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Inefficiency of equilibria in digital mechanisms with continuous valuations

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  • Grigorieva, Elena
  • Herings, P. Jean-Jacques
  • Müller, Rudolf
  • Vermeulen, Dries

Abstract

A digital mechanism is defined as an iterative procedure in which bidders select an action, from a finite set, in each iteration. When bidders have continuous valuations and make strategic reports, we show that any ex post implementation of the Vickrey choice rule via such a mechanism needs infinitely many iterations for almost all realizations of the bidders’ valuations. Thus, when valuations are drawn from a continuous probability distribution, the Vickrey choice rule can only be used at the expense of a running time that is infinite with probability one. This infeasibility result even holds in the case of two bidders and the Vickrey choice rule only being required to be established with probability one. Establishing the efficient allocation when the n bidders’ report truthfully contrasts starkly to the previous setting: a bisection procedure has a finite running time almost always, and an expected number of reports are equal to 2n. Using a Groves payment scheme rather than Vickrey’s second price payment scheme somewhat mitigates the problem. We provide an example mechanism with a Groves payment scheme, in which the running time of the mechanism in equilibrium is finite with probability 12.

Suggested Citation

  • Grigorieva, Elena & Herings, P. Jean-Jacques & Müller, Rudolf & Vermeulen, Dries, 2011. "Inefficiency of equilibria in digital mechanisms with continuous valuations," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 541-544.
    • Handle: RePEc:eee:mateco:v:47:y:2011:i:4:p:541-544
    DOI: 10.1016/j.jmateco.2011.06.002
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    References listed on IDEAS

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    1. Rothkopf, Michael H. & Harstad, Ronald M., 1994. "On the role of discrete bid levels in oral auctions," European Journal of Operational Research, Elsevier, vol. 74(3), pages 572-581, May.
    2. Grigorieva, Elena & Jean-Jacques Herings, P. & Müller, Rudolf & Vermeulen, Dries, 2012. "Fraction auctions: The tradeoff between efficiency and running time," European Journal of Operational Research, Elsevier, vol. 220(2), pages 577-587.
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    6. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
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    1. Grigorieva, Elena & Jean-Jacques Herings, P. & Müller, Rudolf & Vermeulen, Dries, 2012. "Fraction auctions: The tradeoff between efficiency and running time," European Journal of Operational Research, Elsevier, vol. 220(2), pages 577-587.

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