Matrix Bidding in Combinatorial Auctions

Abstract

In a combinational auction in which bidders can bid on any combination of goods, bid data can be of exponential size. We describe an innovative new combinatorial auction format in which bidders submit "matrix bids". The advantage of this approach is that it provides bidders a mechanism to compactly express bids on every possible bundle. We describe many different types of preferences that can be modeled using a matrix bid, which is quite flexible, supporting additive, subadditive, and superadditive preferences simultaneously. To utilize the compactness of the matrix bid format in a more general preference environment, we describe a logical language with matrix bids as "atoms", and show that matrix bids compactly express preferences that require an exponential number of atoms in other bidding languages, and is as expressive as the most sophisticated languages in the literature, (Boutilier, 2002). We model the N P-hard winner-determination problem as a polynomially-sized integer program, specifically an assignment problem with side constraints. We show the strength of this formulation with which we rapidly solve winner determination problems with 72 unique items, indicating that this model may be well suited for practical implementation.

Exact algorithms for the matrix bid auction

“…In Table 3, we give the average computation times Table 1 Average computation times (s) for n bidders and m items using BOI that resulted from solving the assignment based formulation (2)-(6) with the Ilog Cplex 8.1 branch-and-cut algorithm with standard settings (B&C), which is basically the approach followed in Day and Raghavan [13]. Horizontally, the number of bidders n varies from 5 to 100, while the number of items m auctioned ranges from 5 to 50 vertically.…”

“…Indeed, from their computational experiments, Day and Raghavan [13] conclude that the computation time for the general combinatorial auction is higher and grows much faster than for the matrix bid auction. Moreover, they manage to solve the winner determination problem for matrix bid auctions with 72 items, 75 bidders and over 10 23 bids, whereas for the general combinatorial auction, the largest instances that can be solved have 16 items, 25 bidders, and less than 10 9 bids.…”

“…Based on the set packing formulation, we develop two branchand-price algorithms to solve the winner determination problem in Section 4. Finally, in Section 5, we present our computational results and compare them with results from the branch-and-cut approach by Day and Raghavan [13].…”

“…These algorithms are tested on randomly generated instances with up to 50 items and 100 bidders, which they solved within 20 min (on average). Finally, the branch-and-price algorithms withstood the comparison with a branch-and-cut algorithm, based on Day and Raghavan [13]. We noticed that the performance of the various algorithms depends on the bid type used in the instances; the branch-and-price algorithms deal very well with some bid types, while for other bid types, the branch-and-cut algorithms are the better choice.…”

Combinatorial auctions

“…recently proposed an iterative combinatorial exchange mechanism that incorporates the TBBL language. Day and Raghavan (2006) introduced a matrix bidding language and showed that it is as expressive as that proposed by Boutilier and Hoos (2001) with the k − of operator. A bidder expresses its preferences through a matrix B in which a column j represents a fictive "unit-demand" agent and a row i corresponds to an auctioned item.…”