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A Simple Mechanism for a Budget-Constrained Buyer

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Web and Internet Economics (WINE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11316))

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Abstract

We study a classic Bayesian mechanism design setting of monopoly problem for an additive buyer in the presence of budgets. In this setting a monopolist seller with m heterogeneous items faces a single buyer and seeks to maximize her revenue. The buyer has a budget and additive valuations drawn independently for each item from (non-identical) distributions. We show that when the buyer’s budget is publicly known, the better of selling each item separately and selling the grand bundle extracts a constant fraction of the optimal revenue. When the budget is private, we consider a standard Bayesian setting where buyer’s budget b is drawn from a known distribution B. We show that if b is independent of the valuations and distribution B satisfies monotone hazard rate condition, then selling items separately or in a grand bundle is still approximately optimal. We give a complementary example showing that no constant approximation simple mechanism is possible if budget b can be interdependent with valuations.

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Notes

  1. 1.

    A buyer has additive valuations if his value for a set of items is equal to the sum of his values for the items in the set.

  2. 2.

    E.g., the classic VCG mechanism may not be implementable and social efficiency may not be achievable in the budgeted-setting [46].

  3. 3.

    In this paper, we mostly focus on the public budget case. So we define notations and discuss backgrounds assuming the buyer has a public budget.

  4. 4.

    Like previous work on simple and approximately optimal mechanisms, our results extend to continuous types as well (see, e.g., [16] for a more detailed discussion).

  5. 5.

    We use \(x^\top y = \sum _{i=1}^m x_i y_i\) to denote the inner product of two vectors x and y.

  6. 6.

    We do not optimize the constants in our proofs. In Sect. 4, we will give an alternative proof of Theorem 1 that shows \({{ \textsc {Rev}}^{b}}(V) \le 5 {{ \textsc {SRev}}^{b}}(V) + 6 {{ \textsc {BRev}}^{b}}(V)\), thus improving this constant from 32 to 11.

  7. 7.

    Throughout the paper, when we consider the conditional distribution \({\widehat{V}} := V_{|({||v{||}}_1 \le {c})}\), we will always have \({c}> \min _{v \in {\mathrm {supp}}(V)} {||v{||}}_1\), so that the event we condition on happens with non-zero probability.

  8. 8.

    If the distribution is a discrete MHR distribution, similar results still hold. For discrete distributions we have \(\Pr _{b \sim B}[b \ge \lfloor b^* \rfloor ] \ge e^{-1}\) instead of \(\Pr _{b \sim B}[b \ge b^*] \ge e^{-1}\).

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Acknowledgements

Yu Cheng is supported by NSF grants CCF-1527084, CCF-1535972, CCF-1637397, CCF-1704656, IIS-1447554, and NSF CAREER Award CCF-1750140. Kamesh Munagala is supported by NSF grants CCF-1408784, CCF-1637397, and IIS-1447554; and by an Adobe Data Science Research Award. Kangning Wang is supported by NSF grants CCF-1408784 and CCF-1637397.

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Authors and Affiliations

  1. Duke University, Durham, USA

    Yu Cheng, Kamesh Munagala & Kangning Wang

  2. Shanghai University of Finance and Economics, Shanghai, China

    Nick Gravin

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  1. Yu Cheng
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Correspondence to Kangning Wang .

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  1. University of Liverpool, Liverpool, UK

    George Christodoulou

  2. Universität Augsburg, Augsburg, Germany

    Tobias Harks

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Cheng, Y., Gravin, N., Munagala, K., Wang, K. (2018). A Simple Mechanism for a Budget-Constrained Buyer. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_7

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  • DOI: https://doi.org/10.1007/978-3-030-04612-5_7

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