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Solving Strong-Substitutes Product-Mix Auctions

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  • Klemperer, Paul
  • Baldwin, Elizabeth
  • Goldberg, Paul
  • Lock, Edwin

Abstract

This paper develops algorithms to solve strong-substitutes product-mix auctions: it finds competitive equilibrium prices and quantities for agents who use this auction’s bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle. We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.

Suggested Citation

  • Klemperer, Paul & Baldwin, Elizabeth & Goldberg, Paul & Lock, Edwin, 2020. "Solving Strong-Substitutes Product-Mix Auctions," CEPR Discussion Papers 14976, C.E.P.R. Discussion Papers.
    • Handle: RePEc:cpr:ceprdp:14976
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    References listed on IDEAS

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    1. Gul, Faruk & Stacchetti, Ennio, 2000. "The English Auction with Differentiated Commodities," Journal of Economic Theory, Elsevier, vol. 92(1), pages 66-95, May.
    2. Paes Leme, Renato, 2017. "Gross substitutability: An algorithmic survey," Games and Economic Behavior, Elsevier, vol. 106(C), pages 294-316.
    3. Elizabeth Baldwin & Paul Klemperer, 2019. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities," Econometrica, Econometric Society, vol. 87(3), pages 867-932, May.
    4. Paul Klemperer, 2010. "The Product-Mix Auction: A New Auction Design for Differentiated Goods," Journal of the European Economic Association, MIT Press, vol. 8(2-3), pages 526-536, 04-05.
    5. Kazuo Murota, 2016. "Discrete convex analysis: A tool for economics and game theory," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 151-273, December.
    6. Danilov, Vladimir & Koshevoy, Gleb & Murota, Kazuo, 2001. "Discrete convexity and equilibria in economies with indivisible goods and money," Mathematical Social Sciences, Elsevier, vol. 41(3), pages 251-273, May.
    7. Milgrom, Paul & Strulovici, Bruno, 2009. "Substitute goods, auctions, and equilibrium," Journal of Economic Theory, Elsevier, vol. 144(1), pages 212-247, January.
    8. Lawrence M. Ausubel, 2006. "An Efficient Dynamic Auction for Heterogeneous Commodities," American Economic Review, American Economic Association, vol. 96(3), pages 602-629, June.
    9. Kazuo Murota & Akiyoshi Shioura & Zaifu Yang, 2013. "Computing a Walrasian Equilibrium in Iterative Auctions with Multiple Differentiated Items," Discussion Papers 13/13, Department of Economics, University of York.
    10. Kazuo Murota & Akiyoshi Shioura, 1999. "M-Convex Function on Generalized Polymatroid," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 95-105, February.
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    Cited by:

    1. Paul Klemperer, 2018. "Product-Mix Auction," Economics Papers 2018-W07, Economics Group, Nuffield College, University of Oxford.
    2. Klemperer, Paul & Baldwin, Elizabeth & Bichler, Martin & Fichtl, Maximilian, 2021. "Strong Substitutes: Structural Properties, and a New Algorithm for Competitive Equilibrium Prices," CEPR Discussion Papers 15831, C.E.P.R. Discussion Papers.

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    More about this item

    Keywords

    Bidding language; Product-mix auction; Competitive equilibrium; Walrasian equilibrium; Convex optimisation; Strong substitutes; Submodular minimisation;
    All these keywords.

    JEL classification:

    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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