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Liquid Price of Anarchy

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Algorithmic Game Theory (SAGT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 10504))

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Abstract

Incorporating budget constraints into the analysis of auctions has become increasingly important, as they model practical settings more accurately. The social welfare function, which is the standard measure of efficiency in auctions, is inadequate for settings with budgets, since there may be a large disconnect between the value a bidder derives from obtaining an item and what can be liquidated from her. The Liquid Welfare objective function has been suggested as a natural alternative for settings with budgets. Simple auctions, like simultaneous item auctions, are evaluated by their performance at equilibrium using the Price of Anarchy (PoA) measure – the ratio of the objective function value of the optimal outcome to the worst equilibrium. Accordingly, we evaluate the performance of simultaneous item auctions in budgeted settings by the Liquid Price of Anarchy (LPoA) measure – the ratio of the optimal Liquid Welfare to the Liquid Welfare obtained in the worst equilibrium.

For pure Nash equilibria of simultaneous first price auctions, we obtain a bound of 2 on the LPoA for additive buyers. Our results easily extend to the larger class of fractionally-subadditive valuations. Next we show that the LPoA of mixed Nash equilibria for first price auctions with additive bidders is bounded by a constant. Our proofs are robust, and can be extended to achieve similar bounds for Bayesian Nash equilibria. To derive our results, we develop a new technique in which some bidders deviate (surprisingly) toward a non-optimal solution. In particular, this technique goes beyond the smoothness-based approach.

Y. Azar—Supported in part by the Israel Science Foundation (grant No. 1506/16), by the I-CORE program (Center No. 4/11), and by the Blavatnik Fund.

M. Feldman—This work was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 337122.

A. Roytman—This work was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement number 337122, by Thorup’s Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research, by grant number 822/10 from the Israel Science Foundation, and by the Israeli Centers for Research Excellence (ICORE) program.

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Notes

  1. 1.

    The difference is also pointed out in the literature on the design of truthful combinatorial auctions [20, 21].

  2. 2.

    In particular, there are tight PoA bounds of \(\frac{e}{e-1}\) for submodular bidders, and 2 for subadditive bidders.

  3. 3.

    Valuation \(v\) is fractionally-subadditive or equivalently XOS if there is a set of additive valuations \(A = \{a_1,\ldots ,a_\ell \}\) such that \(v_{i}(S) = \max _{a\in A} a(S)\) for every \(S \subseteq [m]\). XOS is a super class of submodular and additive valuations.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Yossi Azar & Michal Feldman

  2. Massachusetts Institute of Technology, Cambridge, USA

    Nick Gravin

  3. University of Copenhagen, Copenhagen, Denmark

    Alan Roytman

Authors
  1. Yossi Azar
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  2. Michal Feldman
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  3. Nick Gravin
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  4. Alan Roytman
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Correspondence to Alan Roytman .

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Editors and Affiliations

  1. University of Salento, Lecce, Italy

    Vittorio Bilò

  2. University of L'Aquila, L'Aquila, Italy

    Michele Flammini

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Azar, Y., Feldman, M., Gravin, N., Roytman, A. (2017). Liquid Price of Anarchy. In: Bilò, V., Flammini, M. (eds) Algorithmic Game Theory. SAGT 2017. Lecture Notes in Computer Science(), vol 10504. Springer, Cham. https://doi.org/10.1007/978-3-319-66700-3_1

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  • DOI: https://doi.org/10.1007/978-3-319-66700-3_1

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