In this chapter we examine different auction formats, such as first-, second-, third- and all-pay auctions. Auctions are a perfect setting in which to apply the Bayesian Nash Equilibrium (BNE) solution concept learned in Chap. 7, since competing bidders are informed about their private valuation for the object but are commonly uninformed about each other’s valuations. Since, in addition, bidders are asked to simultaneously submit their bids under an incomplete information environment; we can use BNE to identify equilibrium behavior, namely, equilibrium bidding strategies.