Affiliation and Positive Dependence in Auctions ∗ Luciano I. de Castro † This version: November 2007 ‡ Abstract 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 char- acterization 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. Keywords: affiliation, positive dependence, dependence of types, auctions, pure strategy equilibrium, revenue ranking. 1 Introduction Private information is a central theme in modern economics. It is often introduced in the economic models through (privately known) random variables. For mathematical * This paper supersedes the first part of “Affiliation, Equilibrium Existence and the Revenue Ranking of Auctions”. The second part of “Affiliation, Equilibrium Existence and the Revenue Ranking of Auctions” (still contained in this version as sections 4 and 5) will belong to a separate paper. I am grateful to semi- nar participants in 2006 Stony Brook Game Theory Festival, Washington University, University of Paris I, Universidad Carlos III, Universidad de Santiago de Compostela, University of Illinois, Universitat Pompeu Fabra, Penn State University and to Aloisio Araujo, Alain Chateauneuf, Maria ´ Angeles de Frutos, ´ Angel Hernando, Vijay Krishna, Humberto Moreira, Stephen Morris, Paulo K. Monteiro, Andreu Mas-Colell, Ser- gio Parreiras and Jeroen Swinkels for helpful suggestions. I am especially grateful to Flavio Menezes and Steven Williams for their comments. Find updated versions of this paper at www.impa.br/∼luciano. † Department of Economics, University of Illinois at Urbana-Champaign. 1206 South Sixth St., Cham- paign, IL, 61822. E-mail: decastro.luciano@gmail.com. ‡ First version: January 2004. 1