Exploring the VCG mechanism in combinatorial auctions: The threshold revenue and the threshold-price rule

We explore interesting potential extensions of the Vickrey-Clarke-Groves (VCG) rule under the assumption of players with independent and private valuations and no budget constraints. First, we apply the VCG rule to a coalition of bidders in order to compute the second price of the coalition. Then, we introduce and formulate the problem of determining that partition of players into coalitions which maximize the auctioneer's revenue in the case whereby such coalitions take part to a VCG auction each one as a single agent; in particular, we provide an integer linear formulation of this problem. We also generalize this issue by allowing players to simultaneously belong to distinct coalitions in the case that players' valuation functions are separable. Finally, we propose some applications of these theoretical results. For instance, we exploit them to provide a class of new payment rules and to decide which bids should be defined as the highest losing ones in combinatorial auctions.