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Bruno Ziliotto - One of the best experts on this subject based on the ideXlab platform.
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Prophet secretary through blind strategies
Mathematical Programming, 2020Co-Authors: José Correa, Raimundo Saona, Bruno ZiliottoAbstract:In the classic prophet inequality, a well-known problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler who knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. In the late seventies, Krengel and Sucheston (Bull Am Math Soc 83(4):745–747, 1977), established that this worst case ratio is 0.5. A particularly interesting variant is the so-called prophet secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of $$1-1/e \approx 0.632$$ 1 - 1 / e ≈ 0.632 and very recently this barrier was slightly improved by Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). In this paper we introduce a new type of multi-threshold strategy, called blind strategy . Such a strategy sets a Nonincreasing Sequence of thresholds that depends only on the distribution of the maximum of the random variables, and the gambler stops the first time a sample surpasses the threshold of the stage. Our main result shows that these strategies can achieve a constant of 0.669 for the prophet secretary problem, improving upon the best known result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018), and even that of Beyhaghi et al. (Improved approximations for posted price and second price mechanisms. CoRR arXiv:1807.03435 , 2018) that works in the case in which the gambler can select the order of the samples. The crux of the result is a very precise analysis of the underlying stopping time distribution for the gambler’s strategy that is inspired by the theory of Schur-convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675. Finally we prove that no algorithm for the gambler can achieve a constant better than $$\sqrt{3}-1 \approx 0.732$$ 3 - 1 ≈ 0.732 , which also improves upon a recent result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). This implies that the upper bound on what the gambler can get in the prophet secretary problem is strictly lower than what she can get in the i.i.d. case. This constitutes the first separation between the prophet secretary problem and the i.i.d. prophet inequality.
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Prophet Secretary Through Blind Strategies
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2019Co-Authors: José Correa, Raimundo Saona, Bruno ZiliottoAbstract:In the classic prophet inequality, samples from independent random variables arrive online. A gambler that knows the distributions must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet, that sees all the realizations in advance, gets. In the late seventies, Krengel and Sucheston, and Gairing (1977) established that this worst case ratio is a universal constant equal to 1/2. In the last decade prophet inequalities has resurged as an important problem due to its connections to posted price mechanisms, frequently used in online sales. A very interesting variant is the Prophet Secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms achieve a constant of 1-1/e and very recently this barrier was slightly improved. This paper analyzes strategies that set a Nonincreasing Sequence of thresholds to be applied at different times. The gambler stops the first time a sample surpasses the corresponding threshold. Specifically we consider a class of strategies called blind quantile strategies. They consist in fixing a function which is used to define a Sequence of thresholds once the instance is revealed. Our main result shows that they can achieve a constant of 0.665, improving upon the best known result of Azar et al. (2018), and on Beyhaghi et al. (2018) (order selection). Our proof analyzes precisely the underlying stopping time distribution, relying on Schur-convexity theory. We further prove that blind strategies cannot achieve better than 0.675. Finally we prove that no nonadaptive algorithm for the gambler can achieve better than 0.732.
Stipulanti Manon - One of the best experts on this subject based on the ideXlab platform.
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Nyldon words
2020Co-Authors: Stipulanti ManonAbstract:The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. In a Mathoverflow post dating from November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order. In a recent collaboration with Emilie Charlier (University of Liège) and Manon Philibert (Aix-Marseille University), we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. In our paper, we investigate this new family of words by presenting some of their properties
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Nyldon words
2020Co-Authors: Stipulanti ManonAbstract:audience: researcher, professional, studentThe Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. In a Mathoverflow post dating from November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order. In a recent collaboration with Emilie Charlier (University of Liège) and Manon Philibert (Aix-Marseille University), we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. In our paper, we investigate this new family of words by presenting some of their properties
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Nyldon words
2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.Peer reviewe
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Nyldon words
'Elsevier BV', 2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:peer reviewedaudience: researcherThe Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set
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Nyldon words
2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.Comment: 28 page
Yan Huang - One of the best experts on this subject based on the ideXlab platform.
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Real-valued Choquet integrals for set-valued mappings
International Journal of Approximate Reasoning, 2014Co-Authors: Yan HuangAbstract:In this paper a new kind of real-valued Choquet integrals for set-valued mappings is introduced, and some elementary properties of this kind of Choquet integrals are studied. Convergence theorems of a Sequence of Choquet integrals for set-valued mappings are shown. However, in the case of the monotone convergence theorem of the Nonincreasing Sequence of Choquet integrals for set-valued mappings, we point out that the integrands must be closed. Specially, this kind of real-valued Choquet integrals for set-valued mappings can be regarded as the Choquet integrals for single-valued functions.
Charlier Emilie - One of the best experts on this subject based on the ideXlab platform.
