Simultaneous auctions with private and common values

We analyze two simultaneous sealed-bid auctions in which n bidders have private values for the good in one auction but a common value for the other, and must choose to participate in at most one auction. The seller in each auction is free to choose whether his good is sold via a first-price or second-price auction. After auction types are announced, bidders simultaneously decide in which auction to participate and their bid in their selected auction. hence bidders bid without knowing the number of other bidders participating in the same auction. Bidders choose auctions according to a cut-off strategy: only those with a sufficiently high private value choose the private auction. For any auction type profile, there is a unique symmetric equilibrium (which is the same for all auction type pro.les) and multiple asymmetric equilibria which vary with auction types. At the unique symmetric equilibrium, revenue equivalence fails to hold in both auctions: the private auction revenue is always strictly higher when it is first-price.