to be nontruthful and suboptimal in terms of social welfare and revenue. In an attempt to explain this tradition, we study a Bayesian setting wherein the valuations of advertisers are drawn independently from a common regular probability distribution. In this setting, it is well known from the work of Myerson [1981] that the optimal revenue is obtained by the VCG mechanism with a particular reserve price that depends on the probability distribution. We show that, by appropriately setting the reserve price, the revenue over any Bayes-Nash equilibrium of the game induced by the GSP auction is at most a small constant factor away from the optimal revenue, improving previous results of Lucier et al. [2012]. Our analysis is based on the Bayes-Nash equilibrium conditions and the improved results are obtained by bounding the utility of each player at equilibrium using infinitely many deviating bids and also by developing novel prophet-like inequalities

Revenue Guarantees in Sponsored Search Auctions

Abstract. Sponsored search auctions are the main source of revenue for search engines. In such an auction, a set of utility-maximizing ad-vertisers compete for a set of ad slots. The assignment of advertisers to slots depends on bids they submit; these bids may be different than the true valuations of the advertisers for the slots. Variants of the celebrated VCG auction mechanism guarantee that advertisers act truthfully and, under mild assumptions, lead to revenue or social welfare maximization. Still, the sponsored search industry mostly uses generalized second price (GSP) auctions; these auctions are known to be non-truthful and sub-optimal in terms of social welfare and revenue. In an attempt to explain this tradition, we study a Bayesian setting where the valuations of adver-tisers are drawn independently from a regular probability distribution. In this setting, it is well known by the work of Myerson (1981) that the optimal revenue is obtained by the VCG mechanism with a particular reserve price that depends on the probability distribution. We show that by appropriately setting the reserve price, the revenue over any Bayes-Nash equilibrium of the game induced by the GSP auction is at most a small constant fraction of the optimal revenue, improving recent re-sults of Lucier, Paes Leme, and Tardos (2012). Our analysis is based on the Bayes-Nash equilibrium conditions and on the properties of regular probability distributions.