This chapter has two main purposes. The first is a technical one: showing how to identify candidate equilibrium strategies in a variety of auction forms and to verify whether the candidate strategies actually form an equilibrium. This part of the analysis uses various single crossing conditions extensively. Researchers have analyzed several different single crossing conditions; in this chapter we describe and relate these conditions and highlight their significance.

The second purpose is to investigate the comparative performance of different auctions when some of the assumptions of chapter 3 do not hold. For example, we show that in a standard symmetric single-good auction model, although expected revenues are the same for the first- and second-price auctions, revenues are riskier in the second-price auction. Consequently, a risk averse seller prefers a first-price auction. In the same model, introducing bidder risk aversion prevents application of the revenue equivalence theorem and leads to higher average prices in the first-price auction than in the second-price auction. Hence, bidder risk aversion also makes sellers favor the first-price auction design. In a procurement auction in which competitive bids determine prices but the buyer afterwards determines quantities, we show that first-price auctions yield lower prices than second-price auctions and that both bidders and buyers may favor the first-price design. On the other hand, introducing a certain type of positive statistical dependence (affiliation) among the buyers' types leads to the conclusion that prices are higher, on average, in a second-price or ascending auction.