### Simplified Bidding and Solution Methodology for Truckload Procurement and Other Vcg Combinatorial Auctions

Damian R. Beil, Amy Cohn, Amitabh Sinha
2007 Social Science Research Network
In theory, combinatorial auctions can provide significant benefits in many real-world applications, such as truckload procurement. In practice, however, the use of such auctions has been greatly limited by the need for bidders to bid on an exponential number of bundles and for the auctioneer to solve an exponentially large winner-determination problem. We address these challenges for VCG combinatorial procurement auctions in which a bidder's cost for each bundle is determined by a cost function
more » ... with an amenable structure. For example, the cost to a trucking company of servicing a bundle of loads is based on the least-cost set of tours covering all of these loads, which can be found by solving a simple minimum cost flow problem. Leveraging the fact that true-cost bidding is a dominant strategy in VCG auctions, we suggest that the bidders' challenges can be overcome by specifying this true-cost function explicitly as a bid, rather than computing and communicating each bid individually. Moreover, we propose to embed this true-cost function directly within the winner-determination problem, using the strength of mathematical programming to solve this problem without ever explicitly enumerating the bids. The research challenge is then to identify this cost function, and formulate and solve the corresponding winner-determination problem. We focus primarily on how this can be done for the truckload procurement problem, outline a more general framework for the approach, and identify a number of other promising applications. Nevertheless, two major hurdles prevent the full realization of the benefits of combinatorial auctions. The first is communication-based: To completely express economies of scale and scope among all items being auctioned, bidders must construct and submit bids on an exponential number of subsets of items (called bundles). For example, in a truckload procurement auction, hundreds or thousands of loads are auctioned simultaneously, which results in a huge number of possible combinations of loads to be considered. The second hurdle is computational: The auctioneer must solve a winner determination problem over the corresponding exponential number of bids, which in general is highly intractable. The implications of this are pointed out, for instance, by de Vries and Vohra (2003) and Pekec and Rothkopf (2003) . Although there has been much recent research into the amelioration of these challenges, which we survey in §2, they continue to present a significant obstacle to the practical use of combinatorial auctions. Both hurdles stem from an underlying assumption that all bids must be explicitly enumerated and communicated, which is true in the most general case Nisan and Segal (2006) , for example, in which costs are exogenously endowed. However, in many situations the bids themselves can naturally be seen to arise from some clearly defined function. In this paper we suggest that rather than enumerating the specific bid for each of the exponentially many bundles, the bidder instead simply communicate the bid-generating function to the auctioneer. This addresses the bidder's hurdle. The auctioneer, on the other hand, is still potentially burdened with the task of computing all bids and solving the resulting exponentially large winner determination problem. Our second insight is that the auctioneer can embed the bid-generating function directly into the winner determination problem. The resulting auction is outcome-equivalent to the explicitly enumerative auction, and under certain conditions permits solution which under the explicitly enumerative auction would be intractable. Surprisingly little work takes advantage of a bid-generating function explicitly in both communicating bids and solving the winner determination problem. One exception we are aware of is Hobbs et al. (2000) , who explicitly use the cost function in bidding in an energy market auction, but do not directly address how the winner determination problem is solved. (We review related combinatorial auctions literature in the next section.) Most work to tackle the hurdles described above use iterative auction or bidding language approaches, which both generally take specific bids as primitives and seek to communicate them to the auctioneer. In contrast, we bypass communicating specific bids and take the bid-generating function as the primitive. Iterative auctions can be shown in some settings to converge to optimality without requiring bidders to bid exhaustively on all bundles (Sandholm and Boutilier 2006) , and results such as polynomial communications for bidding languages can be shown for specific preference structures (Nisan 2006 ). Boutilier (2002 empirically demonstrated that the winner determination problem can be solved much faster using a logic-based bidding language which concisely expresses the bidders' underlying bid structure, compared to solving the auction when all combinatorial bids are explicitly specified. Similarly, our approach is most appropriate if the bid-generating functions can be succinctly communicated to the auctioneer and the winner determination problem which embeds these functions is tractable. In general, no known solution to the hurdles described above applies for all possible problem types and bidder preference structures (Nisan and Segal (2006) proved that with a fully general preference structure, it is impossible to find the optimal allocation without exponential communication in the worst case). However, in this paper we show that a rich and important group of real-world problems can be tackled using mathematical programming techniques to formulate and solve the bid-generating function and the corresponding winner determination problem. For instance, consider the truckload procurement problem: An auctioneer wishes to procure truckload services to transport a set of loads, each load consisting of a full truckload and specified by an origin and a destination. In this case, the cost to a trucking company of serving a bundle of loads is based on the least-cost set of tours covering all of these loads, in recognition of the fact that drivers Beil, Cohn, and Sinha: Simplified Bidding and Solution Methodology for VCG Combinatorial Auctions