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kmth price sealed-bid auctions with general independent values and equilibrium linear mark-ups

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Journal of the Operational Research Society

Abstract

A generalised bidding model is developed to calculate a bidder’s expected profit and auctioners expected revenue/payment for both a General Independent Value and Independent Private Value (IPV) kmth price sealed-bid auction (where the mth bidder wins at the kth bid payment) using a linear (affine) mark-up function. The Common Value (CV) assumption, and highbid and lowbid symmetric and asymmetric First Price Auctions and Second Price Auctions are included as special cases. The optimal n bidder symmetric analytical results are then provided for the uniform IPV and CV models in equilibrium. Final comments concern implications, the assumptions involved and prospects for further research.

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Notes

  1. Cassady (1967) mentions a report by the Greek historian Herodotus, who described the sale by auction of women to be wives in Babylonia around the fifth century BC.

  2. The most notable contributions have come from Griesmer et al (1967), Wilson (1969, 1977), Milgrom (1979, 1981), Riley and Samuelson (1981), Myerson (1981) and Milgrom and Weber (1982)—see Klemperer (1999) for a comprehensive account.

  3. The term value is commonly used for highbid auctions. For lowbid auctions, such as that occurring in tendering for construction work, the equivalent term is cost (Flanagan and Norman 1985). For highbid auctions, the bid denotes the amount of money paid by the bidder to the auctioner and vice versa for lowbid auctions. The amount of money changing hands is determined by the auction payment rule. For example, the first price auction (FPA) payment rule is that the amount of money changing hands is the amount of the winning bid—the highest bid in the case of highbid auction and the lowest (nth highest) bid in the case of lowbid auctions. For the second price auction (SPA), on the other hand, the auction payment rule is that the amount of money changing hands is the amount of second highest bid in highbid auctions and second lowest (n−1th highest) in lowbid auctions. Therefore, the bidder’s profit is obtained by subtracting the payment from the bidder’s value in highbid auctions and by subtracting the bidder’s cost from the payment in lowbid auctions. As both highbid and lowbid auctions are dealt with simultaneously in this paper, the term ‘item value’ is used throughout to denote value/cost and the term ‘payment’ to denote the amount of money changing hands.

  4. The (perfectly) estimated value for IPV auctions is termed the ‘type’ while the (imperfectly) estimated value for CV auctions is termed the ‘signal’. Here we use the term ‘value estimate’ throughout.

  5. Hanson (1992), for example, mentions that ‘researchers report a sizable number of companies that use cost-plus pricing’. There is also a considerable literature recording and advocating the use of a cost plus a percentage or dollar markup or combination of the two in practice. Eichner (1973), for example, refers to ‘the overwhelming empirical evidence that most large business firms set their prices on the basis of a certain percentage mark-up above costs’.

  6. Also worthy of note is that a linear strategy is often the best response to a linear strategy even under full rationality (eg Chatterjee and Samuelson, 1983).

  7. Note the definition 00=1 is used throughout this paper.

  8. See Rothkopf and Harstad (1994) for a general survey.

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  1. Queensland University of Technology, Queensland, Australia

    Martin Skitmore

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Skitmore, M. kmth price sealed-bid auctions with general independent values and equilibrium linear mark-ups. J Oper Res Soc 65, 1864–1875 (2014). https://doi.org/10.1057/jors.2013.163

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