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.
We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25]. In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time. This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [9] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service. Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal @math -OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal @math -random order CRSs for matroids, which are similar to OCRSs but the arrival is chosen uniformly at random. For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers' types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers' valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc). We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an @math -approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than @math of all units of each item, then our generic multi buyer mechanisms are @math -approximation of the optimal multi buyer mechanism, where @math is a constant which is at least @math . Observe that @math is at least 1 2 (for @math ) and approaches 1 as @math . As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore, as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature. We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms, and is used to derive new and improved results for combinatorial markets (with and without complements), multi-dimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees. 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.  We study generalizations of the Prophet Inequality'' and Secretary Problem'', where the algorithm is restricted to an arbitrary downward-closed set system. For 0,1 values, we give O(n)-competitive algorithms for both problems. This is close to the Omega(n log n) lower bound due to Babaioff, Immorlica, and Kleinberg. For general values, our results translate to O(log(n) log(r))-competitive algorithms, where r is the cardinality of the largest feasible set. This resolves (up to the O(loglog(n) log(r)) factor) an open question posed to us by Bobby Kleinberg. 