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Buyers’ welfare maximizing auction design

International Journal of Game Theory (2022)Cite this article

Abstract

I derive the incentive compatible and individually rational mechanism that maximizes the sum of the buyers’ ex-ante expected utilities. I also show that this mechanism minimizes the seller’s revenue. When payments are required to be non-negative, my mechanism takes the form of an arbitrarily weighted lottery with no participation fees, and the resulting allocation is generally not ex-post efficient. This means that before learning their valuations and if side payments from the seller to the buyers are not allowed, buyers as a group would be better off if they gave up ex-post efficiency in order to avoid positive payments. When transfers are allowed, the optimal mechanism is a standard auction with no reserve price and where the seller’s income is redistributed back to the buyers. This mechanism is robust to speculators if a buyer with value 0 never partakes in the redistribution.

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Fig. 1

Notes

  1. Corchón and Serena (2018) provide a good summary of the recent contest theory literature.

  2. Specifically, New Zealand Government’s Explorer Grants and Volkswagen’s Experiment! Grants.

  3. Klemperer (2004), Milgrom (2004), Menezes and Monteiro (2005), Krishna (2009) and Börgers (2015) provide good syntheses of the auction and mechanism design literature.

  4. This notion of ex-post efficiency traces its origins to the works of Vickrey (1961), Clarke (1971) and Groves (1973), who laid the foundations for the ex-post efficient VCG mechanism.

  5. These two equivalences where originally derived by Myerson (1981), and they can be found in, for example, Krishna (2009, sections 5.1.2 and 5.1.3)(Krishna 2009).

  6. I thank an anonymous reviewer for pointing out this particular point.

  7. This is so because:

    $$\begin{aligned} \textrm{E}_{\theta _i}\left[ \frac{1}{\lambda _i\left( \theta _i\right) }\right] =\int _{\varTheta _i}\left( 1-F_i\left( \theta _i\right) \right) d\theta _i=\overline{\theta }-\theta _iF\left( \theta _i\right) \Big |_0^{\overline{\theta }}+\int _{\varTheta _i}\theta _if_i\left( \theta _i\right) d\theta _i=\textrm{E}_{\theta _i}\left[ \theta _i\right] \end{aligned}$$

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Acknowledgements

This paper is a modified version of my master’s thesis at the Torcuato Di Tella University. I am greatly indebted to Leandro Arozamena and Federico Weinschelbaum for their work as thesis advisors. I also thank the valuable comments and suggestions of Susan Athey, Jeremy Bulow, Nick Cao, Matias Cersosimo, Florencia Hnilo, Pedro Martínez Bruera, and of two anonymous referees and an anonymous editor.

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  1. Stanford University Graduate School of Business, 655 Knight Way, Stanford, CA, 94305, USA

    Sebastián D. Bauer

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Correspondence to Sebastián D. Bauer.

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Bauer, S.D. Buyers’ welfare maximizing auction design. Int J Game Theory (2022). https://doi.org/10.1007/s00182-022-00829-w

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Keywords

  • Auction design
  • Buyers’ ex-ante efficiency
  • Buyers’ welfare
  • Non-negative payments
  • Buyer participation