Asymmetric first price auctions
A new approach to asymmetric first price auctions is proposed which circumvents the need to examine bidding strategies directly. Specifically, the ratio of bidders' (endogenous) payoffs is analyzed and compared to the ratio of the (exogenous) distribution functions that describe beliefs. Most of the results are inferred from this comparison. In the existing theoretical literature, assumptions of first order stochastic dominance or stronger imply that the latter ratio has very specific properties, but no such assumptions are imposed here. It is proven that first order stochastic dominance is necessary for bidding strategies not to cross. When this assumption is relaxed in the numerical literature it is done in a manner that leads to exactly one crossing. However, it is straightforward to construct examples with several crossings. Finally, bid distributions will cross in auctions with two bidders whenever second order (but not first order) stochastic dominance applies.