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Optimal Prophet Inequality with Less than One Sample

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Web and Internet Economics (WINE 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13778))

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Abstract

There is a growing interest in studying sample-based prophet inequality with the motivation stemming from the connection between the prophet inequalities and the sequential posted pricing mechanisms. Rubinstein, Wang, and Weinberg (ITCS 2021) established the optimal single-choice prophet inequality with only a single sample per each distribution. Our work considers the sample-based prophet inequality with less than one sample per distribution, i.e., scenarios with no prior information about some of the random variables. Specifically, we propose a p-sample model, where a sample from each distribution is revealed with probability \(p \in [0,1]\) independently across all distributions. This model generalizes the single-sample setting of Rubinstein, Wang, and Weinberg (ITCS 2021), and the i.i.d. prophet inequality with a linear number of samples of Correa et al. (EC 2019). Our main result is the optimal \(\frac{p}{1+p}\) prophet inequality for all \(p\in [0,1]\).

This work is supported by Science and Technology Innovation 2030 - “New Generation of Artificial Intelligence” Major Project No. (2018AAA0100903), Innovation Program of Shanghai Municipal Education Commission, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities. Zhihao Gavin Tang is supported by NSFC grant 61902233. Nick Gravin is supported by NSFC grant 62150610500.

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Notes

  1. 1.

    E.g., \(w_1=w_2+\varepsilon =\ldots =w_k+(k-1)\varepsilon ,\) for some negligibly small \(\varepsilon >0\).

  2. 2.

    We can assume that once \(w_i\) is set to 0 for \(1<i<k\), it is small enough to not appear in the \(\mathcal {E}_{\ell }\). Indeed, we can add a few dummy cards with both sides having negligibly small numbers in the beginning of the sequence that do not affect performance of Max-Sample, but still bigger than \(w_i\leftarrow 0\).

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Authors and Affiliations

  1. ITCS, Shanghai University of Finance and Economics, 100 Wudong Road, Shanghai, 200433, China

    Nick Gravin, Hao Li & Zhihao Gavin Tang

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  1. Nick Gravin
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  2. Hao Li
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  3. Zhihao Gavin Tang
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Correspondence to Nick Gravin .

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Editors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Kristoffer Arnsfelt Hansen

  2. Tsinghua University, Beijing, China

    Tracy Xiao Liu

  3. Rotman School of Management, University of Toronto, Toronto, ON, Canada

    Azarakhsh Malekian

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Gravin, N., Li, H., Tang, Z.G. (2022). Optimal Prophet Inequality with Less than One Sample. In: Hansen, K.A., Liu, T.X., Malekian, A. (eds) Web and Internet Economics. WINE 2022. Lecture Notes in Computer Science, vol 13778. Springer, Cham. https://doi.org/10.1007/978-3-031-22832-2_7

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  • DOI: https://doi.org/10.1007/978-3-031-22832-2_7

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