324 research outputs found
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 Inequalities with Limited Information
In the classical prophet inequality, a gambler observes a sequence of
stochastic rewards and must decide, for each reward ,
whether to keep it and stop the game or to forfeit the reward forever and
reveal the next value . The gambler's goal is to obtain a constant
fraction of the expected reward that the optimal offline algorithm would get.
Recently, prophet inequalities have been generalized to settings where the
gambler can choose items, and, more generally, where he can choose any
independent set in a matroid. However, all the existing algorithms require the
gambler to know the distribution from which the rewards are
drawn.
The assumption that the gambler knows the distribution from which
are drawn is very strong. Instead, we work with the much simpler
assumption that the gambler only knows a few samples from this distribution. We
construct the first single-sample prophet inequalities for many settings of
interest, whose guarantees all match the best possible asymptotically,
\emph{even with full knowledge of the distribution}. Specifically, we provide a
novel single-sample algorithm when the gambler can choose any elements
whose analysis is based on random walks with limited correlation. In addition,
we provide a black-box method for converting specific types of solutions to the
related \emph{secretary problem} to single-sample prophet inequalities, and
apply it to several existing algorithms. Finally, we provide a constant-sample
prophet inequality for constant-degree bipartite matchings.
We apply these results to design the first posted-price and multi-dimensional
auction mechanisms with limited information in settings with asymmetric
bidders
Convex Prophet Inequalities
We introduce a new class of prophet inequalities-convex prophet inequalities-where a gambler observes a sequence of convex cost functions ci (xi ) and is required to assign some fraction 0 ≤ x_i ≤ 1 to each, such that the sum of assigned values is exactly 1. The goal of the gambler is to minimize the sum of the costs. We provide an optimal algorithm for this problem, a dynamic program, and show that it can be implemented in polynomial time when the cost functions are polynomial. We also precisely characterize the competitive ratio of the optimal algorithm in the case where the gambler has an outside option and there are polynomial costs, showing that it grows as θ(n^(p-1)/ℓ), where n is the number of stages, p is the degree of the polynomial costs and the coefficients of the cost functions are bounded by [ℓ,u]
Optimal Single-Choice Prophet Inequalities from Samples
We study the single-choice Prophet Inequality problem when the gambler is
given access to samples. We show that the optimal competitive ratio of
can be achieved with a single sample from each distribution. When the
distributions are identical, we show that for any constant ,
samples from the distribution suffice to achieve the optimal competitive
ratio () within , resolving an open problem of
Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202
Partitioning inequalities and prophet inequalities
Issued as Reports [nos. 1-2], and Final report, Project no. G-37-61
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