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

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm

Comparing Different Information Levels

Given a sequence of random variables ${\bf X}=X_1,X_2,\ldots$ suppose the aim is to maximize one's return by picking a `favorable' $X_i$. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values $X_i=x_i$ and thus receives $E (\sup X_i)$. We will compare this return to the expected payoffs of a number of observers having less information, in particular $\sup_i (EX_i)$, the value of the sequence to a person who only knows the first moments of the random variables. In general, there is a stochastic environment (i.e. a class of random variables $\cal C$), and several levels of information. Given some ${\bf X} \in {\cal C}$, an observer possessing information $j$ obtains $r_j({\bf X})$. We are going to study `information sets' of the form $$R_{\cal C}^{j,k} = \{ (x,y) | x = r_j({\bf X}), y=r_k({\bf X}), {\bf X} \in {\cal C} \},$$ characterizing the advantage of $k$ relative to $j$. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular `prophet-type' inequalities.Comment: 14 pages, 3 figure