Bidding Languages

In a combinatorial auction environment with m goods, a bid that ascribes a real value to any bundle of goods G, b : 2G → R, can require an exponential amount of space to represent, as there are 2m possible bundles. In these notes, we describe several bidding languages that can represent some bids more compactly. While they are called bidding languages, and can be thought of as languages bidders can use to submit bids to an auctioneer, bidding languages can likewise describe valuations, v : 2G → R. The languages we present are general enough to describe any monotone, normalized bid (or valuation).