Polymatroid Prophet Inequalities

Abstract

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.

Approximately Efficient Two-Sided Combinatorial Auctions

“…ey were later studied by Chawla et al [3] for the objective of revenue maximisation. Additionally, Kleinberg and Weinberg [14] and Dü ing and Kleinberg [8] strengthen these results further. Very relevant to our work is the paper of Feldman et al [11], showing that sequential posted price mechanisms can approximate social welfare up to a constant factor of 1/2 for XOS valuation functions if the published price for an item is equal to the expected additive contribution of the item to the social welfare.…”

“…In the remainder of the paper we will proceed to design mechanisms for two-sided markets that do satisfy DSBB. 8 Moreover, two of our results provide an O (1)-approximation even in se ings where M sbb would only provide an approximation factor of logarithmic order.…”

“…For an allocation (X ,Y ) ∈ A, we shall use the notation 7 For two agents, it is straightforward to come up with alternative mechanisms that have the desired properties. 8 We note that the double auctions given in [5] Session 9a: Auctions --Equilibrium EC'17, June 26-30, 2017, Cambridge, MA, USA 1 R. Colini-Baldeschi, P.W. Goldberg, B. de Keijzer, S. Leonardi, T. Roughgarden, and S. Turche a SW B , SW S to respectively denote the buyers' and the sellers' contribution to the social welfare, i.e.,…”

Revenue Guarantees in the Generalized Second Price Auction

“…This approach will give us a second intermediate result (Lemma 3.5) that we will later combine with Lemma 3.3 in Section 3.3 to complete the proof of Theorem 3.1. We remark that the guarantee for high revenue from the advertiser that occupies the first slot is similar in spirit to prophet inequalities in optimal stopping theory [Krengel and Sucheston 1977] (see also Kleinberg and Weinberg [2012] and Dütting and Kleinberg [2013] for more involved settings that include sponsored search auctions) as well as to guarantees for high revenue from single-item auction mechanisms that use posted prices; such mechanisms were introduced in Chawla et al [2007] (see also Chawla et al [2010] for improved and more general results in this direction).…”

“…Prophet inequalities have proved useful in several Bayesian auction settings in the past [Hajiaghayi et al 2007;Chawla et al 2010]. More recently, Kleinberg and Weinberg [2012] provide generalizations of prophet inequalities for the case of matroids, while Dütting and Kleinberg [2013] further extend this study to polymatroid settings and obtain a revenue loss factor of 2 for the Bayesian setting that we consider using an auction mechanism that is different from GSP.…”