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Nyldon words
2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.Peer reviewe
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Nyldon words
'Elsevier BV', 2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:peer reviewedaudience: researcherThe Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set
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Nyldon words
2019Co-Authors: Charlier Emilie, Philibert Manon, Stipulanti ManonAbstract:The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically Nonincreasing Sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing Sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.Comment: 28 page
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Nyldon words
2018Co-Authors: Charlier EmilieAbstract:The theorem of Chen-Fox-Lyndon states that every finite word can be uniquely factorized as a Nonincreasing Sequence of Lyndon words with respect to the lexicographic order. This theorem can be used to define the family of Lyndon words in a recursive way: 1) the letters are Lyndon; 2) a finite word of length greater than one is Lyndon if it cannot be factorized into a Nonincreasing Sequence of shorter Lyndon words. In a post on Mathoverflow in November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order in the previous recursive definition. The class of words so obtained is not, as one might first think, the class of maximal words in their conjugacy classes. Gringberg asks three questions: 1) How many Nyldon words of length n are there? 2) Is there an equivalent to the Chen-Fox-Lyndon theorem for Nyldon words? 3) Is it true that every primitive words admits exactly one Nyldon word in his conjugacy class? In this talk, I will discuss these questions in the more general context of Lazard factorizations of the free monoid and show that each of Grinberg’s questions has an explicit answer. This is a joint work with Manon Philibert (ENS Lyon) and Manon Stipulanti (ULiège
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Nyldon words
2018Co-Authors: Charlier EmilieAbstract:audience: researcherThe theorem of Chen-Fox-Lyndon states that every finite word can be uniquely factorized as a Nonincreasing Sequence of Lyndon words with respect to the lexicographic order. This theorem can be used to define the family of Lyndon words in a recursive way: 1) the letters are Lyndon; 2) a finite word of length greater than one is Lyndon if it cannot be factorized into a Nonincreasing Sequence of shorter Lyndon words. In a post on Mathoverflow in November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order in the previous recursive definition. The class of words so obtained is not, as one might first think, the class of maximal words in their conjugacy classes. Gringberg asks three questions: 1) How many Nyldon words of length n are there? 2) Is there an equivalent to the Chen-Fox-Lyndon theorem for Nyldon words? 3) Is it true that every primitive words admits exactly one Nyldon word in his conjugacy class? In this talk, I will discuss these questions in the more general context of Lazard factorizations of the free monoid and show that each of Grinberg’s questions has an explicit answer. This is a joint work with Manon Philibert (ENS Lyon) and Manon Stipulanti (ULiège
José Correa - One of the best experts on this subject based on the ideXlab platform.
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Prophet secretary through blind strategies
Mathematical Programming, 2020Co-Authors: José Correa, Raimundo Saona, Bruno ZiliottoAbstract:In the classic prophet inequality, a well-known problem in optimal stopping theory, samples from independent random variables (possibly differently distributed) arrive online. A gambler who knows the distributions, but cannot see the future, must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet that sees all the realizations in advance gets. In the late seventies, Krengel and Sucheston (Bull Am Math Soc 83(4):745–747, 1977), established that this worst case ratio is 0.5. A particularly interesting variant is the so-called prophet secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms are known to achieve a constant of $$1-1/e \approx 0.632$$ 1 - 1 / e ≈ 0.632 and very recently this barrier was slightly improved by Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). In this paper we introduce a new type of multi-threshold strategy, called blind strategy . Such a strategy sets a Nonincreasing Sequence of thresholds that depends only on the distribution of the maximum of the random variables, and the gambler stops the first time a sample surpasses the threshold of the stage. Our main result shows that these strategies can achieve a constant of 0.669 for the prophet secretary problem, improving upon the best known result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018), and even that of Beyhaghi et al. (Improved approximations for posted price and second price mechanisms. CoRR arXiv:1807.03435 , 2018) that works in the case in which the gambler can select the order of the samples. The crux of the result is a very precise analysis of the underlying stopping time distribution for the gambler’s strategy that is inspired by the theory of Schur-convex functions. We further prove that our family of blind strategies cannot lead to a constant better than 0.675. Finally we prove that no algorithm for the gambler can achieve a constant better than $$\sqrt{3}-1 \approx 0.732$$ 3 - 1 ≈ 0.732 , which also improves upon a recent result of Azar et al. (in: Proceedings of the ACM conference on economics and computation, EC, 2018). This implies that the upper bound on what the gambler can get in the prophet secretary problem is strictly lower than what she can get in the i.i.d. case. This constitutes the first separation between the prophet secretary problem and the i.i.d. prophet inequality.
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Prophet Secretary Through Blind Strategies
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2019Co-Authors: José Correa, Raimundo Saona, Bruno ZiliottoAbstract:In the classic prophet inequality, samples from independent random variables arrive online. A gambler that knows the distributions must decide at each point in time whether to stop and pick the current sample or to continue and lose that sample forever. The goal of the gambler is to maximize the expected value of what she picks and the performance measure is the worst case ratio between the expected value the gambler gets and what a prophet, that sees all the realizations in advance, gets. In the late seventies, Krengel and Sucheston, and Gairing (1977) established that this worst case ratio is a universal constant equal to 1/2. In the last decade prophet inequalities has resurged as an important problem due to its connections to posted price mechanisms, frequently used in online sales. A very interesting variant is the Prophet Secretary problem, in which the only difference is that the samples arrive in a uniformly random order. For this variant several algorithms achieve a constant of 1-1/e and very recently this barrier was slightly improved. This paper analyzes strategies that set a Nonincreasing Sequence of thresholds to be applied at different times. The gambler stops the first time a sample surpasses the corresponding threshold. Specifically we consider a class of strategies called blind quantile strategies. They consist in fixing a function which is used to define a Sequence of thresholds once the instance is revealed. Our main result shows that they can achieve a constant of 0.665, improving upon the best known result of Azar et al. (2018), and on Beyhaghi et al. (2018) (order selection). Our proof analyzes precisely the underlying stopping time distribution, relying on Schur-convexity theory. We further prove that blind strategies cannot achieve better than 0.675. Finally we prove that no nonadaptive algorithm for the gambler can achieve better than 0.732.