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Equilibria in discrete and continuous second price all-pay auctions, convergence or yoyo phenomena

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  • Gisèle Umbhauer

Abstract

The paper is about mixed strategy Nash equilibria in discrete second price all-pay auctions with a limit budget. Two players fight over a prize of value V. Each player submits a bid lower or equal to M, the limit budget. The prize goes to the highest bidder but both bidders pay the lowest bid. V, M and the bids are integers. The paper studies the convergence of the mixed Nash equilibrium probability distribution in the discrete auction to the mixed Nash equilibrium probability distribution in the more well-known continuous second price all-pay auction –or static war of attrition. We establish that the- expected- convergence between discrete and continuous equilibrium distributions is in no way automatic. Both distributions converge for V odd and large, but, for even values of V, the discrete distribution is quite strange and obeys a singular yoyo phenomenon: the probabilities assigned to two adjacent bids are quite different, one probability being much lower than the continuous one, the adjacent probability being much larger. So the discrete probabilities, for V even, don’t converge to the continuous ones. Yet there is a convergence, when turning to sums: the sums of the discrete probabilities of two adjacent bids converge to the sums of the continuous probabilities of the same two bids for large values of V. It is shown in the paper that the yoyo phenomenon doesn’t disappear - it is even strengthened- when switching to lower natural bid increments, like 0.5 or 0.1. More generally, it is shown that convergence is an exception rather than the rule and that it requires a special link between V, M and the bid increment. It follows a lack of continuity between the discrete Nash equilibria and the continuous Nash equilibria.

Suggested Citation

  • Gisèle Umbhauer, 2017. "Equilibria in discrete and continuous second price all-pay auctions, convergence or yoyo phenomena," Working Papers of BETA 2017-14, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
  • Handle: RePEc:ulp:sbbeta:2017-14
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    File URL: http://www.beta-umr7522.fr/productions/publications/2017/2017-14.pdf
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    References listed on IDEAS

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    1. Lugovskyy, Volodymyr & Puzzello, Daniela & Tucker, Steven, 2010. "An experimental investigation of overdissipation in the all pay auction," European Economic Review, Elsevier, vol. 54(8), pages 974-997, November.
    2. Hendricks, Ken & Weiss, Andrew & Wilson, Charles A, 1988. "The War of Attrition in Continuous Time with Complete Information," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 29(4), pages 663-680, November.
    3. Gneezy, Uri & Smorodinsky, Rann, 2006. "All-pay auctions--an experimental study," Journal of Economic Behavior & Organization, Elsevier, vol. 61(2), pages 255-275, October.
    4. Noussair, Charles & Silver, Jonathon, 2006. "Behavior in all-pay auctions with incomplete information," Games and Economic Behavior, Elsevier, vol. 55(1), pages 189-206, April.
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    Cited by:

    1. Gisèle Umbhauer, 2017. "Second price all-pay auctions, how much money do players get or lose?," Working Papers of BETA 2017-16, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.

    More about this item

    Keywords

    discrete game; continuous game; second price all-pay auction; Nash equilibrium; increment.;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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