TY - GEN
T1 - Prophet inequalities with limited information
AU - Azar, Pablo D.
AU - Kleinberg, Robert
AU - Weinberg, S. Matthew
PY - 2014
Y1 - 2014
N2 - 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 V 1 ...,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.
AB - 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 V 1 ...,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.
UR - http://www.scopus.com/inward/record.url?scp=84902095952&partnerID=8YFLogxK
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U2 - 10.1137/1.9781611973402.100
DO - 10.1137/1.9781611973402.100
M3 - Conference contribution
AN - SCOPUS:84902095952
SN - 9781611973389
T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
SP - 1358
EP - 1377
BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
PB - Association for Computing Machinery
T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014
Y2 - 5 January 2014 through 7 January 2014
ER -