Affiliation, Equilibrium Existence and the Revenue Ranking of Auctions∗

We consider private value auctions where bidders’ types are dependent, a case usually treated by assuming affiliation. As any scientific assumption, affiliation has limitations and it is important to know them. We show that affiliation is topologically and measure-theoretically restrictive. The economic cases where affiliation is justified do not correspond to the intuition usually given for introducing affiliation. Affiliation’s implications do not generalize to other definitions of positive dependence and may be false in general. Nevertheless, we show that some of these implications are true in weaker senses. Also, there are cases where affiliation can be well justified and used in theoretical models. However, since these cases do not cover all economically relevant cases, there is space for a more general approach to dependence in auctions. We propose a new approach that allow both theoretical and numerical characterization of auctions. We treat mainly symmetric auctions, but the approach can be extended to asymmetric auctions with dependence. New results about equilibrium existence and revenue ranking of auctions are provided. JEL Classification Numbers: C62, C72, D44, D82.