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Competitive Weighted Matching in Transversal Matroids
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The ratio index for budgeted learning, with applications
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Multi-armed Bandits with Metric Switching Costs
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Algorithms for Secretary Problems on Graphs and Hypergraphs
ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
Automated online mechanism design and prophet inequalities
AAAI'07 Proceedings of the 22nd national conference on Artificial intelligence - Volume 1
Multi-parameter mechanism design and sequential posted pricing
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Approximation algorithms for restless bandit problems
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On variants of the matroid secretary problem
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Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Improved competitive ratio for the matroid secretary problem
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Matroid secretary problem in the random assignment model
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A stochastic probing problem with applications
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Pricing public goods for private sale
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Stochastic combinatorial optimization via poisson approximation
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On revenue maximization for agents with costly information acquisition: extended abstract
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
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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 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.