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Tight Welfare Guarantees for Pure Nash Equilibria of the Uniform Price Auction

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Abstract

We revisit the inefficiency of the uniform price auction, one of the standard multi-unit auction formats, for allocating multiple units of a single good. In the uniform price auction, each bidder submits a sequence of non-increasing marginal bids, for each additional unit, i.e., a submodular curve. The per unit price is then set to be the highest losing bid. We focus on the pure Nash equilibria of such auctions, for bidders with submodular valuation functions. Our result is a tight upper and lower bound on the inefficiency of equilibria, showing that the Price of Anarchy is bounded by 2.1885. This resolves one of the open questions posed in previous works on multi-unit auctions. We also discuss implications of our bounds for an alternative, more practical form of the auction, employing a “uniform bidding” interface.

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Notes

  1. By Proposition 1, for every \(\ell \leqslant x_{i}(\mathbf {b}^{\prime })\): \(\ell \cdot (v_{i}(x_{i}(\mathbf {b}^{\prime }))/x_{i}(\mathbf {b}^{\prime }))\leqslant v_{i}(\ell )\).

  2. The fact that this assumption can be made without loss of generality is instrumental in the proof of Lemma 3, which lies at the heart of our proof for the upper bound. Although we implicitly made this same assumption in our preliminary conference proceedings version of this work [3], a formal statement of this was omitted.

  3. The following holds: \(\lambda ^{*}\cdot v_{i}(x_{i}(\mathbf {b})) + (1 - \lambda ^{*})\cdot \left ({\sum }_{j=x_{i}+ 1}^{x_{i}^{*}}m_{ij} \right )\cdot \left (1 + {\mathcal {W}}_{0}(-e^{-2}) \right ) = \frac {1 + {\mathcal {W}}_{0}(-e^{-2})}{2 + {\mathcal {W}}_{0}(-e^{-2})} \cdot v_{i}(x_{i}^{*}) \) .

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Acknowledgements

We thank two anonymous reviewers, whose constructive comments helped us significantly in improving the technical and overall presentation of our work.

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Authors and Affiliations

  1. Department of Informatics, Athens University of Economics and Business, Athens, Greece

    Georgios Birmpas, Evangelos Markakis & Artem Tsikiridis

  2. Department of Digital Systems, University of Piraeus, Piraeus, Greece

    Orestis Telelis

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  1. Georgios Birmpas
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  2. Evangelos Markakis
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  3. Orestis Telelis
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  4. Artem Tsikiridis
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Correspondence to Orestis Telelis.

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This article is part of the Topical Collection on Special Issue on Algorithmic Game Theory (SAGT 2017)

A preliminary version of this work appeared in [3].

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Birmpas, G., Markakis, E., Telelis, O. et al. Tight Welfare Guarantees for Pure Nash Equilibria of the Uniform Price Auction. Theory Comput Syst 63, 1451–1469 (2019). https://doi.org/10.1007/s00224-018-9889-7

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