Bidding Languages

This chapter concerns the issue of the representation of bids in combinatorial auctions. Theoretically speaking, bids are simply abstract elements drawn from some space of strategies defined by the auction. Every implementation of a combinatorial auction (and, in fact, every other game) must define how each possible bid is actually represented using the underlying communication technology. As the representation details have no strategic implications, the issue of representation is often ignored. This issue, however, may not be ignored in combinatorial auctions due to the underlying complexity of bids: the space of possible bids in combinatorial auctions is usually huge, as large as the space of possible valuations. Specifying a valuation in a combinatorial auction of m items, requires providing a value for each of the possible 2m−1 non-empty subsets. A naive representation would thus require 2m−1 real numbers to represent each possible bid. It is clear that this would be very inconvenient in practice even for 10-item auctions, and completely impractical for more than about two dozen items. We will thus look for more convenient bidding languages that will allow us to encode “common” bids more succinctly. The exact details of the communication technology are not important, the only point being that communication is always achieved using finite strings of characters. A bidding language is thus simply a mapping from the abstract mathematical space of possible bids into the set of finite strings of characters. The exact choice of the set of characters does not matter much as long as it is finite, and may be taken practically to be the set of 100-or-so ASCII characters, or more theoretically, to be the boolean set {0, 1}. This puts us in a “syntax vs. semantics” situation common in logic and in computer science. A language assigns a semantic meaning (in our case, a bid) to