The setting of the classic prophet inequality is as follows: a gambler is shown the probability distributions of @math independent, non-negative random variables with finite expectations. In their indexed order, a value is drawn from each distribution, and after every draw the gambler may choose to accept the value and end the game, or discard the value permanently and continue the game. What is the best performance that the gambler can achieve in comparison to a prophet who can always choose the highest value? Krengel, Sucheston, and Garling solved this problem in 1978, showing that there exists a strategy for which the gambler can achieve half as much reward as the prophet in expectation. Furthermore, this result is tight. In this work, we consider a setting in which the gambler is allowed much less information. Suppose that the gambler can only take one sample from each of the distributions before playing the game, instead of knowing the full distributions. We provide a simple and intuitive algorithm that recovers the original approximation of @math . Our algorithm works against even an almighty adversary who always chooses a worst-case ordering, rather than the standard offline adversary. The result also has implications for mechanism design -- there is much interest in designing competitive auctions with a finite number of samples from value distributions rather than full distributional knowledge.
The constraint of being able to choose one item has been expanded to many combinatorial domains including multiple choices @cite_1 , matroids @cite_2 , and general down-closed set systems @cite_12 . The connection between prophet inequalities and auction design was first noted by Hajiaghayi, Kleinberg, and Sandholm @cite_11 . Chawla et. al @cite_7 explored this in detail, showing that prophet inequalities could be used to answer questions about the performance of posted price mechanisms. Recently, Correa et. al explored the reverse direction, showing that posted price mechanisms and their guarantees could be turned into results about prophet inequalities @cite_15 . Many threads involving combinatorial prophet inequalities and posted price mechanisms were united and generalized by D " @cite_3 . Another recent connection was noted by Lee and Singla @cite_8 between prophet inequalities and contention resolution schemes, a technique for solving combinatorial optimization problems.
@cite_7 @cite_8 @cite_1 @cite_3 @cite_2 @cite_15 @cite_12 @cite_11
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In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V1, ..., Vn and must decide, for each reward Vi, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value Vi. 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 k 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 V1, ..., Vn are drawn. The assumption that the gambler knows the distribution from which V1, ..., Vn 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, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k 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 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. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called "virtual values" even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities. Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algorithms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a "prophet" who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of @math matroid constraints, the prophet's reward exceeds the gambler's by a factor of at most @math , and this factor is also tight. Beyond their interest as theorems about pure online algoritms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate optimal mechanisms in both single-parameter and multi-parameter Bayesian settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings. Recent work on online auctions for digital goods has explored the role of optimal stopping theory -- particularly secretary problems -- in the design of approximately optimal online mechanisms. This work generally assumes that the size of the market (number of bidders) is known a priori, but that the mechanism designer has no knowledge of the distribution of bid values. However, in many real-world applications (such as online ticket sales), the opposite is true: the seller has distributional knowledge of the bid values (e.g., via the history of past transactions in the market), but there is uncertainty about market size. Adopting the perspective of automated mechanism design, introduced by Conitzer and Sandholm, we develop algorithms that compute an optimal, or approximately optimal, online auction mechanism given access to this distributional knowledge. Our main results are twofold. First, we show that when the seller does not know the market size, no constant-approximation to the optimum efficiency or revenue is achievable in the worst case, even under the very strong assumption that bid values are i.i.d. samples from a distribution known to the seller. Second, we show that when the seller has distributional knowledge of the market size as well as the bid values, one can do well in several senses. Perhaps most interestingly, by combining dynamic programming with prophet inequalities (a technique from optimal stopping theory) we are able to design and analyze online mechanisms which are temporally strategyproof (even with respect to arrival and departure times) and approximately efficiency (revenue)-maximizing. In exploring the interplay between automated mechanism design and prophet inequalities, we prove new prophet inequalities motivated by the auction setting.