Affiliation and Positive Dependence in 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 a restrictive condition, that is, the set of affiliated distributions is small both in topological and measure-theoretical senses. The economic cases where affiliation holds do not correspond to the intuition usually given for affiliation. Affiliation’s implications do not generalize to other definitions of positive dependence and may be frequently false. Nevertheless, some of these implications are true in a weaker sense and there are cases where affiliation can be well justified and used in theoretical models. Since these cases do not cover all economically relevant cases, it is desirable to seek a more general approach to dependence in auctions. We propose a new approach that allow both theoretical and numerical characterization of pure strategy equilibrium of first-price 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.