Welfare Maximization in Combinatorial Auctions

After six long weeks of studying single-parameter auctions to death!, we are ready to move on to the study of multi-parameter auctions. In single-parameter auctions, there was sometimes a single good, and sometimes more than one good (e.g., sponsored search), but the bidders were always characterized by but one parameter. Auctions in which there are multiple (indivisible) goods, and the bidders values for bundles (i.e., subsets) of those goods cannot be characterized by a single parameter, are called combinatorial auctions. Analogous to the single-parameter setting, we will begin our study of combinatorial auctions by focusing on direct mechanisms in which truthful reporting is a dominant strategy (i.e., DSIC mechanisms), and the objective is welfare maximization. Given a set of goods G and a set of bidders N, let Ω denote the set of all possible (feasible)1 allocations of goods to bidders. For 1 No more goods than those available can be allocated to bidders. example, when |G| = 1, there are n + 1 feasible allocations: the good can remain unallocated, or it can be given to any one of the bidders. Generally speaking, each bidder’s preferences are described by a valuation, which is a function from allocations to real numbers: