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kth price auctions and Catalan numbers

Author

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  • Nawar, Abdel-Hameed
  • Sen, Debapriya

Abstract

This paper establishes an interesting link between kth price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction (k is 3 or higher) can be represented as a finite series of k-2 terms whose lth term involves the lth Catalan number. Using an integral representation of Catalan numbers, together with some classical combinatorial identities, we derive the closed form of the unique symmetric, increasing equilibrium of a kth price auction for a non-uniform distribution.

Suggested Citation

  • Nawar, Abdel-Hameed & Sen, Debapriya, 2018. "kth price auctions and Catalan numbers," MPRA Paper 88553, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:88553
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    References listed on IDEAS

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    1. Kagel, John H & Levin, Dan, 1993. "Independent Private Value Auctions: Bidder Behaviour in First-, Second- and Third-Price Auctions with Varying Numbers of Bidders," Economic Journal, Royal Economic Society, vol. 103(419), pages 868-879, July.
    2. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    3. Riley, John G & Samuelson, William F, 1981. "Optimal Auctions," American Economic Review, American Economic Association, vol. 71(3), pages 381-392, June.
    4. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    5. Mathews, Timothy & Schwartz, Jesse A., 2017. "A note on k-price auctions with complete information when mixed strategies are allowed," Economics Letters, Elsevier, vol. 153(C), pages 6-8.
    6. Tauman, Yair, 2002. "A note on k-price auctions with complete information," Games and Economic Behavior, Elsevier, vol. 41(1), pages 161-164, October.
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    Cited by:

    1. Martin Mihelich & Yan Shu, 2019. "Analytical solution of $k$th price auction," Papers 1911.04865, arXiv.org, revised Jun 2020.
    2. Martin Mihelich & Yan Shu, 2020. "Analytical solution of kth price auction," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(3), pages 875-884, September.

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    More about this item

    Keywords

    kth price auction; the revenue equivalence principle; Catalan numbers; Jensen's identity; Hagen-Rothe's identity;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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