Weak Cartels and Collusion-Proof Auctions

We study the problem of collusion in private value auctions by cartels whose members cannot exchange monetary transfers among themselves (i.e. weak cartels). We restrict attention to a large class of auctions that includes standard auctions, which we call winner-payable auctions. Restricting attention to winner payable auctions, we provide a complete characterization of collusion-proof auction mechanisms. We show that an auction is collusion-proof if and only if, whenever for some player the distribution of values is strictly concave in some interval of his value space, then he must obtain a fixed winning probability whenever his value is in that interval. Using this characterization, we are able to identify the optimal collusion-proof auctions for a broad class of value distributions.