2. The method of claim 1 further comprising ordering the plurality of procurement sets by increasing value.
1. A computer-implemented method for an online keyword auction, comprising: dividing, by one or more computing devices, a plurality of bidders for a keyword into a plurality of procurement sets of at least one pair of a bidder and a web page placement; obtaining, by the one or more computing devices, a procurement set from the plurality of procurement sets; determining, by the one or more computing devices, external competition exists for each bidder in the procurement set that also belongs to a set of at least one bidder closed under replacement; determining, by the one or more computing devices, internal competition exists for each bidder in the procurement set that does not also belong to the set of at least one bidder closed under replacement; obtaining, by the one or more computing devices, another procurement set from the plurality of procurement sets; determining, by the one or more computing devices, that external competition does not exist for each bidder in the another procurement set that also belongs to the set of at least one bidder closed under replacement; removing, by the one or more computing devices, each bidder in the another procurement set from buying the web page placement for display of an advertisement; charging, by the one or more computing devices, the at least one bidder in the procurement set for buying the web page placement for display of an advertisement; and updating, by the one or more computing devices, a reserve price for the keyword.