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Numerical simulations of asymmetric first-price auctions

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  • Fibich, Gadi
  • Gavish, Nir

Abstract

The standard method for computing the equilibrium strategies of asymmetric first-price auctions is the backward-shooting method. In this study we show that the backward-shooting method is inherently unstable, and that this instability cannot be eliminated by changing the numerical methodology of the backward solver. Moreover, this instability becomes more severe as the number of players increases. We then present a novel boundary-value method for computing the equilibrium strategies of asymmetric first-price auctions. We demonstrate the robustness and stability of this method for auctions with any number of players, and for players with mixed types of distributions, including distributions with more than one crossing. Finally, we use the boundary-value method to study large auctions with hundreds of players, to compute the asymptotic rate at which large first-price and second-price auctions become revenue equivalent, and to study auctions in which the distributions cannot be ordered according to first-order stochastic dominance.

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  • Fibich, Gadi & Gavish, Nir, 2011. "Numerical simulations of asymmetric first-price auctions," Games and Economic Behavior, Elsevier, vol. 73(2), pages 479-495.
    • Handle: RePEc:eee:gamebe:v:73:y:2011:i:2:p:479-495
    DOI: 10.1016/j.geb.2011.02.010
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    References listed on IDEAS

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    1. Marshall Robert C. & Meurer Michael J. & Richard Jean-Francois & Stromquist Walter, 1994. "Numerical Analysis of Asymmetric First Price Auctions," Games and Economic Behavior, Elsevier, vol. 7(2), pages 193-220, September.
    2. Wayne-Roy Gayle & Jean Richard, 2008. "Numerical Solutions of Asymmetric, First-Price, Independent Private Values Auctions," Computational Economics, Springer;Society for Computational Economics, vol. 32(3), pages 245-278, October.
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    8. Lebrun, Bernard, 2006. "Uniqueness of the equilibrium in first-price auctions," Games and Economic Behavior, Elsevier, vol. 55(1), pages 131-151, April.
    9. Amann, Erwin & Leininger, Wolfgang, 1996. "Asymmetric All-Pay Auctions with Incomplete Information: The Two-Player Case," Games and Economic Behavior, Elsevier, vol. 14(1), pages 1-18, May.
    10. Lebrun, Bernard, 1999. "First Price Auctions in the Asymmetric N Bidder Case," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 40(1), pages 125-142, February.
    11. Gadi Fibich & Arieh Gavious, 2003. "Asymmetric First-Price Auctions---A Perturbation Approach," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 836-852, November.
    12. Fibich, Gadi & Gavious, Arieh & Sela, Aner, 2002. "Low and high types in asymmetric first-price auctions," Economics Letters, Elsevier, vol. 75(2), pages 283-287, April.
    13. Fibich, Gadi & Gavious, Arieh & Sela, Aner, 2004. "Revenue equivalence in asymmetric auctions," Journal of Economic Theory, Elsevier, vol. 115(2), pages 309-321, April.
    14. Kirkegaard, René, 2009. "Asymmetric first price auctions," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1617-1635, July.
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    Cited by:

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    2. Wasser, Cédric, 2013. "Bilateral k+1-price auctions with asymmetric shares and values," Games and Economic Behavior, Elsevier, vol. 82(C), pages 350-368.
    3. Bichler, Martin & Kohring, Nils & Oberlechner, Matthias & Pieroth, Fabian R., 2023. "Learning equilibrium in bilateral bargaining games," European Journal of Operational Research, Elsevier, vol. 311(2), pages 660-678.
    4. Lagerlof, Johan, 2012. "Does Cost Uncertainty in the Bertrand Model Soften Competition?," CEPR Discussion Papers 8817, C.E.P.R. Discussion Papers.
    5. Aryal, Gaurab & Gabrielli, Maria F., 2013. "Testing for collusion in asymmetric first-price auctions," International Journal of Industrial Organization, Elsevier, vol. 31(1), pages 26-35.
    6. Yonatan Gur & Lijian Lu & Gabriel Y. Weintraub, 2017. "Framework Agreements in Procurement: An Auction Model and Design Recommendations," Manufacturing & Service Operations Management, INFORMS, vol. 19(4), pages 586-603, October.
    7. Muhammad Ejaz & Stephen Joe & Chaitanya Joshi, 2021. "Adversarial Risk Analysis for Auctions Using Mirror Equilibrium and Bayes Nash Equilibrium," Decision Analysis, INFORMS, vol. 18(3), pages 185-202, September.
    8. Gadi Fibich & Nir Gavish, 2012. "Asymmetric First-Price Auctions---A Dynamical-Systems Approach," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 219-243, May.
    9. Gabrielli, M. Florencia & Willington, Manuel, 2023. "Estimating damages from bidding rings in first-price auctions," Economic Modelling, Elsevier, vol. 126(C).
    10. Timothy Hubbard & René Kirkegaard & Harry Paarsch, 2013. "Using Economic Theory to Guide Numerical Analysis: Solving for Equilibria in Models of Asymmetric First-Price Auctions," Computational Economics, Springer;Society for Computational Economics, vol. 42(2), pages 241-266, August.
    11. Josheski Dushko & Apostolov Mico, 2023. "The Prospect Theory and First Price Auctions: an Explanation of Overbidding," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 27(1), pages 33-74, March.
    12. Gaurab Aryal & Hanna Charankevich & Seungwon Jeong & Dong-Hyuk Kim, 2021. "Procurements with Bidder Asymmetry in Cost and Risk-Aversion," Papers 2111.04626, arXiv.org, revised Jul 2022.
    13. Timothy C. Au & David Banks & Yi Guo, 2021. "Numerical Solution of Asymmetric Auctions," Decision Analysis, INFORMS, vol. 18(4), pages 321-334, December.

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    More about this item

    Keywords

    Asymmetric auctions; Simulations; Stability; Large auctions; Boundary value method;
    All these keywords.

    JEL classification:

    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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