Annals of Operations Research

, Volume 225, Issue 1, pp 141-160

First online:

A parametric family of two ranked objects auctions: equilibria and associated risk

  • Estrella AlonsoAffiliated withEscuela Tecnica Superior de Ingenieria, Universidad Pontificia Comillas de Madrid Email author 
  • , Joaquin Sanchez-SorianoAffiliated withU.I. Center Operations Research (CIO), Miguel Hernandez University of Elche
  • , Juan TejadaAffiliated withDepartamento de Estadistica e I.O. and Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid

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This paper deals with simultaneous auctions of two commonly ranked objects following the model studied in Menezes and Monteiro (J. Real Estate Finance Econ., 17(3):219–232, 1998). For these problems we introduce a parametric family of auction mechanisms which includes the three classic auctions (discriminatory-price auction, uniform-price auction and Vickrey auction) and we call it the \(\mathcal{DUV}\) family. We provide the unique Bayesian Nash equilibrium for each auction in \(\mathcal{DUV}\) and prove a revenue equivalence theorem for the parametric family. Likewise, we study the value at risk of the auctioneer as a reasonable decision criterion to determine which auctions in \(\mathcal{DUV}\) may be better taking into account the interests of the auctioneer. We show that there are auction mechanisms in \(\mathcal{DUV}\) which are better than the classic auction mechanisms with respect to this criterion.


Multi-object auctions Bayesian Nash equilibrium Expected revenue Value at risk